datafusion-physical-expr 34.0.0

// Licensed to the Apache Software Foundation (ASF) under one
// or more contributor license agreements.  See the NOTICE file
// distributed with this work for additional information
// regarding copyright ownership.  The ASF licenses this file
// to you under the Apache License, Version 2.0 (the
// "License"); you may not use this file except in compliance
// with the License.  You may obtain a copy of the License at
//
//   http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing,
// software distributed under the License is distributed on an
// "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
// KIND, either express or implied.  See the License for the
// specific language governing permissions and limitations
// under the License.

use std::collections::{HashMap, HashSet};
use std::hash::{Hash, Hasher};
use std::sync::Arc;

use crate::expressions::{Column, Literal};
use crate::physical_expr::deduplicate_physical_exprs;
use crate::sort_properties::{ExprOrdering, SortProperties};
use crate::{
    physical_exprs_bag_equal, physical_exprs_contains, LexOrdering, LexOrderingRef,
    LexRequirement, LexRequirementRef, PhysicalExpr, PhysicalSortExpr,
    PhysicalSortRequirement,
};

use arrow::datatypes::SchemaRef;
use arrow_schema::SortOptions;
use datafusion_common::tree_node::{Transformed, TreeNode};
use datafusion_common::{JoinSide, JoinType, Result};

use indexmap::IndexSet;
use itertools::Itertools;

/// An `EquivalenceClass` is a set of [`Arc<dyn PhysicalExpr>`]s that are known
/// to have the same value for all tuples in a relation. These are generated by
/// equality predicates (e.g. `a = b`), typically equi-join conditions and
/// equality conditions in filters.
///
/// Two `EquivalenceClass`es are equal if they contains the same expressions in
/// without any ordering.
#[derive(Debug, Clone)]
pub struct EquivalenceClass {
    /// The expressions in this equivalence class. The order doesn't
    /// matter for equivalence purposes
    ///
    /// TODO: use a HashSet for this instead of a Vec
    exprs: Vec<Arc<dyn PhysicalExpr>>,
}

impl PartialEq for EquivalenceClass {
    /// Returns true if other is equal in the sense
    /// of bags (multi-sets), disregarding their orderings.
    fn eq(&self, other: &Self) -> bool {
        physical_exprs_bag_equal(&self.exprs, &other.exprs)
    }
}

impl EquivalenceClass {
    /// Create a new empty equivalence class
    pub fn new_empty() -> Self {
        Self { exprs: vec![] }
    }

    // Create a new equivalence class from a pre-existing `Vec`
    pub fn new(mut exprs: Vec<Arc<dyn PhysicalExpr>>) -> Self {
        deduplicate_physical_exprs(&mut exprs);
        Self { exprs }
    }

    /// Return the inner vector of expressions
    pub fn into_vec(self) -> Vec<Arc<dyn PhysicalExpr>> {
        self.exprs
    }

    /// Return the "canonical" expression for this class (the first element)
    /// if any
    fn canonical_expr(&self) -> Option<Arc<dyn PhysicalExpr>> {
        self.exprs.first().cloned()
    }

    /// Insert the expression into this class, meaning it is known to be equal to
    /// all other expressions in this class
    pub fn push(&mut self, expr: Arc<dyn PhysicalExpr>) {
        if !self.contains(&expr) {
            self.exprs.push(expr);
        }
    }

    /// Inserts all the expressions from other into this class
    pub fn extend(&mut self, other: Self) {
        for expr in other.exprs {
            // use push so entries are deduplicated
            self.push(expr);
        }
    }

    /// Returns true if this equivalence class contains t expression
    pub fn contains(&self, expr: &Arc<dyn PhysicalExpr>) -> bool {
        physical_exprs_contains(&self.exprs, expr)
    }

    /// Returns true if this equivalence class has any entries in common with `other`
    pub fn contains_any(&self, other: &Self) -> bool {
        self.exprs.iter().any(|e| other.contains(e))
    }

    /// return the number of items in this class
    pub fn len(&self) -> usize {
        self.exprs.len()
    }

    /// return true if this class is empty
    pub fn is_empty(&self) -> bool {
        self.exprs.is_empty()
    }

    /// Iterate over all elements in this class, in some arbitrary order
    pub fn iter(&self) -> impl Iterator<Item = &Arc<dyn PhysicalExpr>> {
        self.exprs.iter()
    }

    /// Return a new equivalence class that have the specified offset added to
    /// each expression (used when schemas are appended such as in joins)
    pub fn with_offset(&self, offset: usize) -> Self {
        let new_exprs = self
            .exprs
            .iter()
            .cloned()
            .map(|e| add_offset_to_expr(e, offset))
            .collect();
        Self::new(new_exprs)
    }
}

/// Stores the mapping between source expressions and target expressions for a
/// projection.
#[derive(Debug, Clone)]
pub struct ProjectionMapping {
    /// Mapping between source expressions and target expressions.
    /// Vector indices correspond to the indices after projection.
    map: Vec<(Arc<dyn PhysicalExpr>, Arc<dyn PhysicalExpr>)>,
}

impl ProjectionMapping {
    /// Constructs the mapping between a projection's input and output
    /// expressions.
    ///
    /// For example, given the input projection expressions (`a + b`, `c + d`)
    /// and an output schema with two columns `"c + d"` and `"a + b"`, the
    /// projection mapping would be:
    ///
    /// ```text
    ///  [0]: (c + d, col("c + d"))
    ///  [1]: (a + b, col("a + b"))
    /// ```
    ///
    /// where `col("c + d")` means the column named `"c + d"`.
    pub fn try_new(
        expr: &[(Arc<dyn PhysicalExpr>, String)],
        input_schema: &SchemaRef,
    ) -> Result<Self> {
        // Construct a map from the input expressions to the output expression of the projection:
        expr.iter()
            .enumerate()
            .map(|(expr_idx, (expression, name))| {
                let target_expr = Arc::new(Column::new(name, expr_idx)) as _;
                expression
                    .clone()
                    .transform_down(&|e| match e.as_any().downcast_ref::<Column>() {
                        Some(col) => {
                            // Sometimes, an expression and its name in the input_schema
                            // doesn't match. This can cause problems, so we make sure
                            // that the expression name matches with the name in `input_schema`.
                            // Conceptually, `source_expr` and `expression` should be the same.
                            let idx = col.index();
                            let matching_input_field = input_schema.field(idx);
                            let matching_input_column =
                                Column::new(matching_input_field.name(), idx);
                            Ok(Transformed::Yes(Arc::new(matching_input_column)))
                        }
                        None => Ok(Transformed::No(e)),
                    })
                    .map(|source_expr| (source_expr, target_expr))
            })
            .collect::<Result<Vec<_>>>()
            .map(|map| Self { map })
    }

    /// Iterate over pairs of (source, target) expressions
    pub fn iter(
        &self,
    ) -> impl Iterator<Item = &(Arc<dyn PhysicalExpr>, Arc<dyn PhysicalExpr>)> + '_ {
        self.map.iter()
    }

    /// This function returns the target expression for a given source expression.
    ///
    /// # Arguments
    ///
    /// * `expr` - Source physical expression.
    ///
    /// # Returns
    ///
    /// An `Option` containing the target for the given source expression,
    /// where a `None` value means that `expr` is not inside the mapping.
    pub fn target_expr(
        &self,
        expr: &Arc<dyn PhysicalExpr>,
    ) -> Option<Arc<dyn PhysicalExpr>> {
        self.map
            .iter()
            .find(|(source, _)| source.eq(expr))
            .map(|(_, target)| target.clone())
    }
}

/// An `EquivalenceGroup` is a collection of `EquivalenceClass`es where each
/// class represents a distinct equivalence class in a relation.
#[derive(Debug, Clone)]
pub struct EquivalenceGroup {
    classes: Vec<EquivalenceClass>,
}

impl EquivalenceGroup {
    /// Creates an empty equivalence group.
    fn empty() -> Self {
        Self { classes: vec![] }
    }

    /// Creates an equivalence group from the given equivalence classes.
    fn new(classes: Vec<EquivalenceClass>) -> Self {
        let mut result = Self { classes };
        result.remove_redundant_entries();
        result
    }

    /// Returns how many equivalence classes there are in this group.
    fn len(&self) -> usize {
        self.classes.len()
    }

    /// Checks whether this equivalence group is empty.
    pub fn is_empty(&self) -> bool {
        self.len() == 0
    }

    /// Returns an iterator over the equivalence classes in this group.
    pub fn iter(&self) -> impl Iterator<Item = &EquivalenceClass> {
        self.classes.iter()
    }

    /// Adds the equality `left` = `right` to this equivalence group.
    /// New equality conditions often arise after steps like `Filter(a = b)`,
    /// `Alias(a, a as b)` etc.
    fn add_equal_conditions(
        &mut self,
        left: &Arc<dyn PhysicalExpr>,
        right: &Arc<dyn PhysicalExpr>,
    ) {
        let mut first_class = None;
        let mut second_class = None;
        for (idx, cls) in self.classes.iter().enumerate() {
            if cls.contains(left) {
                first_class = Some(idx);
            }
            if cls.contains(right) {
                second_class = Some(idx);
            }
        }
        match (first_class, second_class) {
            (Some(mut first_idx), Some(mut second_idx)) => {
                // If the given left and right sides belong to different classes,
                // we should unify/bridge these classes.
                if first_idx != second_idx {
                    // By convention, make sure `second_idx` is larger than `first_idx`.
                    if first_idx > second_idx {
                        (first_idx, second_idx) = (second_idx, first_idx);
                    }
                    // Remove the class at `second_idx` and merge its values with
                    // the class at `first_idx`. The convention above makes sure
                    // that `first_idx` is still valid after removing `second_idx`.
                    let other_class = self.classes.swap_remove(second_idx);
                    self.classes[first_idx].extend(other_class);
                }
            }
            (Some(group_idx), None) => {
                // Right side is new, extend left side's class:
                self.classes[group_idx].push(right.clone());
            }
            (None, Some(group_idx)) => {
                // Left side is new, extend right side's class:
                self.classes[group_idx].push(left.clone());
            }
            (None, None) => {
                // None of the expressions is among existing classes.
                // Create a new equivalence class and extend the group.
                self.classes
                    .push(EquivalenceClass::new(vec![left.clone(), right.clone()]));
            }
        }
    }

    /// Removes redundant entries from this group.
    fn remove_redundant_entries(&mut self) {
        // Remove duplicate entries from each equivalence class:
        self.classes.retain_mut(|cls| {
            // Keep groups that have at least two entries as singleton class is
            // meaningless (i.e. it contains no non-trivial information):
            cls.len() > 1
        });
        // Unify/bridge groups that have common expressions:
        self.bridge_classes()
    }

    /// This utility function unifies/bridges classes that have common expressions.
    /// For example, assume that we have [`EquivalenceClass`]es `[a, b]` and `[b, c]`.
    /// Since both classes contain `b`, columns `a`, `b` and `c` are actually all
    /// equal and belong to one class. This utility converts merges such classes.
    fn bridge_classes(&mut self) {
        let mut idx = 0;
        while idx < self.classes.len() {
            let mut next_idx = idx + 1;
            let start_size = self.classes[idx].len();
            while next_idx < self.classes.len() {
                if self.classes[idx].contains_any(&self.classes[next_idx]) {
                    let extension = self.classes.swap_remove(next_idx);
                    self.classes[idx].extend(extension);
                } else {
                    next_idx += 1;
                }
            }
            if self.classes[idx].len() > start_size {
                continue;
            }
            idx += 1;
        }
    }

    /// Extends this equivalence group with the `other` equivalence group.
    fn extend(&mut self, other: Self) {
        self.classes.extend(other.classes);
        self.remove_redundant_entries();
    }

    /// Normalizes the given physical expression according to this group.
    /// The expression is replaced with the first expression in the equivalence
    /// class it matches with (if any).
    pub fn normalize_expr(&self, expr: Arc<dyn PhysicalExpr>) -> Arc<dyn PhysicalExpr> {
        expr.clone()
            .transform(&|expr| {
                for cls in self.iter() {
                    if cls.contains(&expr) {
                        return Ok(Transformed::Yes(cls.canonical_expr().unwrap()));
                    }
                }
                Ok(Transformed::No(expr))
            })
            .unwrap_or(expr)
    }

    /// Normalizes the given sort expression according to this group.
    /// The underlying physical expression is replaced with the first expression
    /// in the equivalence class it matches with (if any). If the underlying
    /// expression does not belong to any equivalence class in this group, returns
    /// the sort expression as is.
    pub fn normalize_sort_expr(
        &self,
        mut sort_expr: PhysicalSortExpr,
    ) -> PhysicalSortExpr {
        sort_expr.expr = self.normalize_expr(sort_expr.expr);
        sort_expr
    }

    /// Normalizes the given sort requirement according to this group.
    /// The underlying physical expression is replaced with the first expression
    /// in the equivalence class it matches with (if any). If the underlying
    /// expression does not belong to any equivalence class in this group, returns
    /// the given sort requirement as is.
    pub fn normalize_sort_requirement(
        &self,
        mut sort_requirement: PhysicalSortRequirement,
    ) -> PhysicalSortRequirement {
        sort_requirement.expr = self.normalize_expr(sort_requirement.expr);
        sort_requirement
    }

    /// This function applies the `normalize_expr` function for all expressions
    /// in `exprs` and returns the corresponding normalized physical expressions.
    pub fn normalize_exprs(
        &self,
        exprs: impl IntoIterator<Item = Arc<dyn PhysicalExpr>>,
    ) -> Vec<Arc<dyn PhysicalExpr>> {
        exprs
            .into_iter()
            .map(|expr| self.normalize_expr(expr))
            .collect()
    }

    /// This function applies the `normalize_sort_expr` function for all sort
    /// expressions in `sort_exprs` and returns the corresponding normalized
    /// sort expressions.
    pub fn normalize_sort_exprs(&self, sort_exprs: LexOrderingRef) -> LexOrdering {
        // Convert sort expressions to sort requirements:
        let sort_reqs = PhysicalSortRequirement::from_sort_exprs(sort_exprs.iter());
        // Normalize the requirements:
        let normalized_sort_reqs = self.normalize_sort_requirements(&sort_reqs);
        // Convert sort requirements back to sort expressions:
        PhysicalSortRequirement::to_sort_exprs(normalized_sort_reqs)
    }

    /// This function applies the `normalize_sort_requirement` function for all
    /// requirements in `sort_reqs` and returns the corresponding normalized
    /// sort requirements.
    pub fn normalize_sort_requirements(
        &self,
        sort_reqs: LexRequirementRef,
    ) -> LexRequirement {
        collapse_lex_req(
            sort_reqs
                .iter()
                .map(|sort_req| self.normalize_sort_requirement(sort_req.clone()))
                .collect(),
        )
    }

    /// Projects `expr` according to the given projection mapping.
    /// If the resulting expression is invalid after projection, returns `None`.
    fn project_expr(
        &self,
        mapping: &ProjectionMapping,
        expr: &Arc<dyn PhysicalExpr>,
    ) -> Option<Arc<dyn PhysicalExpr>> {
        // First, we try to project expressions with an exact match. If we are
        // unable to do this, we consult equivalence classes.
        if let Some(target) = mapping.target_expr(expr) {
            // If we match the source, we can project directly:
            return Some(target);
        } else {
            // If the given expression is not inside the mapping, try to project
            // expressions considering the equivalence classes.
            for (source, target) in mapping.iter() {
                // If we match an equivalent expression to `source`, then we can
                // project. For example, if we have the mapping `(a as a1, a + c)`
                // and the equivalence class `(a, b)`, expression `b` projects to `a1`.
                if self
                    .get_equivalence_class(source)
                    .map_or(false, |group| group.contains(expr))
                {
                    return Some(target.clone());
                }
            }
        }
        // Project a non-leaf expression by projecting its children.
        let children = expr.children();
        if children.is_empty() {
            // Leaf expression should be inside mapping.
            return None;
        }
        children
            .into_iter()
            .map(|child| self.project_expr(mapping, &child))
            .collect::<Option<Vec<_>>>()
            .map(|children| expr.clone().with_new_children(children).unwrap())
    }

    /// Projects this equivalence group according to the given projection mapping.
    pub fn project(&self, mapping: &ProjectionMapping) -> Self {
        let projected_classes = self.iter().filter_map(|cls| {
            let new_class = cls
                .iter()
                .filter_map(|expr| self.project_expr(mapping, expr))
                .collect::<Vec<_>>();
            (new_class.len() > 1).then_some(EquivalenceClass::new(new_class))
        });
        // TODO: Convert the algorithm below to a version that uses `HashMap`.
        //       once `Arc<dyn PhysicalExpr>` can be stored in `HashMap`.
        // See issue: https://github.com/apache/arrow-datafusion/issues/8027
        let mut new_classes = vec![];
        for (source, target) in mapping.iter() {
            if new_classes.is_empty() {
                new_classes.push((source, vec![target.clone()]));
            }
            if let Some((_, values)) =
                new_classes.iter_mut().find(|(key, _)| key.eq(source))
            {
                if !physical_exprs_contains(values, target) {
                    values.push(target.clone());
                }
            }
        }
        // Only add equivalence classes with at least two members as singleton
        // equivalence classes are meaningless.
        let new_classes = new_classes
            .into_iter()
            .filter_map(|(_, values)| (values.len() > 1).then_some(values))
            .map(EquivalenceClass::new);

        let classes = projected_classes.chain(new_classes).collect();
        Self::new(classes)
    }

    /// Returns the equivalence class containing `expr`. If no equivalence class
    /// contains `expr`, returns `None`.
    fn get_equivalence_class(
        &self,
        expr: &Arc<dyn PhysicalExpr>,
    ) -> Option<&EquivalenceClass> {
        self.iter().find(|cls| cls.contains(expr))
    }

    /// Combine equivalence groups of the given join children.
    pub fn join(
        &self,
        right_equivalences: &Self,
        join_type: &JoinType,
        left_size: usize,
        on: &[(Column, Column)],
    ) -> Self {
        match join_type {
            JoinType::Inner | JoinType::Left | JoinType::Full | JoinType::Right => {
                let mut result = Self::new(
                    self.iter()
                        .cloned()
                        .chain(
                            right_equivalences
                                .iter()
                                .map(|cls| cls.with_offset(left_size)),
                        )
                        .collect(),
                );
                // In we have an inner join, expressions in the "on" condition
                // are equal in the resulting table.
                if join_type == &JoinType::Inner {
                    for (lhs, rhs) in on.iter() {
                        let index = rhs.index() + left_size;
                        let new_lhs = Arc::new(lhs.clone()) as _;
                        let new_rhs = Arc::new(Column::new(rhs.name(), index)) as _;
                        result.add_equal_conditions(&new_lhs, &new_rhs);
                    }
                }
                result
            }
            JoinType::LeftSemi | JoinType::LeftAnti => self.clone(),
            JoinType::RightSemi | JoinType::RightAnti => right_equivalences.clone(),
        }
    }
}

/// This function constructs a duplicate-free `LexOrderingReq` by filtering out
/// duplicate entries that have same physical expression inside. For example,
/// `vec![a Some(ASC), a Some(DESC)]` collapses to `vec![a Some(ASC)]`.
pub fn collapse_lex_req(input: LexRequirement) -> LexRequirement {
    let mut output = Vec::<PhysicalSortRequirement>::new();
    for item in input {
        if !output.iter().any(|req| req.expr.eq(&item.expr)) {
            output.push(item);
        }
    }
    output
}

/// This function constructs a duplicate-free `LexOrdering` by filtering out
/// duplicate entries that have same physical expression inside. For example,
/// `vec![a ASC, a DESC]` collapses to `vec![a ASC]`.
pub fn collapse_lex_ordering(input: LexOrdering) -> LexOrdering {
    let mut output = Vec::<PhysicalSortExpr>::new();
    for item in input {
        if !output.iter().any(|req| req.expr.eq(&item.expr)) {
            output.push(item);
        }
    }
    output
}

/// An `OrderingEquivalenceClass` object keeps track of different alternative
/// orderings than can describe a schema. For example, consider the following table:
///
/// ```text
/// |a|b|c|d|
/// |1|4|3|1|
/// |2|3|3|2|
/// |3|1|2|2|
/// |3|2|1|3|
/// ```
///
/// Here, both `vec![a ASC, b ASC]` and `vec![c DESC, d ASC]` describe the table
/// ordering. In this case, we say that these orderings are equivalent.
#[derive(Debug, Clone, Eq, PartialEq, Hash)]
pub struct OrderingEquivalenceClass {
    orderings: Vec<LexOrdering>,
}

impl OrderingEquivalenceClass {
    /// Creates new empty ordering equivalence class.
    fn empty() -> Self {
        Self { orderings: vec![] }
    }

    /// Clears (empties) this ordering equivalence class.
    pub fn clear(&mut self) {
        self.orderings.clear();
    }

    /// Creates new ordering equivalence class from the given orderings.
    pub fn new(orderings: Vec<LexOrdering>) -> Self {
        let mut result = Self { orderings };
        result.remove_redundant_entries();
        result
    }

    /// Checks whether `ordering` is a member of this equivalence class.
    pub fn contains(&self, ordering: &LexOrdering) -> bool {
        self.orderings.contains(ordering)
    }

    /// Adds `ordering` to this equivalence class.
    #[allow(dead_code)]
    fn push(&mut self, ordering: LexOrdering) {
        self.orderings.push(ordering);
        // Make sure that there are no redundant orderings:
        self.remove_redundant_entries();
    }

    /// Checks whether this ordering equivalence class is empty.
    pub fn is_empty(&self) -> bool {
        self.len() == 0
    }

    /// Returns an iterator over the equivalent orderings in this class.
    pub fn iter(&self) -> impl Iterator<Item = &LexOrdering> {
        self.orderings.iter()
    }

    /// Returns how many equivalent orderings there are in this class.
    pub fn len(&self) -> usize {
        self.orderings.len()
    }

    /// Extend this ordering equivalence class with the `other` class.
    pub fn extend(&mut self, other: Self) {
        self.orderings.extend(other.orderings);
        // Make sure that there are no redundant orderings:
        self.remove_redundant_entries();
    }

    /// Adds new orderings into this ordering equivalence class.
    pub fn add_new_orderings(
        &mut self,
        orderings: impl IntoIterator<Item = LexOrdering>,
    ) {
        self.orderings.extend(orderings);
        // Make sure that there are no redundant orderings:
        self.remove_redundant_entries();
    }

    /// Removes redundant orderings from this equivalence class. For instance,
    /// if we already have the ordering `[a ASC, b ASC, c DESC]`, then there is
    /// no need to keep ordering `[a ASC, b ASC]` in the state.
    fn remove_redundant_entries(&mut self) {
        let mut work = true;
        while work {
            work = false;
            let mut idx = 0;
            while idx < self.orderings.len() {
                let mut ordering_idx = idx + 1;
                let mut removal = self.orderings[idx].is_empty();
                while ordering_idx < self.orderings.len() {
                    work |= resolve_overlap(&mut self.orderings, idx, ordering_idx);
                    if self.orderings[idx].is_empty() {
                        removal = true;
                        break;
                    }
                    work |= resolve_overlap(&mut self.orderings, ordering_idx, idx);
                    if self.orderings[ordering_idx].is_empty() {
                        self.orderings.swap_remove(ordering_idx);
                    } else {
                        ordering_idx += 1;
                    }
                }
                if removal {
                    self.orderings.swap_remove(idx);
                } else {
                    idx += 1;
                }
            }
        }
    }

    /// Returns the concatenation of all the orderings. This enables merge
    /// operations to preserve all equivalent orderings simultaneously.
    pub fn output_ordering(&self) -> Option<LexOrdering> {
        let output_ordering = self.orderings.iter().flatten().cloned().collect();
        let output_ordering = collapse_lex_ordering(output_ordering);
        (!output_ordering.is_empty()).then_some(output_ordering)
    }

    // Append orderings in `other` to all existing orderings in this equivalence
    // class.
    pub fn join_suffix(mut self, other: &Self) -> Self {
        let n_ordering = self.orderings.len();
        // Replicate entries before cross product
        let n_cross = std::cmp::max(n_ordering, other.len() * n_ordering);
        self.orderings = self
            .orderings
            .iter()
            .cloned()
            .cycle()
            .take(n_cross)
            .collect();
        // Suffix orderings of other to the current orderings.
        for (outer_idx, ordering) in other.iter().enumerate() {
            for idx in 0..n_ordering {
                // Calculate cross product index
                let idx = outer_idx * n_ordering + idx;
                self.orderings[idx].extend(ordering.iter().cloned());
            }
        }
        self
    }

    /// Adds `offset` value to the index of each expression inside this
    /// ordering equivalence class.
    pub fn add_offset(&mut self, offset: usize) {
        for ordering in self.orderings.iter_mut() {
            for sort_expr in ordering {
                sort_expr.expr = add_offset_to_expr(sort_expr.expr.clone(), offset);
            }
        }
    }

    /// Gets sort options associated with this expression if it is a leading
    /// ordering expression. Otherwise, returns `None`.
    fn get_options(&self, expr: &Arc<dyn PhysicalExpr>) -> Option<SortOptions> {
        for ordering in self.iter() {
            let leading_ordering = &ordering[0];
            if leading_ordering.expr.eq(expr) {
                return Some(leading_ordering.options);
            }
        }
        None
    }
}

/// Adds the `offset` value to `Column` indices inside `expr`. This function is
/// generally used during the update of the right table schema in join operations.
pub fn add_offset_to_expr(
    expr: Arc<dyn PhysicalExpr>,
    offset: usize,
) -> Arc<dyn PhysicalExpr> {
    expr.transform_down(&|e| match e.as_any().downcast_ref::<Column>() {
        Some(col) => Ok(Transformed::Yes(Arc::new(Column::new(
            col.name(),
            offset + col.index(),
        )))),
        None => Ok(Transformed::No(e)),
    })
    .unwrap()
    // Note that we can safely unwrap here since our transform always returns
    // an `Ok` value.
}

/// Trims `orderings[idx]` if some suffix of it overlaps with a prefix of
/// `orderings[pre_idx]`. Returns `true` if there is any overlap, `false` otherwise.
fn resolve_overlap(orderings: &mut [LexOrdering], idx: usize, pre_idx: usize) -> bool {
    let length = orderings[idx].len();
    let other_length = orderings[pre_idx].len();
    for overlap in 1..=length.min(other_length) {
        if orderings[idx][length - overlap..] == orderings[pre_idx][..overlap] {
            orderings[idx].truncate(length - overlap);
            return true;
        }
    }
    false
}

/// A `EquivalenceProperties` object stores useful information related to a schema.
/// Currently, it keeps track of:
/// - Equivalent expressions, e.g expressions that have same value.
/// - Valid sort expressions (orderings) for the schema.
/// - Constants expressions (e.g expressions that are known to have constant values).
///
/// Consider table below:
///
/// ```text
/// ┌-------┐
/// | a | b |
/// |---|---|
/// | 1 | 9 |
/// | 2 | 8 |
/// | 3 | 7 |
/// | 5 | 5 |
/// └---┴---┘
/// ```
///
/// where both `a ASC` and `b DESC` can describe the table ordering. With
/// `EquivalenceProperties`, we can keep track of these different valid sort
/// expressions and treat `a ASC` and `b DESC` on an equal footing.
///
/// Similarly, consider the table below:
///
/// ```text
/// ┌-------┐
/// | a | b |
/// |---|---|
/// | 1 | 1 |
/// | 2 | 2 |
/// | 3 | 3 |
/// | 5 | 5 |
/// └---┴---┘
/// ```
///
/// where columns `a` and `b` always have the same value. We keep track of such
/// equivalences inside this object. With this information, we can optimize
/// things like partitioning. For example, if the partition requirement is
/// `Hash(a)` and output partitioning is `Hash(b)`, then we can deduce that
/// the existing partitioning satisfies the requirement.
#[derive(Debug, Clone)]
pub struct EquivalenceProperties {
    /// Collection of equivalence classes that store expressions with the same
    /// value.
    eq_group: EquivalenceGroup,
    /// Equivalent sort expressions for this table.
    oeq_class: OrderingEquivalenceClass,
    /// Expressions whose values are constant throughout the table.
    /// TODO: We do not need to track constants separately, they can be tracked
    ///       inside `eq_groups` as `Literal` expressions.
    constants: Vec<Arc<dyn PhysicalExpr>>,
    /// Schema associated with this object.
    schema: SchemaRef,
}

impl EquivalenceProperties {
    /// Creates an empty `EquivalenceProperties` object.
    pub fn new(schema: SchemaRef) -> Self {
        Self {
            eq_group: EquivalenceGroup::empty(),
            oeq_class: OrderingEquivalenceClass::empty(),
            constants: vec![],
            schema,
        }
    }

    /// Creates a new `EquivalenceProperties` object with the given orderings.
    pub fn new_with_orderings(schema: SchemaRef, orderings: &[LexOrdering]) -> Self {
        Self {
            eq_group: EquivalenceGroup::empty(),
            oeq_class: OrderingEquivalenceClass::new(orderings.to_vec()),
            constants: vec![],
            schema,
        }
    }

    /// Returns the associated schema.
    pub fn schema(&self) -> &SchemaRef {
        &self.schema
    }

    /// Returns a reference to the ordering equivalence class within.
    pub fn oeq_class(&self) -> &OrderingEquivalenceClass {
        &self.oeq_class
    }

    /// Returns a reference to the equivalence group within.
    pub fn eq_group(&self) -> &EquivalenceGroup {
        &self.eq_group
    }

    /// Returns a reference to the constant expressions
    pub fn constants(&self) -> &[Arc<dyn PhysicalExpr>] {
        &self.constants
    }

    /// Returns the normalized version of the ordering equivalence class within.
    /// Normalization removes constants and duplicates as well as standardizing
    /// expressions according to the equivalence group within.
    pub fn normalized_oeq_class(&self) -> OrderingEquivalenceClass {
        OrderingEquivalenceClass::new(
            self.oeq_class
                .iter()
                .map(|ordering| self.normalize_sort_exprs(ordering))
                .collect(),
        )
    }

    /// Extends this `EquivalenceProperties` with the `other` object.
    pub fn extend(mut self, other: Self) -> Self {
        self.eq_group.extend(other.eq_group);
        self.oeq_class.extend(other.oeq_class);
        self.add_constants(other.constants)
    }

    /// Clears (empties) the ordering equivalence class within this object.
    /// Call this method when existing orderings are invalidated.
    pub fn clear_orderings(&mut self) {
        self.oeq_class.clear();
    }

    /// Extends this `EquivalenceProperties` by adding the orderings inside the
    /// ordering equivalence class `other`.
    pub fn add_ordering_equivalence_class(&mut self, other: OrderingEquivalenceClass) {
        self.oeq_class.extend(other);
    }

    /// Adds new orderings into the existing ordering equivalence class.
    pub fn add_new_orderings(
        &mut self,
        orderings: impl IntoIterator<Item = LexOrdering>,
    ) {
        self.oeq_class.add_new_orderings(orderings);
    }

    /// Incorporates the given equivalence group to into the existing
    /// equivalence group within.
    pub fn add_equivalence_group(&mut self, other_eq_group: EquivalenceGroup) {
        self.eq_group.extend(other_eq_group);
    }

    /// Adds a new equality condition into the existing equivalence group.
    /// If the given equality defines a new equivalence class, adds this new
    /// equivalence class to the equivalence group.
    pub fn add_equal_conditions(
        &mut self,
        left: &Arc<dyn PhysicalExpr>,
        right: &Arc<dyn PhysicalExpr>,
    ) {
        self.eq_group.add_equal_conditions(left, right);
    }

    /// Track/register physical expressions with constant values.
    pub fn add_constants(
        mut self,
        constants: impl IntoIterator<Item = Arc<dyn PhysicalExpr>>,
    ) -> Self {
        for expr in self.eq_group.normalize_exprs(constants) {
            if !physical_exprs_contains(&self.constants, &expr) {
                self.constants.push(expr);
            }
        }
        self
    }

    /// Updates the ordering equivalence group within assuming that the table
    /// is re-sorted according to the argument `sort_exprs`. Note that constants
    /// and equivalence classes are unchanged as they are unaffected by a re-sort.
    pub fn with_reorder(mut self, sort_exprs: Vec<PhysicalSortExpr>) -> Self {
        // TODO: In some cases, existing ordering equivalences may still be valid add this analysis.
        self.oeq_class = OrderingEquivalenceClass::new(vec![sort_exprs]);
        self
    }

    /// Normalizes the given sort expressions (i.e. `sort_exprs`) using the
    /// equivalence group and the ordering equivalence class within.
    ///
    /// Assume that `self.eq_group` states column `a` and `b` are aliases.
    /// Also assume that `self.oeq_class` states orderings `d ASC` and `a ASC, c ASC`
    /// are equivalent (in the sense that both describe the ordering of the table).
    /// If the `sort_exprs` argument were `vec![b ASC, c ASC, a ASC]`, then this
    /// function would return `vec![a ASC, c ASC]`. Internally, it would first
    /// normalize to `vec![a ASC, c ASC, a ASC]` and end up with the final result
    /// after deduplication.
    fn normalize_sort_exprs(&self, sort_exprs: LexOrderingRef) -> LexOrdering {
        // Convert sort expressions to sort requirements:
        let sort_reqs = PhysicalSortRequirement::from_sort_exprs(sort_exprs.iter());
        // Normalize the requirements:
        let normalized_sort_reqs = self.normalize_sort_requirements(&sort_reqs);
        // Convert sort requirements back to sort expressions:
        PhysicalSortRequirement::to_sort_exprs(normalized_sort_reqs)
    }

    /// Normalizes the given sort requirements (i.e. `sort_reqs`) using the
    /// equivalence group and the ordering equivalence class within. It works by:
    /// - Removing expressions that have a constant value from the given requirement.
    /// - Replacing sections that belong to some equivalence class in the equivalence
    ///   group with the first entry in the matching equivalence class.
    ///
    /// Assume that `self.eq_group` states column `a` and `b` are aliases.
    /// Also assume that `self.oeq_class` states orderings `d ASC` and `a ASC, c ASC`
    /// are equivalent (in the sense that both describe the ordering of the table).
    /// If the `sort_reqs` argument were `vec![b ASC, c ASC, a ASC]`, then this
    /// function would return `vec![a ASC, c ASC]`. Internally, it would first
    /// normalize to `vec![a ASC, c ASC, a ASC]` and end up with the final result
    /// after deduplication.
    fn normalize_sort_requirements(
        &self,
        sort_reqs: LexRequirementRef,
    ) -> LexRequirement {
        let normalized_sort_reqs = self.eq_group.normalize_sort_requirements(sort_reqs);
        let constants_normalized = self.eq_group.normalize_exprs(self.constants.clone());
        // Prune redundant sections in the requirement:
        collapse_lex_req(
            normalized_sort_reqs
                .iter()
                .filter(|&order| {
                    !physical_exprs_contains(&constants_normalized, &order.expr)
                })
                .cloned()
                .collect(),
        )
    }

    /// Checks whether the given ordering is satisfied by any of the existing
    /// orderings.
    pub fn ordering_satisfy(&self, given: LexOrderingRef) -> bool {
        // Convert the given sort expressions to sort requirements:
        let sort_requirements = PhysicalSortRequirement::from_sort_exprs(given.iter());
        self.ordering_satisfy_requirement(&sort_requirements)
    }

    /// Checks whether the given sort requirements are satisfied by any of the
    /// existing orderings.
    pub fn ordering_satisfy_requirement(&self, reqs: LexRequirementRef) -> bool {
        let mut eq_properties = self.clone();
        // First, standardize the given requirement:
        let normalized_reqs = eq_properties.normalize_sort_requirements(reqs);
        for normalized_req in normalized_reqs {
            // Check whether given ordering is satisfied
            if !eq_properties.ordering_satisfy_single(&normalized_req) {
                return false;
            }
            // Treat satisfied keys as constants in subsequent iterations. We
            // can do this because the "next" key only matters in a lexicographical
            // ordering when the keys to its left have the same values.
            //
            // Note that these expressions are not properly "constants". This is just
            // an implementation strategy confined to this function.
            //
            // For example, assume that the requirement is `[a ASC, (b + c) ASC]`,
            // and existing equivalent orderings are `[a ASC, b ASC]` and `[c ASC]`.
            // From the analysis above, we know that `[a ASC]` is satisfied. Then,
            // we add column `a` as constant to the algorithm state. This enables us
            // to deduce that `(b + c) ASC` is satisfied, given `a` is constant.
            eq_properties =
                eq_properties.add_constants(std::iter::once(normalized_req.expr));
        }
        true
    }

    /// Determines whether the ordering specified by the given sort requirement
    /// is satisfied based on the orderings within, equivalence classes, and
    /// constant expressions.
    ///
    /// # Arguments
    ///
    /// - `req`: A reference to a `PhysicalSortRequirement` for which the ordering
    ///   satisfaction check will be done.
    ///
    /// # Returns
    ///
    /// Returns `true` if the specified ordering is satisfied, `false` otherwise.
    fn ordering_satisfy_single(&self, req: &PhysicalSortRequirement) -> bool {
        let expr_ordering = self.get_expr_ordering(req.expr.clone());
        let ExprOrdering { expr, state, .. } = expr_ordering;
        match state {
            SortProperties::Ordered(options) => {
                let sort_expr = PhysicalSortExpr { expr, options };
                sort_expr.satisfy(req, self.schema())
            }
            // Singleton expressions satisfies any ordering.
            SortProperties::Singleton => true,
            SortProperties::Unordered => false,
        }
    }

    /// Checks whether the `given`` sort requirements are equal or more specific
    /// than the `reference` sort requirements.
    pub fn requirements_compatible(
        &self,
        given: LexRequirementRef,
        reference: LexRequirementRef,
    ) -> bool {
        let normalized_given = self.normalize_sort_requirements(given);
        let normalized_reference = self.normalize_sort_requirements(reference);

        (normalized_reference.len() <= normalized_given.len())
            && normalized_reference
                .into_iter()
                .zip(normalized_given)
                .all(|(reference, given)| given.compatible(&reference))
    }

    /// Returns the finer ordering among the orderings `lhs` and `rhs`, breaking
    /// any ties by choosing `lhs`.
    ///
    /// The finer ordering is the ordering that satisfies both of the orderings.
    /// If the orderings are incomparable, returns `None`.
    ///
    /// For example, the finer ordering among `[a ASC]` and `[a ASC, b ASC]` is
    /// the latter.
    pub fn get_finer_ordering(
        &self,
        lhs: LexOrderingRef,
        rhs: LexOrderingRef,
    ) -> Option<LexOrdering> {
        // Convert the given sort expressions to sort requirements:
        let lhs = PhysicalSortRequirement::from_sort_exprs(lhs);
        let rhs = PhysicalSortRequirement::from_sort_exprs(rhs);
        let finer = self.get_finer_requirement(&lhs, &rhs);
        // Convert the chosen sort requirements back to sort expressions:
        finer.map(PhysicalSortRequirement::to_sort_exprs)
    }

    /// Returns the finer ordering among the requirements `lhs` and `rhs`,
    /// breaking any ties by choosing `lhs`.
    ///
    /// The finer requirements are the ones that satisfy both of the given
    /// requirements. If the requirements are incomparable, returns `None`.
    ///
    /// For example, the finer requirements among `[a ASC]` and `[a ASC, b ASC]`
    /// is the latter.
    pub fn get_finer_requirement(
        &self,
        req1: LexRequirementRef,
        req2: LexRequirementRef,
    ) -> Option<LexRequirement> {
        let mut lhs = self.normalize_sort_requirements(req1);
        let mut rhs = self.normalize_sort_requirements(req2);
        lhs.iter_mut()
            .zip(rhs.iter_mut())
            .all(|(lhs, rhs)| {
                lhs.expr.eq(&rhs.expr)
                    && match (lhs.options, rhs.options) {
                        (Some(lhs_opt), Some(rhs_opt)) => lhs_opt == rhs_opt,
                        (Some(options), None) => {
                            rhs.options = Some(options);
                            true
                        }
                        (None, Some(options)) => {
                            lhs.options = Some(options);
                            true
                        }
                        (None, None) => true,
                    }
            })
            .then_some(if lhs.len() >= rhs.len() { lhs } else { rhs })
    }

    /// Calculates the "meet" of the given orderings (`lhs` and `rhs`).
    /// The meet of a set of orderings is the finest ordering that is satisfied
    /// by all the orderings in that set. For details, see:
    ///
    /// <https://en.wikipedia.org/wiki/Join_and_meet>
    ///
    /// If there is no ordering that satisfies both `lhs` and `rhs`, returns
    /// `None`. As an example, the meet of orderings `[a ASC]` and `[a ASC, b ASC]`
    /// is `[a ASC]`.
    pub fn get_meet_ordering(
        &self,
        lhs: LexOrderingRef,
        rhs: LexOrderingRef,
    ) -> Option<LexOrdering> {
        let lhs = self.normalize_sort_exprs(lhs);
        let rhs = self.normalize_sort_exprs(rhs);
        let mut meet = vec![];
        for (lhs, rhs) in lhs.into_iter().zip(rhs.into_iter()) {
            if lhs.eq(&rhs) {
                meet.push(lhs);
            } else {
                break;
            }
        }
        (!meet.is_empty()).then_some(meet)
    }

    /// Projects argument `expr` according to `projection_mapping`, taking
    /// equivalences into account.
    ///
    /// For example, assume that columns `a` and `c` are always equal, and that
    /// `projection_mapping` encodes following mapping:
    ///
    /// ```text
    /// a -> a1
    /// b -> b1
    /// ```
    ///
    /// Then, this function projects `a + b` to `Some(a1 + b1)`, `c + b` to
    /// `Some(a1 + b1)` and `d` to `None`, meaning that it  cannot be projected.
    pub fn project_expr(
        &self,
        expr: &Arc<dyn PhysicalExpr>,
        projection_mapping: &ProjectionMapping,
    ) -> Option<Arc<dyn PhysicalExpr>> {
        self.eq_group.project_expr(projection_mapping, expr)
    }

    /// Constructs a dependency map based on existing orderings referred to in
    /// the projection.
    ///
    /// This function analyzes the orderings in the normalized order-equivalence
    /// class and builds a dependency map. The dependency map captures relationships
    /// between expressions within the orderings, helping to identify dependencies
    /// and construct valid projected orderings during projection operations.
    ///
    /// # Parameters
    ///
    /// - `mapping`: A reference to the `ProjectionMapping` that defines the
    ///   relationship between source and target expressions.
    ///
    /// # Returns
    ///
    /// A [`DependencyMap`] representing the dependency map, where each
    /// [`DependencyNode`] contains dependencies for the key [`PhysicalSortExpr`].
    ///
    /// # Example
    ///
    /// Assume we have two equivalent orderings: `[a ASC, b ASC]` and `[a ASC, c ASC]`,
    /// and the projection mapping is `[a -> a_new, b -> b_new, b + c -> b + c]`.
    /// Then, the dependency map will be:
    ///
    /// ```text
    /// a ASC: Node {Some(a_new ASC), HashSet{}}
    /// b ASC: Node {Some(b_new ASC), HashSet{a ASC}}
    /// c ASC: Node {None, HashSet{a ASC}}
    /// ```
    fn construct_dependency_map(&self, mapping: &ProjectionMapping) -> DependencyMap {
        let mut dependency_map = HashMap::new();
        for ordering in self.normalized_oeq_class().iter() {
            for (idx, sort_expr) in ordering.iter().enumerate() {
                let target_sort_expr =
                    self.project_expr(&sort_expr.expr, mapping).map(|expr| {
                        PhysicalSortExpr {
                            expr,
                            options: sort_expr.options,
                        }
                    });
                let is_projected = target_sort_expr.is_some();
                if is_projected
                    || mapping
                        .iter()
                        .any(|(source, _)| expr_refers(source, &sort_expr.expr))
                {
                    // Previous ordering is a dependency. Note that there is no,
                    // dependency for a leading ordering (i.e. the first sort
                    // expression).
                    let dependency = idx.checked_sub(1).map(|a| &ordering[a]);
                    // Add sort expressions that can be projected or referred to
                    // by any of the projection expressions to the dependency map:
                    dependency_map
                        .entry(sort_expr.clone())
                        .or_insert_with(|| DependencyNode {
                            target_sort_expr: target_sort_expr.clone(),
                            dependencies: HashSet::new(),
                        })
                        .insert_dependency(dependency);
                }
                if !is_projected {
                    // If we can not project, stop constructing the dependency
                    // map as remaining dependencies will be invalid after projection.
                    break;
                }
            }
        }
        dependency_map
    }

    /// Returns a new `ProjectionMapping` where source expressions are normalized.
    ///
    /// This normalization ensures that source expressions are transformed into a
    /// consistent representation. This is beneficial for algorithms that rely on
    /// exact equalities, as it allows for more precise and reliable comparisons.
    ///
    /// # Parameters
    ///
    /// - `mapping`: A reference to the original `ProjectionMapping` to be normalized.
    ///
    /// # Returns
    ///
    /// A new `ProjectionMapping` with normalized source expressions.
    fn normalized_mapping(&self, mapping: &ProjectionMapping) -> ProjectionMapping {
        // Construct the mapping where source expressions are normalized. In this way
        // In the algorithms below we can work on exact equalities
        ProjectionMapping {
            map: mapping
                .iter()
                .map(|(source, target)| {
                    let normalized_source = self.eq_group.normalize_expr(source.clone());
                    (normalized_source, target.clone())
                })
                .collect(),
        }
    }

    /// Computes projected orderings based on a given projection mapping.
    ///
    /// This function takes a `ProjectionMapping` and computes the possible
    /// orderings for the projected expressions. It considers dependencies
    /// between expressions and generates valid orderings according to the
    /// specified sort properties.
    ///
    /// # Parameters
    ///
    /// - `mapping`: A reference to the `ProjectionMapping` that defines the
    ///   relationship between source and target expressions.
    ///
    /// # Returns
    ///
    /// A vector of `LexOrdering` containing all valid orderings after projection.
    fn projected_orderings(&self, mapping: &ProjectionMapping) -> Vec<LexOrdering> {
        let mapping = self.normalized_mapping(mapping);

        // Get dependency map for existing orderings:
        let dependency_map = self.construct_dependency_map(&mapping);

        let orderings = mapping.iter().flat_map(|(source, target)| {
            referred_dependencies(&dependency_map, source)
                .into_iter()
                .filter_map(|relevant_deps| {
                    if let SortProperties::Ordered(options) =
                        get_expr_ordering(source, &relevant_deps)
                    {
                        Some((options, relevant_deps))
                    } else {
                        // Do not consider unordered cases
                        None
                    }
                })
                .flat_map(|(options, relevant_deps)| {
                    let sort_expr = PhysicalSortExpr {
                        expr: target.clone(),
                        options,
                    };
                    // Generate dependent orderings (i.e. prefixes for `sort_expr`):
                    let mut dependency_orderings =
                        generate_dependency_orderings(&relevant_deps, &dependency_map);
                    // Append `sort_expr` to the dependent orderings:
                    for ordering in dependency_orderings.iter_mut() {
                        ordering.push(sort_expr.clone());
                    }
                    dependency_orderings
                })
        });

        // Add valid projected orderings. For example, if existing ordering is
        // `a + b` and projection is `[a -> a_new, b -> b_new]`, we need to
        // preserve `a_new + b_new` as ordered. Please note that `a_new` and
        // `b_new` themselves need not be ordered. Such dependencies cannot be
        // deduced via the pass above.
        let projected_orderings = dependency_map.iter().flat_map(|(sort_expr, node)| {
            let mut prefixes = construct_prefix_orderings(sort_expr, &dependency_map);
            if prefixes.is_empty() {
                // If prefix is empty, there is no dependency. Insert
                // empty ordering:
                prefixes = vec![vec![]];
            }
            // Append current ordering on top its dependencies:
            for ordering in prefixes.iter_mut() {
                if let Some(target) = &node.target_sort_expr {
                    ordering.push(target.clone())
                }
            }
            prefixes
        });

        // Simplify each ordering by removing redundant sections:
        orderings
            .chain(projected_orderings)
            .map(collapse_lex_ordering)
            .collect()
    }

    /// Projects constants based on the provided `ProjectionMapping`.
    ///
    /// This function takes a `ProjectionMapping` and identifies/projects
    /// constants based on the existing constants and the mapping. It ensures
    /// that constants are appropriately propagated through the projection.
    ///
    /// # Arguments
    ///
    /// - `mapping`: A reference to a `ProjectionMapping` representing the
    ///   mapping of source expressions to target expressions in the projection.
    ///
    /// # Returns
    ///
    /// Returns a `Vec<Arc<dyn PhysicalExpr>>` containing the projected constants.
    fn projected_constants(
        &self,
        mapping: &ProjectionMapping,
    ) -> Vec<Arc<dyn PhysicalExpr>> {
        // First, project existing constants. For example, assume that `a + b`
        // is known to be constant. If the projection were `a as a_new`, `b as b_new`,
        // then we would project constant `a + b` as `a_new + b_new`.
        let mut projected_constants = self
            .constants
            .iter()
            .flat_map(|expr| self.eq_group.project_expr(mapping, expr))
            .collect::<Vec<_>>();
        // Add projection expressions that are known to be constant:
        for (source, target) in mapping.iter() {
            if self.is_expr_constant(source)
                && !physical_exprs_contains(&projected_constants, target)
            {
                projected_constants.push(target.clone());
            }
        }
        projected_constants
    }

    /// Projects the equivalences within according to `projection_mapping`
    /// and `output_schema`.
    pub fn project(
        &self,
        projection_mapping: &ProjectionMapping,
        output_schema: SchemaRef,
    ) -> Self {
        let projected_constants = self.projected_constants(projection_mapping);
        let projected_eq_group = self.eq_group.project(projection_mapping);
        let projected_orderings = self.projected_orderings(projection_mapping);
        Self {
            eq_group: projected_eq_group,
            oeq_class: OrderingEquivalenceClass::new(projected_orderings),
            constants: projected_constants,
            schema: output_schema,
        }
    }

    /// Returns the longest (potentially partial) permutation satisfying the
    /// existing ordering. For example, if we have the equivalent orderings
    /// `[a ASC, b ASC]` and `[c DESC]`, with `exprs` containing `[c, b, a, d]`,
    /// then this function returns `([a ASC, b ASC, c DESC], [2, 1, 0])`.
    /// This means that the specification `[a ASC, b ASC, c DESC]` is satisfied
    /// by the existing ordering, and `[a, b, c]` resides at indices: `2, 1, 0`
    /// inside the argument `exprs` (respectively). For the mathematical
    /// definition of "partial permutation", see:
    ///
    /// <https://en.wikipedia.org/wiki/Permutation#k-permutations_of_n>
    pub fn find_longest_permutation(
        &self,
        exprs: &[Arc<dyn PhysicalExpr>],
    ) -> (LexOrdering, Vec<usize>) {
        let mut eq_properties = self.clone();
        let mut result = vec![];
        // The algorithm is as follows:
        // - Iterate over all the expressions and insert ordered expressions
        //   into the result.
        // - Treat inserted expressions as constants (i.e. add them as constants
        //   to the state).
        // - Continue the above procedure until no expression is inserted; i.e.
        //   the algorithm reaches a fixed point.
        // This algorithm should reach a fixed point in at most `exprs.len()`
        // iterations.
        let mut search_indices = (0..exprs.len()).collect::<IndexSet<_>>();
        for _idx in 0..exprs.len() {
            // Get ordered expressions with their indices.
            let ordered_exprs = search_indices
                .iter()
                .flat_map(|&idx| {
                    let ExprOrdering { expr, state, .. } =
                        eq_properties.get_expr_ordering(exprs[idx].clone());
                    if let SortProperties::Ordered(options) = state {
                        Some((PhysicalSortExpr { expr, options }, idx))
                    } else {
                        None
                    }
                })
                .collect::<Vec<_>>();
            // We reached a fixed point, exit.
            if ordered_exprs.is_empty() {
                break;
            }
            // Remove indices that have an ordering from `search_indices`, and
            // treat ordered expressions as constants in subsequent iterations.
            // We can do this because the "next" key only matters in a lexicographical
            // ordering when the keys to its left have the same values.
            //
            // Note that these expressions are not properly "constants". This is just
            // an implementation strategy confined to this function.
            for (PhysicalSortExpr { expr, .. }, idx) in &ordered_exprs {
                eq_properties =
                    eq_properties.add_constants(std::iter::once(expr.clone()));
                search_indices.remove(idx);
            }
            // Add new ordered section to the state.
            result.extend(ordered_exprs);
        }
        result.into_iter().unzip()
    }

    /// This function determines whether the provided expression is constant
    /// based on the known constants.
    ///
    /// # Arguments
    ///
    /// - `expr`: A reference to a `Arc<dyn PhysicalExpr>` representing the
    ///   expression to be checked.
    ///
    /// # Returns
    ///
    /// Returns `true` if the expression is constant according to equivalence
    /// group, `false` otherwise.
    fn is_expr_constant(&self, expr: &Arc<dyn PhysicalExpr>) -> bool {
        // As an example, assume that we know columns `a` and `b` are constant.
        // Then, `a`, `b` and `a + b` will all return `true` whereas `c` will
        // return `false`.
        let normalized_constants = self.eq_group.normalize_exprs(self.constants.to_vec());
        let normalized_expr = self.eq_group.normalize_expr(expr.clone());
        is_constant_recurse(&normalized_constants, &normalized_expr)
    }

    /// Retrieves the ordering information for a given physical expression.
    ///
    /// This function constructs an `ExprOrdering` object for the provided
    /// expression, which encapsulates information about the expression's
    /// ordering, including its [`SortProperties`].
    ///
    /// # Arguments
    ///
    /// - `expr`: An `Arc<dyn PhysicalExpr>` representing the physical expression
    ///   for which ordering information is sought.
    ///
    /// # Returns
    ///
    /// Returns an `ExprOrdering` object containing the ordering information for
    /// the given expression.
    pub fn get_expr_ordering(&self, expr: Arc<dyn PhysicalExpr>) -> ExprOrdering {
        ExprOrdering::new(expr.clone())
            .transform_up(&|expr| Ok(update_ordering(expr, self)))
            // Guaranteed to always return `Ok`.
            .unwrap()
    }
}

/// This function determines whether the provided expression is constant
/// based on the known constants.
///
/// # Arguments
///
/// - `constants`: A `&[Arc<dyn PhysicalExpr>]` containing expressions known to
///   be a constant.
/// - `expr`: A reference to a `Arc<dyn PhysicalExpr>` representing the expression
///   to check.
///
/// # Returns
///
/// Returns `true` if the expression is constant according to equivalence
/// group, `false` otherwise.
fn is_constant_recurse(
    constants: &[Arc<dyn PhysicalExpr>],
    expr: &Arc<dyn PhysicalExpr>,
) -> bool {
    if physical_exprs_contains(constants, expr) {
        return true;
    }
    let children = expr.children();
    !children.is_empty() && children.iter().all(|c| is_constant_recurse(constants, c))
}

/// This function examines whether a referring expression directly refers to a
/// given referred expression or if any of its children in the expression tree
/// refer to the specified expression.
///
/// # Parameters
///
/// - `referring_expr`: A reference to the referring expression (`Arc<dyn PhysicalExpr>`).
/// - `referred_expr`: A reference to the referred expression (`Arc<dyn PhysicalExpr>`)
///
/// # Returns
///
/// A boolean value indicating whether `referring_expr` refers (needs it to evaluate its result)
/// `referred_expr` or not.
fn expr_refers(
    referring_expr: &Arc<dyn PhysicalExpr>,
    referred_expr: &Arc<dyn PhysicalExpr>,
) -> bool {
    referring_expr.eq(referred_expr)
        || referring_expr
            .children()
            .iter()
            .any(|child| expr_refers(child, referred_expr))
}

/// Wrapper struct for `Arc<dyn PhysicalExpr>` to use them as keys in a hash map.
#[derive(Debug, Clone)]
struct ExprWrapper(Arc<dyn PhysicalExpr>);

impl PartialEq<Self> for ExprWrapper {
    fn eq(&self, other: &Self) -> bool {
        self.0.eq(&other.0)
    }
}

impl Eq for ExprWrapper {}

impl Hash for ExprWrapper {
    fn hash<H: Hasher>(&self, state: &mut H) {
        self.0.hash(state);
    }
}

/// This function analyzes the dependency map to collect referred dependencies for
/// a given source expression.
///
/// # Parameters
///
/// - `dependency_map`: A reference to the `DependencyMap` where each
///   `PhysicalSortExpr` is associated with a `DependencyNode`.
/// - `source`: A reference to the source expression (`Arc<dyn PhysicalExpr>`)
///   for which relevant dependencies need to be identified.
///
/// # Returns
///
/// A `Vec<Dependencies>` containing the dependencies for the given source
/// expression. These dependencies are expressions that are referred to by
/// the source expression based on the provided dependency map.
fn referred_dependencies(
    dependency_map: &DependencyMap,
    source: &Arc<dyn PhysicalExpr>,
) -> Vec<Dependencies> {
    // Associate `PhysicalExpr`s with `PhysicalSortExpr`s that contain them:
    let mut expr_to_sort_exprs = HashMap::<ExprWrapper, Dependencies>::new();
    for sort_expr in dependency_map
        .keys()
        .filter(|sort_expr| expr_refers(source, &sort_expr.expr))
    {
        let key = ExprWrapper(sort_expr.expr.clone());
        expr_to_sort_exprs
            .entry(key)
            .or_default()
            .insert(sort_expr.clone());
    }

    // Generate all valid dependencies for the source. For example, if the source
    // is `a + b` and the map is `[a -> (a ASC, a DESC), b -> (b ASC)]`, we get
    // `vec![HashSet(a ASC, b ASC), HashSet(a DESC, b ASC)]`.
    expr_to_sort_exprs
        .values()
        .multi_cartesian_product()
        .map(|referred_deps| referred_deps.into_iter().cloned().collect())
        .collect()
}

/// This function recursively analyzes the dependencies of the given sort
/// expression within the given dependency map to construct lexicographical
/// orderings that include the sort expression and its dependencies.
///
/// # Parameters
///
/// - `referred_sort_expr`: A reference to the sort expression (`PhysicalSortExpr`)
///   for which lexicographical orderings satisfying its dependencies are to be
///   constructed.
/// - `dependency_map`: A reference to the `DependencyMap` that contains
///   dependencies for different `PhysicalSortExpr`s.
///
/// # Returns
///
/// A vector of lexicographical orderings (`Vec<LexOrdering>`) based on the given
/// sort expression and its dependencies.
fn construct_orderings(
    referred_sort_expr: &PhysicalSortExpr,
    dependency_map: &DependencyMap,
) -> Vec<LexOrdering> {
    // We are sure that `referred_sort_expr` is inside `dependency_map`.
    let node = &dependency_map[referred_sort_expr];
    // Since we work on intermediate nodes, we are sure `val.target_sort_expr`
    // exists.
    let target_sort_expr = node.target_sort_expr.clone().unwrap();
    if node.dependencies.is_empty() {
        vec![vec![target_sort_expr]]
    } else {
        node.dependencies
            .iter()
            .flat_map(|dep| {
                let mut orderings = construct_orderings(dep, dependency_map);
                for ordering in orderings.iter_mut() {
                    ordering.push(target_sort_expr.clone())
                }
                orderings
            })
            .collect()
    }
}

/// This function retrieves the dependencies of the given relevant sort expression
/// from the given dependency map. It then constructs prefix orderings by recursively
/// analyzing the dependencies and include them in the orderings.
///
/// # Parameters
///
/// - `relevant_sort_expr`: A reference to the relevant sort expression
///   (`PhysicalSortExpr`) for which prefix orderings are to be constructed.
/// - `dependency_map`: A reference to the `DependencyMap` containing dependencies.
///
/// # Returns
///
/// A vector of prefix orderings (`Vec<LexOrdering>`) based on the given relevant
/// sort expression and its dependencies.
fn construct_prefix_orderings(
    relevant_sort_expr: &PhysicalSortExpr,
    dependency_map: &DependencyMap,
) -> Vec<LexOrdering> {
    dependency_map[relevant_sort_expr]
        .dependencies
        .iter()
        .flat_map(|dep| construct_orderings(dep, dependency_map))
        .collect()
}

/// Given a set of relevant dependencies (`relevant_deps`) and a map of dependencies
/// (`dependency_map`), this function generates all possible prefix orderings
/// based on the given dependencies.
///
/// # Parameters
///
/// * `dependencies` - A reference to the dependencies.
/// * `dependency_map` - A reference to the map of dependencies for expressions.
///
/// # Returns
///
/// A vector of lexical orderings (`Vec<LexOrdering>`) representing all valid orderings
/// based on the given dependencies.
fn generate_dependency_orderings(
    dependencies: &Dependencies,
    dependency_map: &DependencyMap,
) -> Vec<LexOrdering> {
    // Construct all the valid prefix orderings for each expression appearing
    // in the projection:
    let relevant_prefixes = dependencies
        .iter()
        .flat_map(|dep| {
            let prefixes = construct_prefix_orderings(dep, dependency_map);
            (!prefixes.is_empty()).then_some(prefixes)
        })
        .collect::<Vec<_>>();

    // No dependency, dependent is a leading ordering.
    if relevant_prefixes.is_empty() {
        // Return an empty ordering:
        return vec![vec![]];
    }

    // Generate all possible orderings where dependencies are satisfied for the
    // current projection expression. For example, if expression is `a + b ASC`,
    // and the dependency for `a ASC` is `[c ASC]`, the dependency for `b ASC`
    // is `[d DESC]`, then we generate `[c ASC, d DESC, a + b ASC]` and
    // `[d DESC, c ASC, a + b ASC]`.
    relevant_prefixes
        .into_iter()
        .multi_cartesian_product()
        .flat_map(|prefix_orderings| {
            prefix_orderings
                .iter()
                .permutations(prefix_orderings.len())
                .map(|prefixes| prefixes.into_iter().flatten().cloned().collect())
                .collect::<Vec<_>>()
        })
        .collect()
}

/// This function examines the given expression and the sort expressions it
/// refers to determine the ordering properties of the expression.
///
/// # Parameters
///
/// - `expr`: A reference to the source expression (`Arc<dyn PhysicalExpr>`) for
///   which ordering properties need to be determined.
/// - `dependencies`: A reference to `Dependencies`, containing sort expressions
///   referred to by `expr`.
///
/// # Returns
///
/// A `SortProperties` indicating the ordering information of the given expression.
fn get_expr_ordering(
    expr: &Arc<dyn PhysicalExpr>,
    dependencies: &Dependencies,
) -> SortProperties {
    if let Some(column_order) = dependencies.iter().find(|&order| expr.eq(&order.expr)) {
        // If exact match is found, return its ordering.
        SortProperties::Ordered(column_order.options)
    } else {
        // Find orderings of its children
        let child_states = expr
            .children()
            .iter()
            .map(|child| get_expr_ordering(child, dependencies))
            .collect::<Vec<_>>();
        // Calculate expression ordering using ordering of its children.
        expr.get_ordering(&child_states)
    }
}

/// Represents a node in the dependency map used to construct projected orderings.
///
/// A `DependencyNode` contains information about a particular sort expression,
/// including its target sort expression and a set of dependencies on other sort
/// expressions.
///
/// # Fields
///
/// - `target_sort_expr`: An optional `PhysicalSortExpr` representing the target
///   sort expression associated with the node. It is `None` if the sort expression
///   cannot be projected.
/// - `dependencies`: A [`Dependencies`] containing dependencies on other sort
///   expressions that are referred to by the target sort expression.
#[derive(Debug, Clone, PartialEq, Eq)]
struct DependencyNode {
    target_sort_expr: Option<PhysicalSortExpr>,
    dependencies: Dependencies,
}

impl DependencyNode {
    // Insert dependency to the state (if exists).
    fn insert_dependency(&mut self, dependency: Option<&PhysicalSortExpr>) {
        if let Some(dep) = dependency {
            self.dependencies.insert(dep.clone());
        }
    }
}

type DependencyMap = HashMap<PhysicalSortExpr, DependencyNode>;
type Dependencies = HashSet<PhysicalSortExpr>;

/// Calculate ordering equivalence properties for the given join operation.
pub fn join_equivalence_properties(
    left: EquivalenceProperties,
    right: EquivalenceProperties,
    join_type: &JoinType,
    join_schema: SchemaRef,
    maintains_input_order: &[bool],
    probe_side: Option<JoinSide>,
    on: &[(Column, Column)],
) -> EquivalenceProperties {
    let left_size = left.schema.fields.len();
    let mut result = EquivalenceProperties::new(join_schema);
    result.add_equivalence_group(left.eq_group().join(
        right.eq_group(),
        join_type,
        left_size,
        on,
    ));

    let left_oeq_class = left.oeq_class;
    let mut right_oeq_class = right.oeq_class;
    match maintains_input_order {
        [true, false] => {
            // In this special case, right side ordering can be prefixed with
            // the left side ordering.
            if let (Some(JoinSide::Left), JoinType::Inner) = (probe_side, join_type) {
                updated_right_ordering_equivalence_class(
                    &mut right_oeq_class,
                    join_type,
                    left_size,
                );

                // Right side ordering equivalence properties should be prepended
                // with those of the left side while constructing output ordering
                // equivalence properties since stream side is the left side.
                //
                // For example, if the right side ordering equivalences contain
                // `b ASC`, and the left side ordering equivalences contain `a ASC`,
                // then we should add `a ASC, b ASC` to the ordering equivalences
                // of the join output.
                let out_oeq_class = left_oeq_class.join_suffix(&right_oeq_class);
                result.add_ordering_equivalence_class(out_oeq_class);
            } else {
                result.add_ordering_equivalence_class(left_oeq_class);
            }
        }
        [false, true] => {
            updated_right_ordering_equivalence_class(
                &mut right_oeq_class,
                join_type,
                left_size,
            );
            // In this special case, left side ordering can be prefixed with
            // the right side ordering.
            if let (Some(JoinSide::Right), JoinType::Inner) = (probe_side, join_type) {
                // Left side ordering equivalence properties should be prepended
                // with those of the right side while constructing output ordering
                // equivalence properties since stream side is the right side.
                //
                // For example, if the left side ordering equivalences contain
                // `a ASC`, and the right side ordering equivalences contain `b ASC`,
                // then we should add `b ASC, a ASC` to the ordering equivalences
                // of the join output.
                let out_oeq_class = right_oeq_class.join_suffix(&left_oeq_class);
                result.add_ordering_equivalence_class(out_oeq_class);
            } else {
                result.add_ordering_equivalence_class(right_oeq_class);
            }
        }
        [false, false] => {}
        [true, true] => unreachable!("Cannot maintain ordering of both sides"),
        _ => unreachable!("Join operators can not have more than two children"),
    }
    result
}

/// In the context of a join, update the right side `OrderingEquivalenceClass`
/// so that they point to valid indices in the join output schema.
///
/// To do so, we increment column indices by the size of the left table when
/// join schema consists of a combination of the left and right schemas. This
/// is the case for `Inner`, `Left`, `Full` and `Right` joins. For other cases,
/// indices do not change.
fn updated_right_ordering_equivalence_class(
    right_oeq_class: &mut OrderingEquivalenceClass,
    join_type: &JoinType,
    left_size: usize,
) {
    if matches!(
        join_type,
        JoinType::Inner | JoinType::Left | JoinType::Full | JoinType::Right
    ) {
        right_oeq_class.add_offset(left_size);
    }
}

/// Calculates the [`SortProperties`] of a given [`ExprOrdering`] node.
/// The node can either be a leaf node, or an intermediate node:
/// - If it is a leaf node, we directly find the order of the node by looking
/// at the given sort expression and equivalence properties if it is a `Column`
/// leaf, or we mark it as unordered. In the case of a `Literal` leaf, we mark
/// it as singleton so that it can cooperate with all ordered columns.
/// - If it is an intermediate node, the children states matter. Each `PhysicalExpr`
/// and operator has its own rules on how to propagate the children orderings.
/// However, before we engage in recursion, we check whether this intermediate
/// node directly matches with the sort expression. If there is a match, the
/// sort expression emerges at that node immediately, discarding the recursive
/// result coming from its children.
fn update_ordering(
    mut node: ExprOrdering,
    eq_properties: &EquivalenceProperties,
) -> Transformed<ExprOrdering> {
    // We have a Column, which is one of the two possible leaf node types:
    let normalized_expr = eq_properties.eq_group.normalize_expr(node.expr.clone());
    if eq_properties.is_expr_constant(&normalized_expr) {
        node.state = SortProperties::Singleton;
    } else if let Some(options) = eq_properties
        .normalized_oeq_class()
        .get_options(&normalized_expr)
    {
        node.state = SortProperties::Ordered(options);
    } else if !node.expr.children().is_empty() {
        // We have an intermediate (non-leaf) node, account for its children:
        node.state = node.expr.get_ordering(&node.children_state());
    } else if node.expr.as_any().is::<Literal>() {
        // We have a Literal, which is the other possible leaf node type:
        node.state = node.expr.get_ordering(&[]);
    } else {
        return Transformed::No(node);
    }
    Transformed::Yes(node)
}

#[cfg(test)]
mod tests {
    use std::ops::Not;
    use std::sync::Arc;

    use super::*;
    use crate::execution_props::ExecutionProps;
    use crate::expressions::{col, lit, BinaryExpr, Column, Literal};
    use crate::functions::create_physical_expr;

    use arrow::compute::{lexsort_to_indices, SortColumn};
    use arrow::datatypes::{DataType, Field, Schema};
    use arrow_array::{ArrayRef, Float64Array, RecordBatch, UInt32Array};
    use arrow_schema::{Fields, SortOptions, TimeUnit};
    use datafusion_common::{plan_datafusion_err, DataFusionError, Result, ScalarValue};
    use datafusion_expr::{BuiltinScalarFunction, Operator};

    use itertools::{izip, Itertools};
    use rand::rngs::StdRng;
    use rand::seq::SliceRandom;
    use rand::{Rng, SeedableRng};

    fn output_schema(
        mapping: &ProjectionMapping,
        input_schema: &Arc<Schema>,
    ) -> Result<SchemaRef> {
        // Calculate output schema
        let fields: Result<Vec<Field>> = mapping
            .iter()
            .map(|(source, target)| {
                let name = target
                    .as_any()
                    .downcast_ref::<Column>()
                    .ok_or_else(|| plan_datafusion_err!("Expects to have column"))?
                    .name();
                let field = Field::new(
                    name,
                    source.data_type(input_schema)?,
                    source.nullable(input_schema)?,
                );

                Ok(field)
            })
            .collect();

        let output_schema = Arc::new(Schema::new_with_metadata(
            fields?,
            input_schema.metadata().clone(),
        ));

        Ok(output_schema)
    }

    // Generate a schema which consists of 8 columns (a, b, c, d, e, f, g, h)
    fn create_test_schema() -> Result<SchemaRef> {
        let a = Field::new("a", DataType::Int32, true);
        let b = Field::new("b", DataType::Int32, true);
        let c = Field::new("c", DataType::Int32, true);
        let d = Field::new("d", DataType::Int32, true);
        let e = Field::new("e", DataType::Int32, true);
        let f = Field::new("f", DataType::Int32, true);
        let g = Field::new("g", DataType::Int32, true);
        let h = Field::new("h", DataType::Int32, true);
        let schema = Arc::new(Schema::new(vec![a, b, c, d, e, f, g, h]));

        Ok(schema)
    }

    /// Construct a schema with following properties
    /// Schema satisfies following orderings:
    /// [a ASC], [d ASC, b ASC], [e DESC, f ASC, g ASC]
    /// and
    /// Column [a=c] (e.g they are aliases).
    fn create_test_params() -> Result<(SchemaRef, EquivalenceProperties)> {
        let test_schema = create_test_schema()?;
        let col_a = &col("a", &test_schema)?;
        let col_b = &col("b", &test_schema)?;
        let col_c = &col("c", &test_schema)?;
        let col_d = &col("d", &test_schema)?;
        let col_e = &col("e", &test_schema)?;
        let col_f = &col("f", &test_schema)?;
        let col_g = &col("g", &test_schema)?;
        let mut eq_properties = EquivalenceProperties::new(test_schema.clone());
        eq_properties.add_equal_conditions(col_a, col_c);

        let option_asc = SortOptions {
            descending: false,
            nulls_first: false,
        };
        let option_desc = SortOptions {
            descending: true,
            nulls_first: true,
        };
        let orderings = vec![
            // [a ASC]
            vec![(col_a, option_asc)],
            // [d ASC, b ASC]
            vec![(col_d, option_asc), (col_b, option_asc)],
            // [e DESC, f ASC, g ASC]
            vec![
                (col_e, option_desc),
                (col_f, option_asc),
                (col_g, option_asc),
            ],
        ];
        let orderings = convert_to_orderings(&orderings);
        eq_properties.add_new_orderings(orderings);
        Ok((test_schema, eq_properties))
    }

    // Generate a schema which consists of 6 columns (a, b, c, d, e, f)
    fn create_test_schema_2() -> Result<SchemaRef> {
        let a = Field::new("a", DataType::Float64, true);
        let b = Field::new("b", DataType::Float64, true);
        let c = Field::new("c", DataType::Float64, true);
        let d = Field::new("d", DataType::Float64, true);
        let e = Field::new("e", DataType::Float64, true);
        let f = Field::new("f", DataType::Float64, true);
        let schema = Arc::new(Schema::new(vec![a, b, c, d, e, f]));

        Ok(schema)
    }

    /// Construct a schema with random ordering
    /// among column a, b, c, d
    /// where
    /// Column [a=f] (e.g they are aliases).
    /// Column e is constant.
    fn create_random_schema(seed: u64) -> Result<(SchemaRef, EquivalenceProperties)> {
        let test_schema = create_test_schema_2()?;
        let col_a = &col("a", &test_schema)?;
        let col_b = &col("b", &test_schema)?;
        let col_c = &col("c", &test_schema)?;
        let col_d = &col("d", &test_schema)?;
        let col_e = &col("e", &test_schema)?;
        let col_f = &col("f", &test_schema)?;
        let col_exprs = [col_a, col_b, col_c, col_d, col_e, col_f];

        let mut eq_properties = EquivalenceProperties::new(test_schema.clone());
        // Define a and f are aliases
        eq_properties.add_equal_conditions(col_a, col_f);
        // Column e has constant value.
        eq_properties = eq_properties.add_constants([col_e.clone()]);

        // Randomly order columns for sorting
        let mut rng = StdRng::seed_from_u64(seed);
        let mut remaining_exprs = col_exprs[0..4].to_vec(); // only a, b, c, d are sorted

        let options_asc = SortOptions {
            descending: false,
            nulls_first: false,
        };

        while !remaining_exprs.is_empty() {
            let n_sort_expr = rng.gen_range(0..remaining_exprs.len() + 1);
            remaining_exprs.shuffle(&mut rng);

            let ordering = remaining_exprs
                .drain(0..n_sort_expr)
                .map(|expr| PhysicalSortExpr {
                    expr: expr.clone(),
                    options: options_asc,
                })
                .collect();

            eq_properties.add_new_orderings([ordering]);
        }

        Ok((test_schema, eq_properties))
    }

    // Convert each tuple to PhysicalSortRequirement
    fn convert_to_sort_reqs(
        in_data: &[(&Arc<dyn PhysicalExpr>, Option<SortOptions>)],
    ) -> Vec<PhysicalSortRequirement> {
        in_data
            .iter()
            .map(|(expr, options)| {
                PhysicalSortRequirement::new((*expr).clone(), *options)
            })
            .collect()
    }

    // Convert each tuple to PhysicalSortExpr
    fn convert_to_sort_exprs(
        in_data: &[(&Arc<dyn PhysicalExpr>, SortOptions)],
    ) -> Vec<PhysicalSortExpr> {
        in_data
            .iter()
            .map(|(expr, options)| PhysicalSortExpr {
                expr: (*expr).clone(),
                options: *options,
            })
            .collect()
    }

    // Convert each inner tuple to PhysicalSortExpr
    fn convert_to_orderings(
        orderings: &[Vec<(&Arc<dyn PhysicalExpr>, SortOptions)>],
    ) -> Vec<Vec<PhysicalSortExpr>> {
        orderings
            .iter()
            .map(|sort_exprs| convert_to_sort_exprs(sort_exprs))
            .collect()
    }

    // Convert each tuple to PhysicalSortExpr
    fn convert_to_sort_exprs_owned(
        in_data: &[(Arc<dyn PhysicalExpr>, SortOptions)],
    ) -> Vec<PhysicalSortExpr> {
        in_data
            .iter()
            .map(|(expr, options)| PhysicalSortExpr {
                expr: (*expr).clone(),
                options: *options,
            })
            .collect()
    }

    // Convert each inner tuple to PhysicalSortExpr
    fn convert_to_orderings_owned(
        orderings: &[Vec<(Arc<dyn PhysicalExpr>, SortOptions)>],
    ) -> Vec<Vec<PhysicalSortExpr>> {
        orderings
            .iter()
            .map(|sort_exprs| convert_to_sort_exprs_owned(sort_exprs))
            .collect()
    }

    // Apply projection to the input_data, return projected equivalence properties and record batch
    fn apply_projection(
        proj_exprs: Vec<(Arc<dyn PhysicalExpr>, String)>,
        input_data: &RecordBatch,
        input_eq_properties: &EquivalenceProperties,
    ) -> Result<(RecordBatch, EquivalenceProperties)> {
        let input_schema = input_data.schema();
        let projection_mapping = ProjectionMapping::try_new(&proj_exprs, &input_schema)?;

        let output_schema = output_schema(&projection_mapping, &input_schema)?;
        let num_rows = input_data.num_rows();
        // Apply projection to the input record batch.
        let projected_values = projection_mapping
            .iter()
            .map(|(source, _target)| source.evaluate(input_data)?.into_array(num_rows))
            .collect::<Result<Vec<_>>>()?;
        let projected_batch = if projected_values.is_empty() {
            RecordBatch::new_empty(output_schema.clone())
        } else {
            RecordBatch::try_new(output_schema.clone(), projected_values)?
        };

        let projected_eq =
            input_eq_properties.project(&projection_mapping, output_schema);
        Ok((projected_batch, projected_eq))
    }

    #[test]
    fn add_equal_conditions_test() -> Result<()> {
        let schema = Arc::new(Schema::new(vec![
            Field::new("a", DataType::Int64, true),
            Field::new("b", DataType::Int64, true),
            Field::new("c", DataType::Int64, true),
            Field::new("x", DataType::Int64, true),
            Field::new("y", DataType::Int64, true),
        ]));

        let mut eq_properties = EquivalenceProperties::new(schema);
        let col_a_expr = Arc::new(Column::new("a", 0)) as Arc<dyn PhysicalExpr>;
        let col_b_expr = Arc::new(Column::new("b", 1)) as Arc<dyn PhysicalExpr>;
        let col_c_expr = Arc::new(Column::new("c", 2)) as Arc<dyn PhysicalExpr>;
        let col_x_expr = Arc::new(Column::new("x", 3)) as Arc<dyn PhysicalExpr>;
        let col_y_expr = Arc::new(Column::new("y", 4)) as Arc<dyn PhysicalExpr>;

        // a and b are aliases
        eq_properties.add_equal_conditions(&col_a_expr, &col_b_expr);
        assert_eq!(eq_properties.eq_group().len(), 1);

        // This new entry is redundant, size shouldn't increase
        eq_properties.add_equal_conditions(&col_b_expr, &col_a_expr);
        assert_eq!(eq_properties.eq_group().len(), 1);
        let eq_groups = &eq_properties.eq_group().classes[0];
        assert_eq!(eq_groups.len(), 2);
        assert!(eq_groups.contains(&col_a_expr));
        assert!(eq_groups.contains(&col_b_expr));

        // b and c are aliases. Exising equivalence class should expand,
        // however there shouldn't be any new equivalence class
        eq_properties.add_equal_conditions(&col_b_expr, &col_c_expr);
        assert_eq!(eq_properties.eq_group().len(), 1);
        let eq_groups = &eq_properties.eq_group().classes[0];
        assert_eq!(eq_groups.len(), 3);
        assert!(eq_groups.contains(&col_a_expr));
        assert!(eq_groups.contains(&col_b_expr));
        assert!(eq_groups.contains(&col_c_expr));

        // This is a new set of equality. Hence equivalent class count should be 2.
        eq_properties.add_equal_conditions(&col_x_expr, &col_y_expr);
        assert_eq!(eq_properties.eq_group().len(), 2);

        // This equality bridges distinct equality sets.
        // Hence equivalent class count should decrease from 2 to 1.
        eq_properties.add_equal_conditions(&col_x_expr, &col_a_expr);
        assert_eq!(eq_properties.eq_group().len(), 1);
        let eq_groups = &eq_properties.eq_group().classes[0];
        assert_eq!(eq_groups.len(), 5);
        assert!(eq_groups.contains(&col_a_expr));
        assert!(eq_groups.contains(&col_b_expr));
        assert!(eq_groups.contains(&col_c_expr));
        assert!(eq_groups.contains(&col_x_expr));
        assert!(eq_groups.contains(&col_y_expr));

        Ok(())
    }

    #[test]
    fn project_equivalence_properties_test() -> Result<()> {
        let input_schema = Arc::new(Schema::new(vec![
            Field::new("a", DataType::Int64, true),
            Field::new("b", DataType::Int64, true),
            Field::new("c", DataType::Int64, true),
        ]));

        let input_properties = EquivalenceProperties::new(input_schema.clone());
        let col_a = col("a", &input_schema)?;

        // a as a1, a as a2, a as a3, a as a3
        let proj_exprs = vec![
            (col_a.clone(), "a1".to_string()),
            (col_a.clone(), "a2".to_string()),
            (col_a.clone(), "a3".to_string()),
            (col_a.clone(), "a4".to_string()),
        ];
        let projection_mapping = ProjectionMapping::try_new(&proj_exprs, &input_schema)?;

        let out_schema = output_schema(&projection_mapping, &input_schema)?;
        // a as a1, a as a2, a as a3, a as a3
        let proj_exprs = vec![
            (col_a.clone(), "a1".to_string()),
            (col_a.clone(), "a2".to_string()),
            (col_a.clone(), "a3".to_string()),
            (col_a.clone(), "a4".to_string()),
        ];
        let projection_mapping = ProjectionMapping::try_new(&proj_exprs, &input_schema)?;

        // a as a1, a as a2, a as a3, a as a3
        let col_a1 = &col("a1", &out_schema)?;
        let col_a2 = &col("a2", &out_schema)?;
        let col_a3 = &col("a3", &out_schema)?;
        let col_a4 = &col("a4", &out_schema)?;
        let out_properties = input_properties.project(&projection_mapping, out_schema);

        // At the output a1=a2=a3=a4
        assert_eq!(out_properties.eq_group().len(), 1);
        let eq_class = &out_properties.eq_group().classes[0];
        assert_eq!(eq_class.len(), 4);
        assert!(eq_class.contains(col_a1));
        assert!(eq_class.contains(col_a2));
        assert!(eq_class.contains(col_a3));
        assert!(eq_class.contains(col_a4));

        Ok(())
    }

    #[test]
    fn test_ordering_satisfy() -> Result<()> {
        let input_schema = Arc::new(Schema::new(vec![
            Field::new("a", DataType::Int64, true),
            Field::new("b", DataType::Int64, true),
        ]));
        let crude = vec![PhysicalSortExpr {
            expr: Arc::new(Column::new("a", 0)),
            options: SortOptions::default(),
        }];
        let finer = vec![
            PhysicalSortExpr {
                expr: Arc::new(Column::new("a", 0)),
                options: SortOptions::default(),
            },
            PhysicalSortExpr {
                expr: Arc::new(Column::new("b", 1)),
                options: SortOptions::default(),
            },
        ];
        // finer ordering satisfies, crude ordering should return true
        let mut eq_properties_finer = EquivalenceProperties::new(input_schema.clone());
        eq_properties_finer.oeq_class.push(finer.clone());
        assert!(eq_properties_finer.ordering_satisfy(&crude));

        // Crude ordering doesn't satisfy finer ordering. should return false
        let mut eq_properties_crude = EquivalenceProperties::new(input_schema.clone());
        eq_properties_crude.oeq_class.push(crude.clone());
        assert!(!eq_properties_crude.ordering_satisfy(&finer));
        Ok(())
    }

    #[test]
    fn test_ordering_satisfy_with_equivalence() -> Result<()> {
        // Schema satisfies following orderings:
        // [a ASC], [d ASC, b ASC], [e DESC, f ASC, g ASC]
        // and
        // Column [a=c] (e.g they are aliases).
        let (test_schema, eq_properties) = create_test_params()?;
        let col_a = &col("a", &test_schema)?;
        let col_b = &col("b", &test_schema)?;
        let col_c = &col("c", &test_schema)?;
        let col_d = &col("d", &test_schema)?;
        let col_e = &col("e", &test_schema)?;
        let col_f = &col("f", &test_schema)?;
        let col_g = &col("g", &test_schema)?;
        let option_asc = SortOptions {
            descending: false,
            nulls_first: false,
        };
        let option_desc = SortOptions {
            descending: true,
            nulls_first: true,
        };
        let table_data_with_properties =
            generate_table_for_eq_properties(&eq_properties, 625, 5)?;

        // First element in the tuple stores vector of requirement, second element is the expected return value for ordering_satisfy function
        let requirements = vec![
            // `a ASC NULLS LAST`, expects `ordering_satisfy` to be `true`, since existing ordering `a ASC NULLS LAST, b ASC NULLS LAST` satisfies it
            (vec![(col_a, option_asc)], true),
            (vec![(col_a, option_desc)], false),
            // Test whether equivalence works as expected
            (vec![(col_c, option_asc)], true),
            (vec![(col_c, option_desc)], false),
            // Test whether ordering equivalence works as expected
            (vec![(col_d, option_asc)], true),
            (vec![(col_d, option_asc), (col_b, option_asc)], true),
            (vec![(col_d, option_desc), (col_b, option_asc)], false),
            (
                vec![
                    (col_e, option_desc),
                    (col_f, option_asc),
                    (col_g, option_asc),
                ],
                true,
            ),
            (vec![(col_e, option_desc), (col_f, option_asc)], true),
            (vec![(col_e, option_asc), (col_f, option_asc)], false),
            (vec![(col_e, option_desc), (col_b, option_asc)], false),
            (vec![(col_e, option_asc), (col_b, option_asc)], false),
            (
                vec![
                    (col_d, option_asc),
                    (col_b, option_asc),
                    (col_d, option_asc),
                    (col_b, option_asc),
                ],
                true,
            ),
            (
                vec![
                    (col_d, option_asc),
                    (col_b, option_asc),
                    (col_e, option_desc),
                    (col_f, option_asc),
                ],
                true,
            ),
            (
                vec![
                    (col_d, option_asc),
                    (col_b, option_asc),
                    (col_e, option_desc),
                    (col_b, option_asc),
                ],
                true,
            ),
            (
                vec![
                    (col_d, option_asc),
                    (col_b, option_asc),
                    (col_d, option_desc),
                    (col_b, option_asc),
                ],
                true,
            ),
            (
                vec![
                    (col_d, option_asc),
                    (col_b, option_asc),
                    (col_e, option_asc),
                    (col_f, option_asc),
                ],
                false,
            ),
            (
                vec![
                    (col_d, option_asc),
                    (col_b, option_asc),
                    (col_e, option_asc),
                    (col_b, option_asc),
                ],
                false,
            ),
            (vec![(col_d, option_asc), (col_e, option_desc)], true),
            (
                vec![
                    (col_d, option_asc),
                    (col_c, option_asc),
                    (col_b, option_asc),
                ],
                true,
            ),
            (
                vec![
                    (col_d, option_asc),
                    (col_e, option_desc),
                    (col_f, option_asc),
                    (col_b, option_asc),
                ],
                true,
            ),
            (
                vec![
                    (col_d, option_asc),
                    (col_e, option_desc),
                    (col_c, option_asc),
                    (col_b, option_asc),
                ],
                true,
            ),
            (
                vec![
                    (col_d, option_asc),
                    (col_e, option_desc),
                    (col_b, option_asc),
                    (col_f, option_asc),
                ],
                true,
            ),
        ];

        for (cols, expected) in requirements {
            let err_msg = format!("Error in test case:{cols:?}");
            let required = cols
                .into_iter()
                .map(|(expr, options)| PhysicalSortExpr {
                    expr: expr.clone(),
                    options,
                })
                .collect::<Vec<_>>();

            // Check expected result with experimental result.
            assert_eq!(
                is_table_same_after_sort(
                    required.clone(),
                    table_data_with_properties.clone()
                )?,
                expected
            );
            assert_eq!(
                eq_properties.ordering_satisfy(&required),
                expected,
                "{err_msg}"
            );
        }
        Ok(())
    }

    #[test]
    fn test_ordering_satisfy_with_equivalence2() -> Result<()> {
        let test_schema = create_test_schema()?;
        let col_a = &col("a", &test_schema)?;
        let col_b = &col("b", &test_schema)?;
        let col_c = &col("c", &test_schema)?;
        let col_d = &col("d", &test_schema)?;
        let col_e = &col("e", &test_schema)?;
        let col_f = &col("f", &test_schema)?;
        let floor_a = &create_physical_expr(
            &BuiltinScalarFunction::Floor,
            &[col("a", &test_schema)?],
            &test_schema,
            &ExecutionProps::default(),
        )?;
        let floor_f = &create_physical_expr(
            &BuiltinScalarFunction::Floor,
            &[col("f", &test_schema)?],
            &test_schema,
            &ExecutionProps::default(),
        )?;
        let exp_a = &create_physical_expr(
            &BuiltinScalarFunction::Exp,
            &[col("a", &test_schema)?],
            &test_schema,
            &ExecutionProps::default(),
        )?;
        let a_plus_b = Arc::new(BinaryExpr::new(
            col_a.clone(),
            Operator::Plus,
            col_b.clone(),
        )) as Arc<dyn PhysicalExpr>;
        let options = SortOptions {
            descending: false,
            nulls_first: false,
        };

        let test_cases = vec![
            // ------------ TEST CASE 1 ------------
            (
                // orderings
                vec![
                    // [a ASC, d ASC, b ASC]
                    vec![(col_a, options), (col_d, options), (col_b, options)],
                    // [c ASC]
                    vec![(col_c, options)],
                ],
                // equivalence classes
                vec![vec![col_a, col_f]],
                // constants
                vec![col_e],
                // requirement [a ASC, b ASC], requirement is not satisfied.
                vec![(col_a, options), (col_b, options)],
                // expected: requirement is not satisfied.
                false,
            ),
            // ------------ TEST CASE 2 ------------
            (
                // orderings
                vec![
                    // [a ASC, c ASC, b ASC]
                    vec![(col_a, options), (col_c, options), (col_b, options)],
                    // [d ASC]
                    vec![(col_d, options)],
                ],
                // equivalence classes
                vec![vec![col_a, col_f]],
                // constants
                vec![col_e],
                // requirement [floor(a) ASC],
                vec![(floor_a, options)],
                // expected: requirement is satisfied.
                true,
            ),
            // ------------ TEST CASE 2.1 ------------
            (
                // orderings
                vec![
                    // [a ASC, c ASC, b ASC]
                    vec![(col_a, options), (col_c, options), (col_b, options)],
                    // [d ASC]
                    vec![(col_d, options)],
                ],
                // equivalence classes
                vec![vec![col_a, col_f]],
                // constants
                vec![col_e],
                // requirement [floor(f) ASC], (Please note that a=f)
                vec![(floor_f, options)],
                // expected: requirement is satisfied.
                true,
            ),
            // ------------ TEST CASE 3 ------------
            (
                // orderings
                vec![
                    // [a ASC, c ASC, b ASC]
                    vec![(col_a, options), (col_c, options), (col_b, options)],
                    // [d ASC]
                    vec![(col_d, options)],
                ],
                // equivalence classes
                vec![vec![col_a, col_f]],
                // constants
                vec![col_e],
                // requirement [a ASC, c ASC, a+b ASC],
                vec![(col_a, options), (col_c, options), (&a_plus_b, options)],
                // expected: requirement is satisfied.
                true,
            ),
            // ------------ TEST CASE 4 ------------
            (
                // orderings
                vec![
                    // [a ASC, b ASC, c ASC, d ASC]
                    vec![
                        (col_a, options),
                        (col_b, options),
                        (col_c, options),
                        (col_d, options),
                    ],
                ],
                // equivalence classes
                vec![vec![col_a, col_f]],
                // constants
                vec![col_e],
                // requirement [floor(a) ASC, a+b ASC],
                vec![(floor_a, options), (&a_plus_b, options)],
                // expected: requirement is satisfied.
                false,
            ),
            // ------------ TEST CASE 5 ------------
            (
                // orderings
                vec![
                    // [a ASC, b ASC, c ASC, d ASC]
                    vec![
                        (col_a, options),
                        (col_b, options),
                        (col_c, options),
                        (col_d, options),
                    ],
                ],
                // equivalence classes
                vec![vec![col_a, col_f]],
                // constants
                vec![col_e],
                // requirement [exp(a) ASC, a+b ASC],
                vec![(exp_a, options), (&a_plus_b, options)],
                // expected: requirement is not satisfied.
                // TODO: If we know that exp function is 1-to-1 function.
                //  we could have deduced that above requirement is satisfied.
                false,
            ),
            // ------------ TEST CASE 6 ------------
            (
                // orderings
                vec![
                    // [a ASC, d ASC, b ASC]
                    vec![(col_a, options), (col_d, options), (col_b, options)],
                    // [c ASC]
                    vec![(col_c, options)],
                ],
                // equivalence classes
                vec![vec![col_a, col_f]],
                // constants
                vec![col_e],
                // requirement [a ASC, d ASC, floor(a) ASC],
                vec![(col_a, options), (col_d, options), (floor_a, options)],
                // expected: requirement is satisfied.
                true,
            ),
            // ------------ TEST CASE 7 ------------
            (
                // orderings
                vec![
                    // [a ASC, c ASC, b ASC]
                    vec![(col_a, options), (col_c, options), (col_b, options)],
                    // [d ASC]
                    vec![(col_d, options)],
                ],
                // equivalence classes
                vec![vec![col_a, col_f]],
                // constants
                vec![col_e],
                // requirement [a ASC, floor(a) ASC, a + b ASC],
                vec![(col_a, options), (floor_a, options), (&a_plus_b, options)],
                // expected: requirement is not satisfied.
                false,
            ),
            // ------------ TEST CASE 8 ------------
            (
                // orderings
                vec![
                    // [a ASC, b ASC, c ASC]
                    vec![(col_a, options), (col_b, options), (col_c, options)],
                    // [d ASC]
                    vec![(col_d, options)],
                ],
                // equivalence classes
                vec![vec![col_a, col_f]],
                // constants
                vec![col_e],
                // requirement [a ASC, c ASC, floor(a) ASC, a + b ASC],
                vec![
                    (col_a, options),
                    (col_c, options),
                    (&floor_a, options),
                    (&a_plus_b, options),
                ],
                // expected: requirement is not satisfied.
                false,
            ),
            // ------------ TEST CASE 9 ------------
            (
                // orderings
                vec![
                    // [a ASC, b ASC, c ASC, d ASC]
                    vec![
                        (col_a, options),
                        (col_b, options),
                        (col_c, options),
                        (col_d, options),
                    ],
                ],
                // equivalence classes
                vec![vec![col_a, col_f]],
                // constants
                vec![col_e],
                // requirement [a ASC, b ASC, c ASC, floor(a) ASC],
                vec![
                    (col_a, options),
                    (col_b, options),
                    (&col_c, options),
                    (&floor_a, options),
                ],
                // expected: requirement is satisfied.
                true,
            ),
            // ------------ TEST CASE 10 ------------
            (
                // orderings
                vec![
                    // [d ASC, b ASC]
                    vec![(col_d, options), (col_b, options)],
                    // [c ASC, a ASC]
                    vec![(col_c, options), (col_a, options)],
                ],
                // equivalence classes
                vec![vec![col_a, col_f]],
                // constants
                vec![col_e],
                // requirement [c ASC, d ASC, a + b ASC],
                vec![(col_c, options), (col_d, options), (&a_plus_b, options)],
                // expected: requirement is satisfied.
                true,
            ),
        ];

        for (orderings, eq_group, constants, reqs, expected) in test_cases {
            let err_msg =
                format!("error in test orderings: {orderings:?}, eq_group: {eq_group:?}, constants: {constants:?}, reqs: {reqs:?}, expected: {expected:?}");
            let mut eq_properties = EquivalenceProperties::new(test_schema.clone());
            let orderings = convert_to_orderings(&orderings);
            eq_properties.add_new_orderings(orderings);
            let eq_group = eq_group
                .into_iter()
                .map(|eq_class| {
                    let eq_classes = eq_class.into_iter().cloned().collect::<Vec<_>>();
                    EquivalenceClass::new(eq_classes)
                })
                .collect::<Vec<_>>();
            let eq_group = EquivalenceGroup::new(eq_group);
            eq_properties.add_equivalence_group(eq_group);

            let constants = constants.into_iter().cloned();
            eq_properties = eq_properties.add_constants(constants);

            let reqs = convert_to_sort_exprs(&reqs);
            assert_eq!(
                eq_properties.ordering_satisfy(&reqs),
                expected,
                "{}",
                err_msg
            );
        }

        Ok(())
    }

    #[test]
    fn test_ordering_satisfy_with_equivalence_random() -> Result<()> {
        const N_RANDOM_SCHEMA: usize = 5;
        const N_ELEMENTS: usize = 125;
        const N_DISTINCT: usize = 5;
        const SORT_OPTIONS: SortOptions = SortOptions {
            descending: false,
            nulls_first: false,
        };

        for seed in 0..N_RANDOM_SCHEMA {
            // Create a random schema with random properties
            let (test_schema, eq_properties) = create_random_schema(seed as u64)?;
            // Generate a data that satisfies properties given
            let table_data_with_properties =
                generate_table_for_eq_properties(&eq_properties, N_ELEMENTS, N_DISTINCT)?;
            let col_exprs = vec![
                col("a", &test_schema)?,
                col("b", &test_schema)?,
                col("c", &test_schema)?,
                col("d", &test_schema)?,
                col("e", &test_schema)?,
                col("f", &test_schema)?,
            ];

            for n_req in 0..=col_exprs.len() {
                for exprs in col_exprs.iter().combinations(n_req) {
                    let requirement = exprs
                        .into_iter()
                        .map(|expr| PhysicalSortExpr {
                            expr: expr.clone(),
                            options: SORT_OPTIONS,
                        })
                        .collect::<Vec<_>>();
                    let expected = is_table_same_after_sort(
                        requirement.clone(),
                        table_data_with_properties.clone(),
                    )?;
                    let err_msg = format!(
                        "Error in test case requirement:{:?}, expected: {:?}, eq_properties.oeq_class: {:?}, eq_properties.eq_group: {:?}, eq_properties.constants: {:?}",
                        requirement, expected, eq_properties.oeq_class, eq_properties.eq_group, eq_properties.constants
                    );
                    // Check whether ordering_satisfy API result and
                    // experimental result matches.
                    assert_eq!(
                        eq_properties.ordering_satisfy(&requirement),
                        expected,
                        "{}",
                        err_msg
                    );
                }
            }
        }

        Ok(())
    }

    #[test]
    fn test_ordering_satisfy_with_equivalence_complex_random() -> Result<()> {
        const N_RANDOM_SCHEMA: usize = 100;
        const N_ELEMENTS: usize = 125;
        const N_DISTINCT: usize = 5;
        const SORT_OPTIONS: SortOptions = SortOptions {
            descending: false,
            nulls_first: false,
        };

        for seed in 0..N_RANDOM_SCHEMA {
            // Create a random schema with random properties
            let (test_schema, eq_properties) = create_random_schema(seed as u64)?;
            // Generate a data that satisfies properties given
            let table_data_with_properties =
                generate_table_for_eq_properties(&eq_properties, N_ELEMENTS, N_DISTINCT)?;

            let floor_a = create_physical_expr(
                &BuiltinScalarFunction::Floor,
                &[col("a", &test_schema)?],
                &test_schema,
                &ExecutionProps::default(),
            )?;
            let a_plus_b = Arc::new(BinaryExpr::new(
                col("a", &test_schema)?,
                Operator::Plus,
                col("b", &test_schema)?,
            )) as Arc<dyn PhysicalExpr>;
            let exprs = vec![
                col("a", &test_schema)?,
                col("b", &test_schema)?,
                col("c", &test_schema)?,
                col("d", &test_schema)?,
                col("e", &test_schema)?,
                col("f", &test_schema)?,
                floor_a,
                a_plus_b,
            ];

            for n_req in 0..=exprs.len() {
                for exprs in exprs.iter().combinations(n_req) {
                    let requirement = exprs
                        .into_iter()
                        .map(|expr| PhysicalSortExpr {
                            expr: expr.clone(),
                            options: SORT_OPTIONS,
                        })
                        .collect::<Vec<_>>();
                    let expected = is_table_same_after_sort(
                        requirement.clone(),
                        table_data_with_properties.clone(),
                    )?;
                    let err_msg = format!(
                        "Error in test case requirement:{:?}, expected: {:?}, eq_properties.oeq_class: {:?}, eq_properties.eq_group: {:?}, eq_properties.constants: {:?}",
                        requirement, expected, eq_properties.oeq_class, eq_properties.eq_group, eq_properties.constants
                    );
                    // Check whether ordering_satisfy API result and
                    // experimental result matches.

                    assert_eq!(
                        eq_properties.ordering_satisfy(&requirement),
                        (expected | false),
                        "{}",
                        err_msg
                    );
                }
            }
        }

        Ok(())
    }

    #[test]
    fn test_ordering_satisfy_different_lengths() -> Result<()> {
        let test_schema = create_test_schema()?;
        let col_a = &col("a", &test_schema)?;
        let col_b = &col("b", &test_schema)?;
        let col_c = &col("c", &test_schema)?;
        let col_d = &col("d", &test_schema)?;
        let col_e = &col("e", &test_schema)?;
        let col_f = &col("f", &test_schema)?;
        let options = SortOptions {
            descending: false,
            nulls_first: false,
        };
        // a=c (e.g they are aliases).
        let mut eq_properties = EquivalenceProperties::new(test_schema);
        eq_properties.add_equal_conditions(col_a, col_c);

        let orderings = vec![
            vec![(col_a, options)],
            vec![(col_e, options)],
            vec![(col_d, options), (col_f, options)],
        ];
        let orderings = convert_to_orderings(&orderings);

        // Column [a ASC], [e ASC], [d ASC, f ASC] are all valid orderings for the schema.
        eq_properties.add_new_orderings(orderings);

        // First entry in the tuple is required ordering, second entry is the expected flag
        // that indicates whether this required ordering is satisfied.
        // ([a ASC], true) indicate a ASC requirement is already satisfied by existing orderings.
        let test_cases = vec![
            // [c ASC, a ASC, e ASC], expected represents this requirement is satisfied
            (
                vec![(col_c, options), (col_a, options), (col_e, options)],
                true,
            ),
            (vec![(col_c, options), (col_b, options)], false),
            (vec![(col_c, options), (col_d, options)], true),
            (
                vec![(col_d, options), (col_f, options), (col_b, options)],
                false,
            ),
            (vec![(col_d, options), (col_f, options)], true),
        ];

        for (reqs, expected) in test_cases {
            let err_msg =
                format!("error in test reqs: {:?}, expected: {:?}", reqs, expected,);
            let reqs = convert_to_sort_exprs(&reqs);
            assert_eq!(
                eq_properties.ordering_satisfy(&reqs),
                expected,
                "{}",
                err_msg
            );
        }

        Ok(())
    }

    #[test]
    fn test_bridge_groups() -> Result<()> {
        // First entry in the tuple is argument, second entry is the bridged result
        let test_cases = vec![
            // ------- TEST CASE 1 -----------//
            (
                vec![vec![1, 2, 3], vec![2, 4, 5], vec![11, 12, 9], vec![7, 6, 5]],
                // Expected is compared with set equality. Order of the specific results may change.
                vec![vec![1, 2, 3, 4, 5, 6, 7], vec![9, 11, 12]],
            ),
            // ------- TEST CASE 2 -----------//
            (
                vec![vec![1, 2, 3], vec![3, 4, 5], vec![9, 8, 7], vec![7, 6, 5]],
                // Expected
                vec![vec![1, 2, 3, 4, 5, 6, 7, 8, 9]],
            ),
        ];
        for (entries, expected) in test_cases {
            let entries = entries
                .into_iter()
                .map(|entry| entry.into_iter().map(lit).collect::<Vec<_>>())
                .map(EquivalenceClass::new)
                .collect::<Vec<_>>();
            let expected = expected
                .into_iter()
                .map(|entry| entry.into_iter().map(lit).collect::<Vec<_>>())
                .map(EquivalenceClass::new)
                .collect::<Vec<_>>();
            let mut eq_groups = EquivalenceGroup::new(entries.clone());
            eq_groups.bridge_classes();
            let eq_groups = eq_groups.classes;
            let err_msg = format!(
                "error in test entries: {:?}, expected: {:?}, actual:{:?}",
                entries, expected, eq_groups
            );
            assert_eq!(eq_groups.len(), expected.len(), "{}", err_msg);
            for idx in 0..eq_groups.len() {
                assert_eq!(&eq_groups[idx], &expected[idx], "{}", err_msg);
            }
        }
        Ok(())
    }

    #[test]
    fn test_remove_redundant_entries_eq_group() -> Result<()> {
        let entries = vec![
            EquivalenceClass::new(vec![lit(1), lit(1), lit(2)]),
            // This group is meaningless should be removed
            EquivalenceClass::new(vec![lit(3), lit(3)]),
            EquivalenceClass::new(vec![lit(4), lit(5), lit(6)]),
        ];
        // Given equivalences classes are not in succinct form.
        // Expected form is the most plain representation that is functionally same.
        let expected = vec![
            EquivalenceClass::new(vec![lit(1), lit(2)]),
            EquivalenceClass::new(vec![lit(4), lit(5), lit(6)]),
        ];
        let mut eq_groups = EquivalenceGroup::new(entries);
        eq_groups.remove_redundant_entries();

        let eq_groups = eq_groups.classes;
        assert_eq!(eq_groups.len(), expected.len());
        assert_eq!(eq_groups.len(), 2);

        assert_eq!(eq_groups[0], expected[0]);
        assert_eq!(eq_groups[1], expected[1]);
        Ok(())
    }

    #[test]
    fn test_remove_redundant_entries_oeq_class() -> Result<()> {
        let schema = create_test_schema()?;
        let col_a = &col("a", &schema)?;
        let col_b = &col("b", &schema)?;
        let col_c = &col("c", &schema)?;
        let col_d = &col("d", &schema)?;
        let col_e = &col("e", &schema)?;

        let option_asc = SortOptions {
            descending: false,
            nulls_first: false,
        };
        let option_desc = SortOptions {
            descending: true,
            nulls_first: true,
        };

        // First entry in the tuple is the given orderings for the table
        // Second entry is the simplest version of the given orderings that is functionally equivalent.
        let test_cases = vec![
            // ------- TEST CASE 1 ---------
            (
                // ORDERINGS GIVEN
                vec![
                    // [a ASC, b ASC]
                    vec![(col_a, option_asc), (col_b, option_asc)],
                ],
                // EXPECTED orderings that is succinct.
                vec![
                    // [a ASC, b ASC]
                    vec![(col_a, option_asc), (col_b, option_asc)],
                ],
            ),
            // ------- TEST CASE 2 ---------
            (
                // ORDERINGS GIVEN
                vec![
                    // [a ASC, b ASC]
                    vec![(col_a, option_asc), (col_b, option_asc)],
                    // [a ASC, b ASC, c ASC]
                    vec![
                        (col_a, option_asc),
                        (col_b, option_asc),
                        (col_c, option_asc),
                    ],
                ],
                // EXPECTED orderings that is succinct.
                vec![
                    // [a ASC, b ASC, c ASC]
                    vec![
                        (col_a, option_asc),
                        (col_b, option_asc),
                        (col_c, option_asc),
                    ],
                ],
            ),
            // ------- TEST CASE 3 ---------
            (
                // ORDERINGS GIVEN
                vec![
                    // [a ASC, b DESC]
                    vec![(col_a, option_asc), (col_b, option_desc)],
                    // [a ASC]
                    vec![(col_a, option_asc)],
                    // [a ASC, c ASC]
                    vec![(col_a, option_asc), (col_c, option_asc)],
                ],
                // EXPECTED orderings that is succinct.
                vec![
                    // [a ASC, b DESC]
                    vec![(col_a, option_asc), (col_b, option_desc)],
                    // [a ASC, c ASC]
                    vec![(col_a, option_asc), (col_c, option_asc)],
                ],
            ),
            // ------- TEST CASE 4 ---------
            (
                // ORDERINGS GIVEN
                vec![
                    // [a ASC, b ASC]
                    vec![(col_a, option_asc), (col_b, option_asc)],
                    // [a ASC, b ASC, c ASC]
                    vec![
                        (col_a, option_asc),
                        (col_b, option_asc),
                        (col_c, option_asc),
                    ],
                    // [a ASC]
                    vec![(col_a, option_asc)],
                ],
                // EXPECTED orderings that is succinct.
                vec![
                    // [a ASC, b ASC, c ASC]
                    vec![
                        (col_a, option_asc),
                        (col_b, option_asc),
                        (col_c, option_asc),
                    ],
                ],
            ),
            // ------- TEST CASE 5 ---------
            // Empty ordering
            (
                vec![vec![]],
                // No ordering in the state (empty ordering is ignored).
                vec![],
            ),
            // ------- TEST CASE 6 ---------
            (
                // ORDERINGS GIVEN
                vec![
                    // [a ASC, b ASC]
                    vec![(col_a, option_asc), (col_b, option_asc)],
                    // [b ASC]
                    vec![(col_b, option_asc)],
                ],
                // EXPECTED orderings that is succinct.
                vec![
                    // [a ASC]
                    vec![(col_a, option_asc)],
                    // [b ASC]
                    vec![(col_b, option_asc)],
                ],
            ),
            // ------- TEST CASE 7 ---------
            // b, a
            // c, a
            // d, b, c
            (
                // ORDERINGS GIVEN
                vec![
                    // [b ASC, a ASC]
                    vec![(col_b, option_asc), (col_a, option_asc)],
                    // [c ASC, a ASC]
                    vec![(col_c, option_asc), (col_a, option_asc)],
                    // [d ASC, b ASC, c ASC]
                    vec![
                        (col_d, option_asc),
                        (col_b, option_asc),
                        (col_c, option_asc),
                    ],
                ],
                // EXPECTED orderings that is succinct.
                vec![
                    // [b ASC, a ASC]
                    vec![(col_b, option_asc), (col_a, option_asc)],
                    // [c ASC, a ASC]
                    vec![(col_c, option_asc), (col_a, option_asc)],
                    // [d ASC]
                    vec![(col_d, option_asc)],
                ],
            ),
            // ------- TEST CASE 8 ---------
            // b, e
            // c, a
            // d, b, e, c, a
            (
                // ORDERINGS GIVEN
                vec![
                    // [b ASC, e ASC]
                    vec![(col_b, option_asc), (col_e, option_asc)],
                    // [c ASC, a ASC]
                    vec![(col_c, option_asc), (col_a, option_asc)],
                    // [d ASC, b ASC, e ASC, c ASC, a ASC]
                    vec![
                        (col_d, option_asc),
                        (col_b, option_asc),
                        (col_e, option_asc),
                        (col_c, option_asc),
                        (col_a, option_asc),
                    ],
                ],
                // EXPECTED orderings that is succinct.
                vec![
                    // [b ASC, e ASC]
                    vec![(col_b, option_asc), (col_e, option_asc)],
                    // [c ASC, a ASC]
                    vec![(col_c, option_asc), (col_a, option_asc)],
                    // [d ASC]
                    vec![(col_d, option_asc)],
                ],
            ),
            // ------- TEST CASE 9 ---------
            // b
            // a, b, c
            // d, a, b
            (
                // ORDERINGS GIVEN
                vec![
                    // [b ASC]
                    vec![(col_b, option_asc)],
                    // [a ASC, b ASC, c ASC]
                    vec![
                        (col_a, option_asc),
                        (col_b, option_asc),
                        (col_c, option_asc),
                    ],
                    // [d ASC, a ASC, b ASC]
                    vec![
                        (col_d, option_asc),
                        (col_a, option_asc),
                        (col_b, option_asc),
                    ],
                ],
                // EXPECTED orderings that is succinct.
                vec![
                    // [b ASC]
                    vec![(col_b, option_asc)],
                    // [a ASC, b ASC, c ASC]
                    vec![
                        (col_a, option_asc),
                        (col_b, option_asc),
                        (col_c, option_asc),
                    ],
                    // [d ASC]
                    vec![(col_d, option_asc)],
                ],
            ),
        ];
        for (orderings, expected) in test_cases {
            let orderings = convert_to_orderings(&orderings);
            let expected = convert_to_orderings(&expected);
            let actual = OrderingEquivalenceClass::new(orderings.clone());
            let actual = actual.orderings;
            let err_msg = format!(
                "orderings: {:?}, expected: {:?}, actual :{:?}",
                orderings, expected, actual
            );
            assert_eq!(actual.len(), expected.len(), "{}", err_msg);
            for elem in actual {
                assert!(expected.contains(&elem), "{}", err_msg);
            }
        }

        Ok(())
    }

    #[test]
    fn test_get_updated_right_ordering_equivalence_properties() -> Result<()> {
        let join_type = JoinType::Inner;
        // Join right child schema
        let child_fields: Fields = ["x", "y", "z", "w"]
            .into_iter()
            .map(|name| Field::new(name, DataType::Int32, true))
            .collect();
        let child_schema = Schema::new(child_fields);
        let col_x = &col("x", &child_schema)?;
        let col_y = &col("y", &child_schema)?;
        let col_z = &col("z", &child_schema)?;
        let col_w = &col("w", &child_schema)?;
        let option_asc = SortOptions {
            descending: false,
            nulls_first: false,
        };
        // [x ASC, y ASC], [z ASC, w ASC]
        let orderings = vec![
            vec![(col_x, option_asc), (col_y, option_asc)],
            vec![(col_z, option_asc), (col_w, option_asc)],
        ];
        let orderings = convert_to_orderings(&orderings);
        // Right child ordering equivalences
        let mut right_oeq_class = OrderingEquivalenceClass::new(orderings);

        let left_columns_len = 4;

        let fields: Fields = ["a", "b", "c", "d", "x", "y", "z", "w"]
            .into_iter()
            .map(|name| Field::new(name, DataType::Int32, true))
            .collect();

        // Join Schema
        let schema = Schema::new(fields);
        let col_a = &col("a", &schema)?;
        let col_d = &col("d", &schema)?;
        let col_x = &col("x", &schema)?;
        let col_y = &col("y", &schema)?;
        let col_z = &col("z", &schema)?;
        let col_w = &col("w", &schema)?;

        let mut join_eq_properties = EquivalenceProperties::new(Arc::new(schema));
        // a=x and d=w
        join_eq_properties.add_equal_conditions(col_a, col_x);
        join_eq_properties.add_equal_conditions(col_d, col_w);

        updated_right_ordering_equivalence_class(
            &mut right_oeq_class,
            &join_type,
            left_columns_len,
        );
        join_eq_properties.add_ordering_equivalence_class(right_oeq_class);
        let result = join_eq_properties.oeq_class().clone();

        // [x ASC, y ASC], [z ASC, w ASC]
        let orderings = vec![
            vec![(col_x, option_asc), (col_y, option_asc)],
            vec![(col_z, option_asc), (col_w, option_asc)],
        ];
        let orderings = convert_to_orderings(&orderings);
        let expected = OrderingEquivalenceClass::new(orderings);

        assert_eq!(result, expected);

        Ok(())
    }

    /// Checks if the table (RecordBatch) remains unchanged when sorted according to the provided `required_ordering`.
    ///
    /// The function works by adding a unique column of ascending integers to the original table. This column ensures
    /// that rows that are otherwise indistinguishable (e.g., if they have the same values in all other columns) can
    /// still be differentiated. When sorting the extended table, the unique column acts as a tie-breaker to produce
    /// deterministic sorting results.
    ///
    /// If the table remains the same after sorting with the added unique column, it indicates that the table was
    /// already sorted according to `required_ordering` to begin with.
    fn is_table_same_after_sort(
        mut required_ordering: Vec<PhysicalSortExpr>,
        batch: RecordBatch,
    ) -> Result<bool> {
        // Clone the original schema and columns
        let original_schema = batch.schema();
        let mut columns = batch.columns().to_vec();

        // Create a new unique column
        let n_row = batch.num_rows();
        let vals: Vec<usize> = (0..n_row).collect::<Vec<_>>();
        let vals: Vec<f64> = vals.into_iter().map(|val| val as f64).collect();
        let unique_col = Arc::new(Float64Array::from_iter_values(vals)) as ArrayRef;
        columns.push(unique_col.clone());

        // Create a new schema with the added unique column
        let unique_col_name = "unique";
        let unique_field =
            Arc::new(Field::new(unique_col_name, DataType::Float64, false));
        let fields: Vec<_> = original_schema
            .fields()
            .iter()
            .cloned()
            .chain(std::iter::once(unique_field))
            .collect();
        let schema = Arc::new(Schema::new(fields));

        // Create a new batch with the added column
        let new_batch = RecordBatch::try_new(schema.clone(), columns)?;

        // Add the unique column to the required ordering to ensure deterministic results
        required_ordering.push(PhysicalSortExpr {
            expr: Arc::new(Column::new(unique_col_name, original_schema.fields().len())),
            options: Default::default(),
        });

        // Convert the required ordering to a list of SortColumn
        let sort_columns = required_ordering
            .iter()
            .map(|order_expr| {
                let expr_result = order_expr.expr.evaluate(&new_batch)?;
                let values = expr_result.into_array(new_batch.num_rows())?;
                Ok(SortColumn {
                    values,
                    options: Some(order_expr.options),
                })
            })
            .collect::<Result<Vec<_>>>()?;

        // Check if the indices after sorting match the initial ordering
        let sorted_indices = lexsort_to_indices(&sort_columns, None)?;
        let original_indices = UInt32Array::from_iter_values(0..n_row as u32);

        Ok(sorted_indices == original_indices)
    }

    // If we already generated a random result for one of the
    // expressions in the equivalence classes. For other expressions in the same
    // equivalence class use same result. This util gets already calculated result, when available.
    fn get_representative_arr(
        eq_group: &EquivalenceClass,
        existing_vec: &[Option<ArrayRef>],
        schema: SchemaRef,
    ) -> Option<ArrayRef> {
        for expr in eq_group.iter() {
            let col = expr.as_any().downcast_ref::<Column>().unwrap();
            let (idx, _field) = schema.column_with_name(col.name()).unwrap();
            if let Some(res) = &existing_vec[idx] {
                return Some(res.clone());
            }
        }
        None
    }

    // Generate a table that satisfies the given equivalence properties; i.e.
    // equivalences, ordering equivalences, and constants.
    fn generate_table_for_eq_properties(
        eq_properties: &EquivalenceProperties,
        n_elem: usize,
        n_distinct: usize,
    ) -> Result<RecordBatch> {
        let mut rng = StdRng::seed_from_u64(23);

        let schema = eq_properties.schema();
        let mut schema_vec = vec![None; schema.fields.len()];

        // Utility closure to generate random array
        let mut generate_random_array = |num_elems: usize, max_val: usize| -> ArrayRef {
            let values: Vec<f64> = (0..num_elems)
                .map(|_| rng.gen_range(0..max_val) as f64 / 2.0)
                .collect();
            Arc::new(Float64Array::from_iter_values(values))
        };

        // Fill constant columns
        for constant in &eq_properties.constants {
            let col = constant.as_any().downcast_ref::<Column>().unwrap();
            let (idx, _field) = schema.column_with_name(col.name()).unwrap();
            let arr = Arc::new(Float64Array::from_iter_values(vec![0 as f64; n_elem]))
                as ArrayRef;
            schema_vec[idx] = Some(arr);
        }

        // Fill columns based on ordering equivalences
        for ordering in eq_properties.oeq_class.iter() {
            let (sort_columns, indices): (Vec<_>, Vec<_>) = ordering
                .iter()
                .map(|PhysicalSortExpr { expr, options }| {
                    let col = expr.as_any().downcast_ref::<Column>().unwrap();
                    let (idx, _field) = schema.column_with_name(col.name()).unwrap();
                    let arr = generate_random_array(n_elem, n_distinct);
                    (
                        SortColumn {
                            values: arr,
                            options: Some(*options),
                        },
                        idx,
                    )
                })
                .unzip();

            let sort_arrs = arrow::compute::lexsort(&sort_columns, None)?;
            for (idx, arr) in izip!(indices, sort_arrs) {
                schema_vec[idx] = Some(arr);
            }
        }

        // Fill columns based on equivalence groups
        for eq_group in eq_properties.eq_group.iter() {
            let representative_array =
                get_representative_arr(eq_group, &schema_vec, schema.clone())
                    .unwrap_or_else(|| generate_random_array(n_elem, n_distinct));

            for expr in eq_group.iter() {
                let col = expr.as_any().downcast_ref::<Column>().unwrap();
                let (idx, _field) = schema.column_with_name(col.name()).unwrap();
                schema_vec[idx] = Some(representative_array.clone());
            }
        }

        let res: Vec<_> = schema_vec
            .into_iter()
            .zip(schema.fields.iter())
            .map(|(elem, field)| {
                (
                    field.name(),
                    // Generate random values for columns that do not occur in any of the groups (equivalence, ordering equivalence, constants)
                    elem.unwrap_or_else(|| generate_random_array(n_elem, n_distinct)),
                )
            })
            .collect();

        Ok(RecordBatch::try_from_iter(res)?)
    }

    #[test]
    fn test_schema_normalize_expr_with_equivalence() -> Result<()> {
        let col_a = &Column::new("a", 0);
        let col_b = &Column::new("b", 1);
        let col_c = &Column::new("c", 2);
        // Assume that column a and c are aliases.
        let (_test_schema, eq_properties) = create_test_params()?;

        let col_a_expr = Arc::new(col_a.clone()) as Arc<dyn PhysicalExpr>;
        let col_b_expr = Arc::new(col_b.clone()) as Arc<dyn PhysicalExpr>;
        let col_c_expr = Arc::new(col_c.clone()) as Arc<dyn PhysicalExpr>;
        // Test cases for equivalence normalization,
        // First entry in the tuple is argument, second entry is expected result after normalization.
        let expressions = vec![
            // Normalized version of the column a and c should go to a
            // (by convention all the expressions inside equivalence class are mapped to the first entry
            // in this case a is the first entry in the equivalence class.)
            (&col_a_expr, &col_a_expr),
            (&col_c_expr, &col_a_expr),
            // Cannot normalize column b
            (&col_b_expr, &col_b_expr),
        ];
        let eq_group = eq_properties.eq_group();
        for (expr, expected_eq) in expressions {
            assert!(
                expected_eq.eq(&eq_group.normalize_expr(expr.clone())),
                "error in test: expr: {expr:?}"
            );
        }

        Ok(())
    }

    #[test]
    fn test_schema_normalize_sort_requirement_with_equivalence() -> Result<()> {
        let option1 = SortOptions {
            descending: false,
            nulls_first: false,
        };
        // Assume that column a and c are aliases.
        let (test_schema, eq_properties) = create_test_params()?;
        let col_a = &col("a", &test_schema)?;
        let col_c = &col("c", &test_schema)?;
        let col_d = &col("d", &test_schema)?;

        // Test cases for equivalence normalization
        // First entry in the tuple is PhysicalSortRequirement, second entry in the tuple is
        // expected PhysicalSortRequirement after normalization.
        let test_cases = vec![
            (vec![(col_a, Some(option1))], vec![(col_a, Some(option1))]),
            // In the normalized version column c should be replace with column a
            (vec![(col_c, Some(option1))], vec![(col_a, Some(option1))]),
            (vec![(col_c, None)], vec![(col_a, None)]),
            (vec![(col_d, Some(option1))], vec![(col_d, Some(option1))]),
        ];
        for (reqs, expected) in test_cases.into_iter() {
            let reqs = convert_to_sort_reqs(&reqs);
            let expected = convert_to_sort_reqs(&expected);

            let normalized = eq_properties.normalize_sort_requirements(&reqs);
            assert!(
                expected.eq(&normalized),
                "error in test: reqs: {reqs:?}, expected: {expected:?}, normalized: {normalized:?}"
            );
        }

        Ok(())
    }

    #[test]
    fn test_normalize_sort_reqs() -> Result<()> {
        // Schema satisfies following properties
        // a=c
        // and following orderings are valid
        // [a ASC], [d ASC, b ASC], [e DESC, f ASC, g ASC]
        let (test_schema, eq_properties) = create_test_params()?;
        let col_a = &col("a", &test_schema)?;
        let col_b = &col("b", &test_schema)?;
        let col_c = &col("c", &test_schema)?;
        let col_d = &col("d", &test_schema)?;
        let col_e = &col("e", &test_schema)?;
        let col_f = &col("f", &test_schema)?;
        let option_asc = SortOptions {
            descending: false,
            nulls_first: false,
        };
        let option_desc = SortOptions {
            descending: true,
            nulls_first: true,
        };
        // First element in the tuple stores vector of requirement, second element is the expected return value for ordering_satisfy function
        let requirements = vec![
            (
                vec![(col_a, Some(option_asc))],
                vec![(col_a, Some(option_asc))],
            ),
            (
                vec![(col_a, Some(option_desc))],
                vec![(col_a, Some(option_desc))],
            ),
            (vec![(col_a, None)], vec![(col_a, None)]),
            // Test whether equivalence works as expected
            (
                vec![(col_c, Some(option_asc))],
                vec![(col_a, Some(option_asc))],
            ),
            (vec![(col_c, None)], vec![(col_a, None)]),
            // Test whether ordering equivalence works as expected
            (
                vec![(col_d, Some(option_asc)), (col_b, Some(option_asc))],
                vec![(col_d, Some(option_asc)), (col_b, Some(option_asc))],
            ),
            (
                vec![(col_d, None), (col_b, None)],
                vec![(col_d, None), (col_b, None)],
            ),
            (
                vec![(col_e, Some(option_desc)), (col_f, Some(option_asc))],
                vec![(col_e, Some(option_desc)), (col_f, Some(option_asc))],
            ),
            // We should be able to normalize in compatible requirements also (not exactly equal)
            (
                vec![(col_e, Some(option_desc)), (col_f, None)],
                vec![(col_e, Some(option_desc)), (col_f, None)],
            ),
            (
                vec![(col_e, None), (col_f, None)],
                vec![(col_e, None), (col_f, None)],
            ),
        ];

        for (reqs, expected_normalized) in requirements.into_iter() {
            let req = convert_to_sort_reqs(&reqs);
            let expected_normalized = convert_to_sort_reqs(&expected_normalized);

            assert_eq!(
                eq_properties.normalize_sort_requirements(&req),
                expected_normalized
            );
        }

        Ok(())
    }

    #[test]
    fn test_get_finer() -> Result<()> {
        let schema = create_test_schema()?;
        let col_a = &col("a", &schema)?;
        let col_b = &col("b", &schema)?;
        let col_c = &col("c", &schema)?;
        let eq_properties = EquivalenceProperties::new(schema);
        let option_asc = SortOptions {
            descending: false,
            nulls_first: false,
        };
        let option_desc = SortOptions {
            descending: true,
            nulls_first: true,
        };
        // First entry, and second entry are the physical sort requirement that are argument for get_finer_requirement.
        // Third entry is the expected result.
        let tests_cases = vec![
            // Get finer requirement between [a Some(ASC)] and [a None, b Some(ASC)]
            // result should be [a Some(ASC), b Some(ASC)]
            (
                vec![(col_a, Some(option_asc))],
                vec![(col_a, None), (col_b, Some(option_asc))],
                Some(vec![(col_a, Some(option_asc)), (col_b, Some(option_asc))]),
            ),
            // Get finer requirement between [a Some(ASC), b Some(ASC), c Some(ASC)] and [a Some(ASC), b Some(ASC)]
            // result should be [a Some(ASC), b Some(ASC), c Some(ASC)]
            (
                vec![
                    (col_a, Some(option_asc)),
                    (col_b, Some(option_asc)),
                    (col_c, Some(option_asc)),
                ],
                vec![(col_a, Some(option_asc)), (col_b, Some(option_asc))],
                Some(vec![
                    (col_a, Some(option_asc)),
                    (col_b, Some(option_asc)),
                    (col_c, Some(option_asc)),
                ]),
            ),
            // Get finer requirement between [a Some(ASC), b Some(ASC)] and [a Some(ASC), b Some(DESC)]
            // result should be None
            (
                vec![(col_a, Some(option_asc)), (col_b, Some(option_asc))],
                vec![(col_a, Some(option_asc)), (col_b, Some(option_desc))],
                None,
            ),
        ];
        for (lhs, rhs, expected) in tests_cases {
            let lhs = convert_to_sort_reqs(&lhs);
            let rhs = convert_to_sort_reqs(&rhs);
            let expected = expected.map(|expected| convert_to_sort_reqs(&expected));
            let finer = eq_properties.get_finer_requirement(&lhs, &rhs);
            assert_eq!(finer, expected)
        }

        Ok(())
    }

    #[test]
    fn test_get_meet_ordering() -> Result<()> {
        let schema = create_test_schema()?;
        let col_a = &col("a", &schema)?;
        let col_b = &col("b", &schema)?;
        let eq_properties = EquivalenceProperties::new(schema);
        let option_asc = SortOptions {
            descending: false,
            nulls_first: false,
        };
        let option_desc = SortOptions {
            descending: true,
            nulls_first: true,
        };
        let tests_cases = vec![
            // Get meet ordering between [a ASC] and [a ASC, b ASC]
            // result should be [a ASC]
            (
                vec![(col_a, option_asc)],
                vec![(col_a, option_asc), (col_b, option_asc)],
                Some(vec![(col_a, option_asc)]),
            ),
            // Get meet ordering between [a ASC] and [a DESC]
            // result should be None.
            (vec![(col_a, option_asc)], vec![(col_a, option_desc)], None),
            // Get meet ordering between [a ASC, b ASC] and [a ASC, b DESC]
            // result should be [a ASC].
            (
                vec![(col_a, option_asc), (col_b, option_asc)],
                vec![(col_a, option_asc), (col_b, option_desc)],
                Some(vec![(col_a, option_asc)]),
            ),
        ];
        for (lhs, rhs, expected) in tests_cases {
            let lhs = convert_to_sort_exprs(&lhs);
            let rhs = convert_to_sort_exprs(&rhs);
            let expected = expected.map(|expected| convert_to_sort_exprs(&expected));
            let finer = eq_properties.get_meet_ordering(&lhs, &rhs);
            assert_eq!(finer, expected)
        }

        Ok(())
    }

    #[test]
    fn test_find_longest_permutation() -> Result<()> {
        // Schema satisfies following orderings:
        // [a ASC], [d ASC, b ASC], [e DESC, f ASC, g ASC]
        // and
        // Column [a=c] (e.g they are aliases).
        // At below we add [d ASC, h DESC] also, for test purposes
        let (test_schema, mut eq_properties) = create_test_params()?;
        let col_a = &col("a", &test_schema)?;
        let col_b = &col("b", &test_schema)?;
        let col_c = &col("c", &test_schema)?;
        let col_d = &col("d", &test_schema)?;
        let col_e = &col("e", &test_schema)?;
        let col_h = &col("h", &test_schema)?;
        // a + d
        let a_plus_d = Arc::new(BinaryExpr::new(
            col_a.clone(),
            Operator::Plus,
            col_d.clone(),
        )) as Arc<dyn PhysicalExpr>;

        let option_asc = SortOptions {
            descending: false,
            nulls_first: false,
        };
        let option_desc = SortOptions {
            descending: true,
            nulls_first: true,
        };
        // [d ASC, h ASC] also satisfies schema.
        eq_properties.add_new_orderings([vec![
            PhysicalSortExpr {
                expr: col_d.clone(),
                options: option_asc,
            },
            PhysicalSortExpr {
                expr: col_h.clone(),
                options: option_desc,
            },
        ]]);
        let test_cases = vec![
            // TEST CASE 1
            (vec![col_a], vec![(col_a, option_asc)]),
            // TEST CASE 2
            (vec![col_c], vec![(col_c, option_asc)]),
            // TEST CASE 3
            (
                vec![col_d, col_e, col_b],
                vec![
                    (col_d, option_asc),
                    (col_e, option_desc),
                    (col_b, option_asc),
                ],
            ),
            // TEST CASE 4
            (vec![col_b], vec![]),
            // TEST CASE 5
            (vec![col_d], vec![(col_d, option_asc)]),
            // TEST CASE 5
            (vec![&a_plus_d], vec![(&a_plus_d, option_asc)]),
            // TEST CASE 6
            (
                vec![col_b, col_d],
                vec![(col_d, option_asc), (col_b, option_asc)],
            ),
            // TEST CASE 6
            (
                vec![col_c, col_e],
                vec![(col_c, option_asc), (col_e, option_desc)],
            ),
        ];
        for (exprs, expected) in test_cases {
            let exprs = exprs.into_iter().cloned().collect::<Vec<_>>();
            let expected = convert_to_sort_exprs(&expected);
            let (actual, _) = eq_properties.find_longest_permutation(&exprs);
            assert_eq!(actual, expected);
        }

        Ok(())
    }

    #[test]
    fn test_find_longest_permutation_random() -> Result<()> {
        const N_RANDOM_SCHEMA: usize = 100;
        const N_ELEMENTS: usize = 125;
        const N_DISTINCT: usize = 5;

        for seed in 0..N_RANDOM_SCHEMA {
            // Create a random schema with random properties
            let (test_schema, eq_properties) = create_random_schema(seed as u64)?;
            // Generate a data that satisfies properties given
            let table_data_with_properties =
                generate_table_for_eq_properties(&eq_properties, N_ELEMENTS, N_DISTINCT)?;

            let floor_a = create_physical_expr(
                &BuiltinScalarFunction::Floor,
                &[col("a", &test_schema)?],
                &test_schema,
                &ExecutionProps::default(),
            )?;
            let a_plus_b = Arc::new(BinaryExpr::new(
                col("a", &test_schema)?,
                Operator::Plus,
                col("b", &test_schema)?,
            )) as Arc<dyn PhysicalExpr>;
            let exprs = vec![
                col("a", &test_schema)?,
                col("b", &test_schema)?,
                col("c", &test_schema)?,
                col("d", &test_schema)?,
                col("e", &test_schema)?,
                col("f", &test_schema)?,
                floor_a,
                a_plus_b,
            ];

            for n_req in 0..=exprs.len() {
                for exprs in exprs.iter().combinations(n_req) {
                    let exprs = exprs.into_iter().cloned().collect::<Vec<_>>();
                    let (ordering, indices) =
                        eq_properties.find_longest_permutation(&exprs);
                    // Make sure that find_longest_permutation return values are consistent
                    let ordering2 = indices
                        .iter()
                        .zip(ordering.iter())
                        .map(|(&idx, sort_expr)| PhysicalSortExpr {
                            expr: exprs[idx].clone(),
                            options: sort_expr.options,
                        })
                        .collect::<Vec<_>>();
                    assert_eq!(
                        ordering, ordering2,
                        "indices and lexicographical ordering do not match"
                    );

                    let err_msg = format!(
                        "Error in test case ordering:{:?}, eq_properties.oeq_class: {:?}, eq_properties.eq_group: {:?}, eq_properties.constants: {:?}",
                        ordering, eq_properties.oeq_class, eq_properties.eq_group, eq_properties.constants
                    );
                    assert_eq!(ordering.len(), indices.len(), "{}", err_msg);
                    // Since ordered section satisfies schema, we expect
                    // that result will be same after sort (e.g sort was unnecessary).
                    assert!(
                        is_table_same_after_sort(
                            ordering.clone(),
                            table_data_with_properties.clone(),
                        )?,
                        "{}",
                        err_msg
                    );
                }
            }
        }

        Ok(())
    }

    #[test]
    fn test_update_ordering() -> Result<()> {
        let schema = Schema::new(vec![
            Field::new("a", DataType::Int32, true),
            Field::new("b", DataType::Int32, true),
            Field::new("c", DataType::Int32, true),
            Field::new("d", DataType::Int32, true),
        ]);

        let mut eq_properties = EquivalenceProperties::new(Arc::new(schema.clone()));
        let col_a = &col("a", &schema)?;
        let col_b = &col("b", &schema)?;
        let col_c = &col("c", &schema)?;
        let col_d = &col("d", &schema)?;
        let option_asc = SortOptions {
            descending: false,
            nulls_first: false,
        };
        // b=a (e.g they are aliases)
        eq_properties.add_equal_conditions(col_b, col_a);
        // [b ASC], [d ASC]
        eq_properties.add_new_orderings(vec![
            vec![PhysicalSortExpr {
                expr: col_b.clone(),
                options: option_asc,
            }],
            vec![PhysicalSortExpr {
                expr: col_d.clone(),
                options: option_asc,
            }],
        ]);

        let test_cases = vec![
            // d + b
            (
                Arc::new(BinaryExpr::new(
                    col_d.clone(),
                    Operator::Plus,
                    col_b.clone(),
                )) as Arc<dyn PhysicalExpr>,
                SortProperties::Ordered(option_asc),
            ),
            // b
            (col_b.clone(), SortProperties::Ordered(option_asc)),
            // a
            (col_a.clone(), SortProperties::Ordered(option_asc)),
            // a + c
            (
                Arc::new(BinaryExpr::new(
                    col_a.clone(),
                    Operator::Plus,
                    col_c.clone(),
                )),
                SortProperties::Unordered,
            ),
        ];
        for (expr, expected) in test_cases {
            let leading_orderings = eq_properties
                .oeq_class()
                .iter()
                .flat_map(|ordering| ordering.first().cloned())
                .collect::<Vec<_>>();
            let expr_ordering = eq_properties.get_expr_ordering(expr.clone());
            let err_msg = format!(
                "expr:{:?}, expected: {:?}, actual: {:?}, leading_orderings: {leading_orderings:?}",
                expr, expected, expr_ordering.state
            );
            assert_eq!(expr_ordering.state, expected, "{}", err_msg);
        }

        Ok(())
    }

    #[test]
    fn test_contains_any() {
        let lit_true = Arc::new(Literal::new(ScalarValue::Boolean(Some(true))))
            as Arc<dyn PhysicalExpr>;
        let lit_false = Arc::new(Literal::new(ScalarValue::Boolean(Some(false))))
            as Arc<dyn PhysicalExpr>;
        let lit2 =
            Arc::new(Literal::new(ScalarValue::Int32(Some(2)))) as Arc<dyn PhysicalExpr>;
        let lit1 =
            Arc::new(Literal::new(ScalarValue::Int32(Some(1)))) as Arc<dyn PhysicalExpr>;
        let col_b_expr = Arc::new(Column::new("b", 1)) as Arc<dyn PhysicalExpr>;

        let cls1 = EquivalenceClass::new(vec![lit_true.clone(), lit_false.clone()]);
        let cls2 = EquivalenceClass::new(vec![lit_true.clone(), col_b_expr.clone()]);
        let cls3 = EquivalenceClass::new(vec![lit2.clone(), lit1.clone()]);

        // lit_true is common
        assert!(cls1.contains_any(&cls2));
        // there is no common entry
        assert!(!cls1.contains_any(&cls3));
        assert!(!cls2.contains_any(&cls3));
    }

    #[test]
    fn test_get_indices_of_matching_sort_exprs_with_order_eq() -> Result<()> {
        let sort_options = SortOptions::default();
        let sort_options_not = SortOptions::default().not();

        let schema = Schema::new(vec![
            Field::new("a", DataType::Int32, true),
            Field::new("b", DataType::Int32, true),
        ]);
        let col_a = &col("a", &schema)?;
        let col_b = &col("b", &schema)?;
        let required_columns = [col_b.clone(), col_a.clone()];
        let mut eq_properties = EquivalenceProperties::new(Arc::new(schema));
        eq_properties.add_new_orderings([vec![
            PhysicalSortExpr {
                expr: Arc::new(Column::new("b", 1)),
                options: sort_options_not,
            },
            PhysicalSortExpr {
                expr: Arc::new(Column::new("a", 0)),
                options: sort_options,
            },
        ]]);
        let (result, idxs) = eq_properties.find_longest_permutation(&required_columns);
        assert_eq!(idxs, vec![0, 1]);
        assert_eq!(
            result,
            vec![
                PhysicalSortExpr {
                    expr: col_b.clone(),
                    options: sort_options_not
                },
                PhysicalSortExpr {
                    expr: col_a.clone(),
                    options: sort_options
                }
            ]
        );

        let schema = Schema::new(vec![
            Field::new("a", DataType::Int32, true),
            Field::new("b", DataType::Int32, true),
            Field::new("c", DataType::Int32, true),
        ]);
        let col_a = &col("a", &schema)?;
        let col_b = &col("b", &schema)?;
        let required_columns = [col_b.clone(), col_a.clone()];
        let mut eq_properties = EquivalenceProperties::new(Arc::new(schema));
        eq_properties.add_new_orderings([
            vec![PhysicalSortExpr {
                expr: Arc::new(Column::new("c", 2)),
                options: sort_options,
            }],
            vec![
                PhysicalSortExpr {
                    expr: Arc::new(Column::new("b", 1)),
                    options: sort_options_not,
                },
                PhysicalSortExpr {
                    expr: Arc::new(Column::new("a", 0)),
                    options: sort_options,
                },
            ],
        ]);
        let (result, idxs) = eq_properties.find_longest_permutation(&required_columns);
        assert_eq!(idxs, vec![0, 1]);
        assert_eq!(
            result,
            vec![
                PhysicalSortExpr {
                    expr: col_b.clone(),
                    options: sort_options_not
                },
                PhysicalSortExpr {
                    expr: col_a.clone(),
                    options: sort_options
                }
            ]
        );

        let required_columns = [
            Arc::new(Column::new("b", 1)) as _,
            Arc::new(Column::new("a", 0)) as _,
        ];
        let schema = Schema::new(vec![
            Field::new("a", DataType::Int32, true),
            Field::new("b", DataType::Int32, true),
            Field::new("c", DataType::Int32, true),
        ]);
        let mut eq_properties = EquivalenceProperties::new(Arc::new(schema));

        // not satisfied orders
        eq_properties.add_new_orderings([vec![
            PhysicalSortExpr {
                expr: Arc::new(Column::new("b", 1)),
                options: sort_options_not,
            },
            PhysicalSortExpr {
                expr: Arc::new(Column::new("c", 2)),
                options: sort_options,
            },
            PhysicalSortExpr {
                expr: Arc::new(Column::new("a", 0)),
                options: sort_options,
            },
        ]]);
        let (_, idxs) = eq_properties.find_longest_permutation(&required_columns);
        assert_eq!(idxs, vec![0]);

        Ok(())
    }

    #[test]
    fn test_normalize_ordering_equivalence_classes() -> Result<()> {
        let sort_options = SortOptions::default();

        let schema = Schema::new(vec![
            Field::new("a", DataType::Int32, true),
            Field::new("b", DataType::Int32, true),
            Field::new("c", DataType::Int32, true),
        ]);
        let col_a_expr = col("a", &schema)?;
        let col_b_expr = col("b", &schema)?;
        let col_c_expr = col("c", &schema)?;
        let mut eq_properties = EquivalenceProperties::new(Arc::new(schema.clone()));

        eq_properties.add_equal_conditions(&col_a_expr, &col_c_expr);
        let others = vec![
            vec![PhysicalSortExpr {
                expr: col_b_expr.clone(),
                options: sort_options,
            }],
            vec![PhysicalSortExpr {
                expr: col_c_expr.clone(),
                options: sort_options,
            }],
        ];
        eq_properties.add_new_orderings(others);

        let mut expected_eqs = EquivalenceProperties::new(Arc::new(schema));
        expected_eqs.add_new_orderings([
            vec![PhysicalSortExpr {
                expr: col_b_expr.clone(),
                options: sort_options,
            }],
            vec![PhysicalSortExpr {
                expr: col_c_expr.clone(),
                options: sort_options,
            }],
        ]);

        let oeq_class = eq_properties.oeq_class().clone();
        let expected = expected_eqs.oeq_class();
        assert!(oeq_class.eq(expected));

        Ok(())
    }

    #[test]
    fn project_orderings() -> Result<()> {
        let schema = Arc::new(Schema::new(vec![
            Field::new("a", DataType::Int32, true),
            Field::new("b", DataType::Int32, true),
            Field::new("c", DataType::Int32, true),
            Field::new("d", DataType::Int32, true),
            Field::new("e", DataType::Int32, true),
            Field::new("ts", DataType::Timestamp(TimeUnit::Nanosecond, None), true),
        ]));
        let col_a = &col("a", &schema)?;
        let col_b = &col("b", &schema)?;
        let col_c = &col("c", &schema)?;
        let col_d = &col("d", &schema)?;
        let col_e = &col("e", &schema)?;
        let col_ts = &col("ts", &schema)?;
        let interval = Arc::new(Literal::new(ScalarValue::IntervalDayTime(Some(2))))
            as Arc<dyn PhysicalExpr>;
        let date_bin_func = &create_physical_expr(
            &BuiltinScalarFunction::DateBin,
            &[interval, col_ts.clone()],
            &schema,
            &ExecutionProps::default(),
        )?;
        let a_plus_b = Arc::new(BinaryExpr::new(
            col_a.clone(),
            Operator::Plus,
            col_b.clone(),
        )) as Arc<dyn PhysicalExpr>;
        let b_plus_d = Arc::new(BinaryExpr::new(
            col_b.clone(),
            Operator::Plus,
            col_d.clone(),
        )) as Arc<dyn PhysicalExpr>;
        let b_plus_e = Arc::new(BinaryExpr::new(
            col_b.clone(),
            Operator::Plus,
            col_e.clone(),
        )) as Arc<dyn PhysicalExpr>;
        let c_plus_d = Arc::new(BinaryExpr::new(
            col_c.clone(),
            Operator::Plus,
            col_d.clone(),
        )) as Arc<dyn PhysicalExpr>;

        let option_asc = SortOptions {
            descending: false,
            nulls_first: false,
        };
        let option_desc = SortOptions {
            descending: true,
            nulls_first: true,
        };

        let test_cases = vec![
            // ---------- TEST CASE 1 ------------
            (
                // orderings
                vec![
                    // [b ASC]
                    vec![(col_b, option_asc)],
                ],
                // projection exprs
                vec![(col_b, "b_new".to_string()), (col_a, "a_new".to_string())],
                // expected
                vec![
                    // [b_new ASC]
                    vec![("b_new", option_asc)],
                ],
            ),
            // ---------- TEST CASE 2 ------------
            (
                // orderings
                vec![
                    // empty ordering
                ],
                // projection exprs
                vec![(col_c, "c_new".to_string()), (col_b, "b_new".to_string())],
                // expected
                vec![
                    // no ordering at the output
                ],
            ),
            // ---------- TEST CASE 3 ------------
            (
                // orderings
                vec![
                    // [ts ASC]
                    vec![(col_ts, option_asc)],
                ],
                // projection exprs
                vec![
                    (col_b, "b_new".to_string()),
                    (col_a, "a_new".to_string()),
                    (col_ts, "ts_new".to_string()),
                    (date_bin_func, "date_bin_res".to_string()),
                ],
                // expected
                vec![
                    // [date_bin_res ASC]
                    vec![("date_bin_res", option_asc)],
                    // [ts_new ASC]
                    vec![("ts_new", option_asc)],
                ],
            ),
            // ---------- TEST CASE 4 ------------
            (
                // orderings
                vec![
                    // [a ASC, ts ASC]
                    vec![(col_a, option_asc), (col_ts, option_asc)],
                    // [b ASC, ts ASC]
                    vec![(col_b, option_asc), (col_ts, option_asc)],
                ],
                // projection exprs
                vec![
                    (col_b, "b_new".to_string()),
                    (col_a, "a_new".to_string()),
                    (col_ts, "ts_new".to_string()),
                    (date_bin_func, "date_bin_res".to_string()),
                ],
                // expected
                vec![
                    // [a_new ASC, ts_new ASC]
                    vec![("a_new", option_asc), ("ts_new", option_asc)],
                    // [a_new ASC, date_bin_res ASC]
                    vec![("a_new", option_asc), ("date_bin_res", option_asc)],
                    // [b_new ASC, ts_new ASC]
                    vec![("b_new", option_asc), ("ts_new", option_asc)],
                    // [b_new ASC, date_bin_res ASC]
                    vec![("b_new", option_asc), ("date_bin_res", option_asc)],
                ],
            ),
            // ---------- TEST CASE 5 ------------
            (
                // orderings
                vec![
                    // [a + b ASC]
                    vec![(&a_plus_b, option_asc)],
                ],
                // projection exprs
                vec![
                    (col_b, "b_new".to_string()),
                    (col_a, "a_new".to_string()),
                    (&a_plus_b, "a+b".to_string()),
                ],
                // expected
                vec![
                    // [a + b ASC]
                    vec![("a+b", option_asc)],
                ],
            ),
            // ---------- TEST CASE 6 ------------
            (
                // orderings
                vec![
                    // [a + b ASC, c ASC]
                    vec![(&a_plus_b, option_asc), (&col_c, option_asc)],
                ],
                // projection exprs
                vec![
                    (col_b, "b_new".to_string()),
                    (col_a, "a_new".to_string()),
                    (col_c, "c_new".to_string()),
                    (&a_plus_b, "a+b".to_string()),
                ],
                // expected
                vec![
                    // [a + b ASC, c_new ASC]
                    vec![("a+b", option_asc), ("c_new", option_asc)],
                ],
            ),
            // ------- TEST CASE 7 ----------
            (
                vec![
                    // [a ASC, b ASC, c ASC]
                    vec![(col_a, option_asc), (col_b, option_asc)],
                    // [a ASC, d ASC]
                    vec![(col_a, option_asc), (col_d, option_asc)],
                ],
                // b as b_new, a as a_new, d as d_new b+d
                vec![
                    (col_b, "b_new".to_string()),
                    (col_a, "a_new".to_string()),
                    (col_d, "d_new".to_string()),
                    (&b_plus_d, "b+d".to_string()),
                ],
                // expected
                vec![
                    // [a_new ASC, b_new ASC]
                    vec![("a_new", option_asc), ("b_new", option_asc)],
                    // [a_new ASC, d_new ASC]
                    vec![("a_new", option_asc), ("d_new", option_asc)],
                    // [a_new ASC, b+d ASC]
                    vec![("a_new", option_asc), ("b+d", option_asc)],
                ],
            ),
            // ------- TEST CASE 8 ----------
            (
                // orderings
                vec![
                    // [b+d ASC]
                    vec![(&b_plus_d, option_asc)],
                ],
                // proj exprs
                vec![
                    (col_b, "b_new".to_string()),
                    (col_a, "a_new".to_string()),
                    (col_d, "d_new".to_string()),
                    (&b_plus_d, "b+d".to_string()),
                ],
                // expected
                vec![
                    // [b+d ASC]
                    vec![("b+d", option_asc)],
                ],
            ),
            // ------- TEST CASE 9 ----------
            (
                // orderings
                vec![
                    // [a ASC, d ASC, b ASC]
                    vec![
                        (col_a, option_asc),
                        (col_d, option_asc),
                        (col_b, option_asc),
                    ],
                    // [c ASC]
                    vec![(col_c, option_asc)],
                ],
                // proj exprs
                vec![
                    (col_b, "b_new".to_string()),
                    (col_a, "a_new".to_string()),
                    (col_d, "d_new".to_string()),
                    (col_c, "c_new".to_string()),
                ],
                // expected
                vec![
                    // [a_new ASC, d_new ASC, b_new ASC]
                    vec![
                        ("a_new", option_asc),
                        ("d_new", option_asc),
                        ("b_new", option_asc),
                    ],
                    // [c_new ASC],
                    vec![("c_new", option_asc)],
                ],
            ),
            // ------- TEST CASE 10 ----------
            (
                vec![
                    // [a ASC, b ASC, c ASC]
                    vec![
                        (col_a, option_asc),
                        (col_b, option_asc),
                        (col_c, option_asc),
                    ],
                    // [a ASC, d ASC]
                    vec![(col_a, option_asc), (col_d, option_asc)],
                ],
                // proj exprs
                vec![
                    (col_b, "b_new".to_string()),
                    (col_a, "a_new".to_string()),
                    (col_c, "c_new".to_string()),
                    (&c_plus_d, "c+d".to_string()),
                ],
                // expected
                vec![
                    // [a_new ASC, b_new ASC, c_new ASC]
                    vec![
                        ("a_new", option_asc),
                        ("b_new", option_asc),
                        ("c_new", option_asc),
                    ],
                    // [a_new ASC, b_new ASC, c+d ASC]
                    vec![
                        ("a_new", option_asc),
                        ("b_new", option_asc),
                        ("c+d", option_asc),
                    ],
                ],
            ),
            // ------- TEST CASE 11 ----------
            (
                // orderings
                vec![
                    // [a ASC, b ASC]
                    vec![(col_a, option_asc), (col_b, option_asc)],
                    // [a ASC, d ASC]
                    vec![(col_a, option_asc), (col_d, option_asc)],
                ],
                // proj exprs
                vec![
                    (col_b, "b_new".to_string()),
                    (col_a, "a_new".to_string()),
                    (&b_plus_d, "b+d".to_string()),
                ],
                // expected
                vec![
                    // [a_new ASC, b_new ASC]
                    vec![("a_new", option_asc), ("b_new", option_asc)],
                    // [a_new ASC, b + d ASC]
                    vec![("a_new", option_asc), ("b+d", option_asc)],
                ],
            ),
            // ------- TEST CASE 12 ----------
            (
                // orderings
                vec![
                    // [a ASC, b ASC, c ASC]
                    vec![
                        (col_a, option_asc),
                        (col_b, option_asc),
                        (col_c, option_asc),
                    ],
                ],
                // proj exprs
                vec![(col_c, "c_new".to_string()), (col_a, "a_new".to_string())],
                // expected
                vec![
                    // [a_new ASC]
                    vec![("a_new", option_asc)],
                ],
            ),
            // ------- TEST CASE 13 ----------
            (
                // orderings
                vec![
                    // [a ASC, b ASC, c ASC]
                    vec![
                        (col_a, option_asc),
                        (col_b, option_asc),
                        (col_c, option_asc),
                    ],
                    // [a ASC, a + b ASC, c ASC]
                    vec![
                        (col_a, option_asc),
                        (&a_plus_b, option_asc),
                        (col_c, option_asc),
                    ],
                ],
                // proj exprs
                vec![
                    (col_c, "c_new".to_string()),
                    (col_b, "b_new".to_string()),
                    (col_a, "a_new".to_string()),
                    (&a_plus_b, "a+b".to_string()),
                ],
                // expected
                vec![
                    // [a_new ASC, b_new ASC, c_new ASC]
                    vec![
                        ("a_new", option_asc),
                        ("b_new", option_asc),
                        ("c_new", option_asc),
                    ],
                    // [a_new ASC, a+b ASC, c_new ASC]
                    vec![
                        ("a_new", option_asc),
                        ("a+b", option_asc),
                        ("c_new", option_asc),
                    ],
                ],
            ),
            // ------- TEST CASE 14 ----------
            (
                // orderings
                vec![
                    // [a ASC, b ASC]
                    vec![(col_a, option_asc), (col_b, option_asc)],
                    // [c ASC, b ASC]
                    vec![(col_c, option_asc), (col_b, option_asc)],
                    // [d ASC, e ASC]
                    vec![(col_d, option_asc), (col_e, option_asc)],
                ],
                // proj exprs
                vec![
                    (col_c, "c_new".to_string()),
                    (col_d, "d_new".to_string()),
                    (col_a, "a_new".to_string()),
                    (&b_plus_e, "b+e".to_string()),
                ],
                // expected
                vec![
                    // [a_new ASC, d_new ASC, b+e ASC]
                    vec![
                        ("a_new", option_asc),
                        ("d_new", option_asc),
                        ("b+e", option_asc),
                    ],
                    // [d_new ASC, a_new ASC, b+e ASC]
                    vec![
                        ("d_new", option_asc),
                        ("a_new", option_asc),
                        ("b+e", option_asc),
                    ],
                    // [c_new ASC, d_new ASC, b+e ASC]
                    vec![
                        ("c_new", option_asc),
                        ("d_new", option_asc),
                        ("b+e", option_asc),
                    ],
                    // [d_new ASC, c_new ASC, b+e ASC]
                    vec![
                        ("d_new", option_asc),
                        ("c_new", option_asc),
                        ("b+e", option_asc),
                    ],
                ],
            ),
            // ------- TEST CASE 15 ----------
            (
                // orderings
                vec![
                    // [a ASC, c ASC, b ASC]
                    vec![
                        (col_a, option_asc),
                        (col_c, option_asc),
                        (&col_b, option_asc),
                    ],
                ],
                // proj exprs
                vec![
                    (col_c, "c_new".to_string()),
                    (col_a, "a_new".to_string()),
                    (&a_plus_b, "a+b".to_string()),
                ],
                // expected
                vec![
                    // [a_new ASC, d_new ASC, b+e ASC]
                    vec![
                        ("a_new", option_asc),
                        ("c_new", option_asc),
                        ("a+b", option_asc),
                    ],
                ],
            ),
            // ------- TEST CASE 16 ----------
            (
                // orderings
                vec![
                    // [a ASC, b ASC]
                    vec![(col_a, option_asc), (col_b, option_asc)],
                    // [c ASC, b DESC]
                    vec![(col_c, option_asc), (col_b, option_desc)],
                    // [e ASC]
                    vec![(col_e, option_asc)],
                ],
                // proj exprs
                vec![
                    (col_c, "c_new".to_string()),
                    (col_a, "a_new".to_string()),
                    (col_b, "b_new".to_string()),
                    (&b_plus_e, "b+e".to_string()),
                ],
                // expected
                vec![
                    // [a_new ASC, b_new ASC]
                    vec![("a_new", option_asc), ("b_new", option_asc)],
                    // [a_new ASC, b_new ASC]
                    vec![("a_new", option_asc), ("b+e", option_asc)],
                    // [c_new ASC, b_new DESC]
                    vec![("c_new", option_asc), ("b_new", option_desc)],
                ],
            ),
        ];

        for (idx, (orderings, proj_exprs, expected)) in test_cases.into_iter().enumerate()
        {
            let mut eq_properties = EquivalenceProperties::new(schema.clone());

            let orderings = convert_to_orderings(&orderings);
            eq_properties.add_new_orderings(orderings);

            let proj_exprs = proj_exprs
                .into_iter()
                .map(|(expr, name)| (expr.clone(), name))
                .collect::<Vec<_>>();
            let projection_mapping = ProjectionMapping::try_new(&proj_exprs, &schema)?;
            let output_schema = output_schema(&projection_mapping, &schema)?;

            let expected = expected
                .into_iter()
                .map(|ordering| {
                    ordering
                        .into_iter()
                        .map(|(name, options)| {
                            (col(name, &output_schema).unwrap(), options)
                        })
                        .collect::<Vec<_>>()
                })
                .collect::<Vec<_>>();
            let expected = convert_to_orderings_owned(&expected);

            let projected_eq = eq_properties.project(&projection_mapping, output_schema);
            let orderings = projected_eq.oeq_class();

            let err_msg = format!(
                "test_idx: {:?}, actual: {:?}, expected: {:?}, projection_mapping: {:?}",
                idx, orderings.orderings, expected, projection_mapping
            );

            assert_eq!(orderings.len(), expected.len(), "{}", err_msg);
            for expected_ordering in &expected {
                assert!(orderings.contains(expected_ordering), "{}", err_msg)
            }
        }

        Ok(())
    }

    #[test]
    fn project_orderings2() -> Result<()> {
        let schema = Arc::new(Schema::new(vec![
            Field::new("a", DataType::Int32, true),
            Field::new("b", DataType::Int32, true),
            Field::new("c", DataType::Int32, true),
            Field::new("d", DataType::Int32, true),
            Field::new("ts", DataType::Timestamp(TimeUnit::Nanosecond, None), true),
        ]));
        let col_a = &col("a", &schema)?;
        let col_b = &col("b", &schema)?;
        let col_c = &col("c", &schema)?;
        let col_ts = &col("ts", &schema)?;
        let a_plus_b = Arc::new(BinaryExpr::new(
            col_a.clone(),
            Operator::Plus,
            col_b.clone(),
        )) as Arc<dyn PhysicalExpr>;
        let interval = Arc::new(Literal::new(ScalarValue::IntervalDayTime(Some(2))))
            as Arc<dyn PhysicalExpr>;
        let date_bin_ts = &create_physical_expr(
            &BuiltinScalarFunction::DateBin,
            &[interval, col_ts.clone()],
            &schema,
            &ExecutionProps::default(),
        )?;

        let round_c = &create_physical_expr(
            &BuiltinScalarFunction::Round,
            &[col_c.clone()],
            &schema,
            &ExecutionProps::default(),
        )?;

        let option_asc = SortOptions {
            descending: false,
            nulls_first: false,
        };

        let proj_exprs = vec![
            (col_b, "b_new".to_string()),
            (col_a, "a_new".to_string()),
            (col_c, "c_new".to_string()),
            (date_bin_ts, "date_bin_res".to_string()),
            (round_c, "round_c_res".to_string()),
        ];
        let proj_exprs = proj_exprs
            .into_iter()
            .map(|(expr, name)| (expr.clone(), name))
            .collect::<Vec<_>>();
        let projection_mapping = ProjectionMapping::try_new(&proj_exprs, &schema)?;
        let output_schema = output_schema(&projection_mapping, &schema)?;

        let col_a_new = &col("a_new", &output_schema)?;
        let col_b_new = &col("b_new", &output_schema)?;
        let col_c_new = &col("c_new", &output_schema)?;
        let col_date_bin_res = &col("date_bin_res", &output_schema)?;
        let col_round_c_res = &col("round_c_res", &output_schema)?;
        let a_new_plus_b_new = Arc::new(BinaryExpr::new(
            col_a_new.clone(),
            Operator::Plus,
            col_b_new.clone(),
        )) as Arc<dyn PhysicalExpr>;

        let test_cases = vec![
            // ---------- TEST CASE 1 ------------
            (
                // orderings
                vec![
                    // [a ASC]
                    vec![(col_a, option_asc)],
                ],
                // expected
                vec![
                    // [b_new ASC]
                    vec![(col_a_new, option_asc)],
                ],
            ),
            // ---------- TEST CASE 2 ------------
            (
                // orderings
                vec![
                    // [a+b ASC]
                    vec![(&a_plus_b, option_asc)],
                ],
                // expected
                vec![
                    // [b_new ASC]
                    vec![(&a_new_plus_b_new, option_asc)],
                ],
            ),
            // ---------- TEST CASE 3 ------------
            (
                // orderings
                vec![
                    // [a ASC, ts ASC]
                    vec![(col_a, option_asc), (col_ts, option_asc)],
                ],
                // expected
                vec![
                    // [a_new ASC, date_bin_res ASC]
                    vec![(col_a_new, option_asc), (col_date_bin_res, option_asc)],
                ],
            ),
            // ---------- TEST CASE 4 ------------
            (
                // orderings
                vec![
                    // [a ASC, ts ASC, b ASC]
                    vec![
                        (col_a, option_asc),
                        (col_ts, option_asc),
                        (col_b, option_asc),
                    ],
                ],
                // expected
                vec![
                    // [a_new ASC, date_bin_res ASC]
                    // Please note that result is not [a_new ASC, date_bin_res ASC, b_new ASC]
                    // because, datebin_res may not be 1-1 function. Hence without introducing ts
                    // dependency we cannot guarantee any ordering after date_bin_res column.
                    vec![(col_a_new, option_asc), (col_date_bin_res, option_asc)],
                ],
            ),
            // ---------- TEST CASE 5 ------------
            (
                // orderings
                vec![
                    // [a ASC, c ASC]
                    vec![(col_a, option_asc), (col_c, option_asc)],
                ],
                // expected
                vec![
                    // [a_new ASC, round_c_res ASC, c_new ASC]
                    vec![(col_a_new, option_asc), (col_round_c_res, option_asc)],
                    // [a_new ASC, c_new ASC]
                    vec![(col_a_new, option_asc), (col_c_new, option_asc)],
                ],
            ),
            // ---------- TEST CASE 6 ------------
            (
                // orderings
                vec![
                    // [c ASC, b ASC]
                    vec![(col_c, option_asc), (col_b, option_asc)],
                ],
                // expected
                vec![
                    // [round_c_res ASC]
                    vec![(col_round_c_res, option_asc)],
                    // [c_new ASC, b_new ASC]
                    vec![(col_c_new, option_asc), (col_b_new, option_asc)],
                ],
            ),
            // ---------- TEST CASE 7 ------------
            (
                // orderings
                vec![
                    // [a+b ASC, c ASC]
                    vec![(&a_plus_b, option_asc), (col_c, option_asc)],
                ],
                // expected
                vec![
                    // [a+b ASC, round(c) ASC, c_new ASC]
                    vec![
                        (&a_new_plus_b_new, option_asc),
                        (&col_round_c_res, option_asc),
                    ],
                    // [a+b ASC, c_new ASC]
                    vec![(&a_new_plus_b_new, option_asc), (col_c_new, option_asc)],
                ],
            ),
        ];

        for (idx, (orderings, expected)) in test_cases.iter().enumerate() {
            let mut eq_properties = EquivalenceProperties::new(schema.clone());

            let orderings = convert_to_orderings(orderings);
            eq_properties.add_new_orderings(orderings);

            let expected = convert_to_orderings(expected);

            let projected_eq =
                eq_properties.project(&projection_mapping, output_schema.clone());
            let orderings = projected_eq.oeq_class();

            let err_msg = format!(
                "test idx: {:?}, actual: {:?}, expected: {:?}, projection_mapping: {:?}",
                idx, orderings.orderings, expected, projection_mapping
            );

            assert_eq!(orderings.len(), expected.len(), "{}", err_msg);
            for expected_ordering in &expected {
                assert!(orderings.contains(expected_ordering), "{}", err_msg)
            }
        }
        Ok(())
    }

    #[test]
    fn project_orderings3() -> Result<()> {
        let schema = Arc::new(Schema::new(vec![
            Field::new("a", DataType::Int32, true),
            Field::new("b", DataType::Int32, true),
            Field::new("c", DataType::Int32, true),
            Field::new("d", DataType::Int32, true),
            Field::new("e", DataType::Int32, true),
            Field::new("f", DataType::Int32, true),
        ]));
        let col_a = &col("a", &schema)?;
        let col_b = &col("b", &schema)?;
        let col_c = &col("c", &schema)?;
        let col_d = &col("d", &schema)?;
        let col_e = &col("e", &schema)?;
        let col_f = &col("f", &schema)?;
        let a_plus_b = Arc::new(BinaryExpr::new(
            col_a.clone(),
            Operator::Plus,
            col_b.clone(),
        )) as Arc<dyn PhysicalExpr>;

        let option_asc = SortOptions {
            descending: false,
            nulls_first: false,
        };

        let proj_exprs = vec![
            (col_c, "c_new".to_string()),
            (col_d, "d_new".to_string()),
            (&a_plus_b, "a+b".to_string()),
        ];
        let proj_exprs = proj_exprs
            .into_iter()
            .map(|(expr, name)| (expr.clone(), name))
            .collect::<Vec<_>>();
        let projection_mapping = ProjectionMapping::try_new(&proj_exprs, &schema)?;
        let output_schema = output_schema(&projection_mapping, &schema)?;

        let col_a_plus_b_new = &col("a+b", &output_schema)?;
        let col_c_new = &col("c_new", &output_schema)?;
        let col_d_new = &col("d_new", &output_schema)?;

        let test_cases = vec![
            // ---------- TEST CASE 1 ------------
            (
                // orderings
                vec![
                    // [d ASC, b ASC]
                    vec![(col_d, option_asc), (col_b, option_asc)],
                    // [c ASC, a ASC]
                    vec![(col_c, option_asc), (col_a, option_asc)],
                ],
                // equal conditions
                vec![],
                // expected
                vec![
                    // [d_new ASC, c_new ASC, a+b ASC]
                    vec![
                        (col_d_new, option_asc),
                        (col_c_new, option_asc),
                        (col_a_plus_b_new, option_asc),
                    ],
                    // [c_new ASC, d_new ASC, a+b ASC]
                    vec![
                        (col_c_new, option_asc),
                        (col_d_new, option_asc),
                        (col_a_plus_b_new, option_asc),
                    ],
                ],
            ),
            // ---------- TEST CASE 2 ------------
            (
                // orderings
                vec![
                    // [d ASC, b ASC]
                    vec![(col_d, option_asc), (col_b, option_asc)],
                    // [c ASC, e ASC], Please note that a=e
                    vec![(col_c, option_asc), (col_e, option_asc)],
                ],
                // equal conditions
                vec![(col_e, col_a)],
                // expected
                vec![
                    // [d_new ASC, c_new ASC, a+b ASC]
                    vec![
                        (col_d_new, option_asc),
                        (col_c_new, option_asc),
                        (col_a_plus_b_new, option_asc),
                    ],
                    // [c_new ASC, d_new ASC, a+b ASC]
                    vec![
                        (col_c_new, option_asc),
                        (col_d_new, option_asc),
                        (col_a_plus_b_new, option_asc),
                    ],
                ],
            ),
            // ---------- TEST CASE 3 ------------
            (
                // orderings
                vec![
                    // [d ASC, b ASC]
                    vec![(col_d, option_asc), (col_b, option_asc)],
                    // [c ASC, e ASC], Please note that a=f
                    vec![(col_c, option_asc), (col_e, option_asc)],
                ],
                // equal conditions
                vec![(col_a, col_f)],
                // expected
                vec![
                    // [d_new ASC]
                    vec![(col_d_new, option_asc)],
                    // [c_new ASC]
                    vec![(col_c_new, option_asc)],
                ],
            ),
        ];
        for (orderings, equal_columns, expected) in test_cases {
            let mut eq_properties = EquivalenceProperties::new(schema.clone());
            for (lhs, rhs) in equal_columns {
                eq_properties.add_equal_conditions(lhs, rhs);
            }

            let orderings = convert_to_orderings(&orderings);
            eq_properties.add_new_orderings(orderings);

            let expected = convert_to_orderings(&expected);

            let projected_eq =
                eq_properties.project(&projection_mapping, output_schema.clone());
            let orderings = projected_eq.oeq_class();

            let err_msg = format!(
                "actual: {:?}, expected: {:?}, projection_mapping: {:?}",
                orderings.orderings, expected, projection_mapping
            );

            assert_eq!(orderings.len(), expected.len(), "{}", err_msg);
            for expected_ordering in &expected {
                assert!(orderings.contains(expected_ordering), "{}", err_msg)
            }
        }

        Ok(())
    }

    #[test]
    fn project_orderings_random() -> Result<()> {
        const N_RANDOM_SCHEMA: usize = 20;
        const N_ELEMENTS: usize = 125;
        const N_DISTINCT: usize = 5;

        for seed in 0..N_RANDOM_SCHEMA {
            // Create a random schema with random properties
            let (test_schema, eq_properties) = create_random_schema(seed as u64)?;
            // Generate a data that satisfies properties given
            let table_data_with_properties =
                generate_table_for_eq_properties(&eq_properties, N_ELEMENTS, N_DISTINCT)?;
            // Floor(a)
            let floor_a = create_physical_expr(
                &BuiltinScalarFunction::Floor,
                &[col("a", &test_schema)?],
                &test_schema,
                &ExecutionProps::default(),
            )?;
            // a + b
            let a_plus_b = Arc::new(BinaryExpr::new(
                col("a", &test_schema)?,
                Operator::Plus,
                col("b", &test_schema)?,
            )) as Arc<dyn PhysicalExpr>;
            let proj_exprs = vec![
                (col("a", &test_schema)?, "a_new"),
                (col("b", &test_schema)?, "b_new"),
                (col("c", &test_schema)?, "c_new"),
                (col("d", &test_schema)?, "d_new"),
                (col("e", &test_schema)?, "e_new"),
                (col("f", &test_schema)?, "f_new"),
                (floor_a, "floor(a)"),
                (a_plus_b, "a+b"),
            ];

            for n_req in 0..=proj_exprs.len() {
                for proj_exprs in proj_exprs.iter().combinations(n_req) {
                    let proj_exprs = proj_exprs
                        .into_iter()
                        .map(|(expr, name)| (expr.clone(), name.to_string()))
                        .collect::<Vec<_>>();
                    let (projected_batch, projected_eq) = apply_projection(
                        proj_exprs.clone(),
                        &table_data_with_properties,
                        &eq_properties,
                    )?;

                    // Make sure each ordering after projection is valid.
                    for ordering in projected_eq.oeq_class().iter() {
                        let err_msg = format!(
                            "Error in test case ordering:{:?}, eq_properties.oeq_class: {:?}, eq_properties.eq_group: {:?}, eq_properties.constants: {:?}, proj_exprs: {:?}",
                            ordering, eq_properties.oeq_class, eq_properties.eq_group, eq_properties.constants, proj_exprs
                        );
                        // Since ordered section satisfies schema, we expect
                        // that result will be same after sort (e.g sort was unnecessary).
                        assert!(
                            is_table_same_after_sort(
                                ordering.clone(),
                                projected_batch.clone(),
                            )?,
                            "{}",
                            err_msg
                        );
                    }
                }
            }
        }

        Ok(())
    }

    #[test]
    fn ordering_satisfy_after_projection_random() -> Result<()> {
        const N_RANDOM_SCHEMA: usize = 20;
        const N_ELEMENTS: usize = 125;
        const N_DISTINCT: usize = 5;
        const SORT_OPTIONS: SortOptions = SortOptions {
            descending: false,
            nulls_first: false,
        };

        for seed in 0..N_RANDOM_SCHEMA {
            // Create a random schema with random properties
            let (test_schema, eq_properties) = create_random_schema(seed as u64)?;
            // Generate a data that satisfies properties given
            let table_data_with_properties =
                generate_table_for_eq_properties(&eq_properties, N_ELEMENTS, N_DISTINCT)?;
            // Floor(a)
            let floor_a = create_physical_expr(
                &BuiltinScalarFunction::Floor,
                &[col("a", &test_schema)?],
                &test_schema,
                &ExecutionProps::default(),
            )?;
            // a + b
            let a_plus_b = Arc::new(BinaryExpr::new(
                col("a", &test_schema)?,
                Operator::Plus,
                col("b", &test_schema)?,
            )) as Arc<dyn PhysicalExpr>;
            let proj_exprs = vec![
                (col("a", &test_schema)?, "a_new"),
                (col("b", &test_schema)?, "b_new"),
                (col("c", &test_schema)?, "c_new"),
                (col("d", &test_schema)?, "d_new"),
                (col("e", &test_schema)?, "e_new"),
                (col("f", &test_schema)?, "f_new"),
                (floor_a, "floor(a)"),
                (a_plus_b, "a+b"),
            ];

            for n_req in 0..=proj_exprs.len() {
                for proj_exprs in proj_exprs.iter().combinations(n_req) {
                    let proj_exprs = proj_exprs
                        .into_iter()
                        .map(|(expr, name)| (expr.clone(), name.to_string()))
                        .collect::<Vec<_>>();
                    let (projected_batch, projected_eq) = apply_projection(
                        proj_exprs.clone(),
                        &table_data_with_properties,
                        &eq_properties,
                    )?;

                    let projection_mapping =
                        ProjectionMapping::try_new(&proj_exprs, &test_schema)?;

                    let projected_exprs = projection_mapping
                        .iter()
                        .map(|(_source, target)| target.clone())
                        .collect::<Vec<_>>();

                    for n_req in 0..=projected_exprs.len() {
                        for exprs in projected_exprs.iter().combinations(n_req) {
                            let requirement = exprs
                                .into_iter()
                                .map(|expr| PhysicalSortExpr {
                                    expr: expr.clone(),
                                    options: SORT_OPTIONS,
                                })
                                .collect::<Vec<_>>();
                            let expected = is_table_same_after_sort(
                                requirement.clone(),
                                projected_batch.clone(),
                            )?;
                            let err_msg = format!(
                                "Error in test case requirement:{:?}, expected: {:?}, eq_properties.oeq_class: {:?}, eq_properties.eq_group: {:?}, eq_properties.constants: {:?}, projected_eq.oeq_class: {:?}, projected_eq.eq_group: {:?}, projected_eq.constants: {:?}, projection_mapping: {:?}",
                                requirement, expected, eq_properties.oeq_class, eq_properties.eq_group, eq_properties.constants, projected_eq.oeq_class, projected_eq.eq_group, projected_eq.constants, projection_mapping
                            );
                            // Check whether ordering_satisfy API result and
                            // experimental result matches.
                            assert_eq!(
                                projected_eq.ordering_satisfy(&requirement),
                                expected,
                                "{}",
                                err_msg
                            );
                        }
                    }
                }
            }
        }

        Ok(())
    }

    #[test]
    fn test_expr_consists_of_constants() -> Result<()> {
        let schema = Arc::new(Schema::new(vec![
            Field::new("a", DataType::Int32, true),
            Field::new("b", DataType::Int32, true),
            Field::new("c", DataType::Int32, true),
            Field::new("d", DataType::Int32, true),
            Field::new("ts", DataType::Timestamp(TimeUnit::Nanosecond, None), true),
        ]));
        let col_a = col("a", &schema)?;
        let col_b = col("b", &schema)?;
        let col_d = col("d", &schema)?;
        let b_plus_d = Arc::new(BinaryExpr::new(
            col_b.clone(),
            Operator::Plus,
            col_d.clone(),
        )) as Arc<dyn PhysicalExpr>;

        let constants = vec![col_a.clone(), col_b.clone()];
        let expr = b_plus_d.clone();
        assert!(!is_constant_recurse(&constants, &expr));

        let constants = vec![col_a.clone(), col_b.clone(), col_d.clone()];
        let expr = b_plus_d.clone();
        assert!(is_constant_recurse(&constants, &expr));
        Ok(())
    }

    #[test]
    fn test_join_equivalence_properties() -> Result<()> {
        let schema = create_test_schema()?;
        let col_a = &col("a", &schema)?;
        let col_b = &col("b", &schema)?;
        let col_c = &col("c", &schema)?;
        let offset = schema.fields.len();
        let col_a2 = &add_offset_to_expr(col_a.clone(), offset);
        let col_b2 = &add_offset_to_expr(col_b.clone(), offset);
        let option_asc = SortOptions {
            descending: false,
            nulls_first: false,
        };
        let test_cases = vec![
            // ------- TEST CASE 1 --------
            // [a ASC], [b ASC]
            (
                // [a ASC], [b ASC]
                vec![vec![(col_a, option_asc)], vec![(col_b, option_asc)]],
                // [a ASC], [b ASC]
                vec![vec![(col_a, option_asc)], vec![(col_b, option_asc)]],
                // expected [a ASC, a2 ASC], [a ASC, b2 ASC], [b ASC, a2 ASC], [b ASC, b2 ASC]
                vec![
                    vec![(col_a, option_asc), (col_a2, option_asc)],
                    vec![(col_a, option_asc), (col_b2, option_asc)],
                    vec![(col_b, option_asc), (col_a2, option_asc)],
                    vec![(col_b, option_asc), (col_b2, option_asc)],
                ],
            ),
            // ------- TEST CASE 2 --------
            // [a ASC], [b ASC]
            (
                // [a ASC], [b ASC], [c ASC]
                vec![
                    vec![(col_a, option_asc)],
                    vec![(col_b, option_asc)],
                    vec![(col_c, option_asc)],
                ],
                // [a ASC], [b ASC]
                vec![vec![(col_a, option_asc)], vec![(col_b, option_asc)]],
                // expected [a ASC, a2 ASC], [a ASC, b2 ASC], [b ASC, a2 ASC], [b ASC, b2 ASC], [c ASC, a2 ASC], [c ASC, b2 ASC]
                vec![
                    vec![(col_a, option_asc), (col_a2, option_asc)],
                    vec![(col_a, option_asc), (col_b2, option_asc)],
                    vec![(col_b, option_asc), (col_a2, option_asc)],
                    vec![(col_b, option_asc), (col_b2, option_asc)],
                    vec![(col_c, option_asc), (col_a2, option_asc)],
                    vec![(col_c, option_asc), (col_b2, option_asc)],
                ],
            ),
        ];
        for (left_orderings, right_orderings, expected) in test_cases {
            let mut left_eq_properties = EquivalenceProperties::new(schema.clone());
            let mut right_eq_properties = EquivalenceProperties::new(schema.clone());
            let left_orderings = convert_to_orderings(&left_orderings);
            let right_orderings = convert_to_orderings(&right_orderings);
            let expected = convert_to_orderings(&expected);
            left_eq_properties.add_new_orderings(left_orderings);
            right_eq_properties.add_new_orderings(right_orderings);
            let join_eq = join_equivalence_properties(
                left_eq_properties,
                right_eq_properties,
                &JoinType::Inner,
                Arc::new(Schema::empty()),
                &[true, false],
                Some(JoinSide::Left),
                &[],
            );
            let orderings = &join_eq.oeq_class.orderings;
            let err_msg = format!("expected: {:?}, actual:{:?}", expected, orderings);
            assert_eq!(
                join_eq.oeq_class.orderings.len(),
                expected.len(),
                "{}",
                err_msg
            );
            for ordering in orderings {
                assert!(
                    expected.contains(ordering),
                    "{}, ordering: {:?}",
                    err_msg,
                    ordering
                );
            }
        }
        Ok(())
    }
}