array_range_query

A high-performance, generic implementation of segment trees and lazy segment trees in Rust for efficient range queries and range updates.

Includes helpers for:

• Range sum, min, max queries
• Range add, range assignment (replace), and lazy propagation for sum/min/max

Features

• Generic Segment Tree (SegTree): Supports any associative operation (monoid) with O(log n) point updates and range queries
• Generic Lazy Segment Tree (LazySegTree): Supports range updates and range queries in O(log n) time with lazy propagation
• Type-safe design: Uses the type system to ensure correctness and prevent misuse
• Comprehensive helper types: Pre-built implementations for common operations (sum, min, max)
• Zero-cost abstractions: Generic design with no runtime overhead
• Well-tested: Extensive test coverage including edge cases

Quick Start

Add this to your Cargo.toml:

[dependencies]
array_range_query = "0.1.0"

Basic Usage

Segment Tree for Range Sum Queries

use array_range_query::{SegTree, SegTreeSpec};

// Define a spec for sum operations
struct SumSpec;

impl SegTreeSpec for SumSpec {
    type T = i64;
    const ID: Self::T = 0; // Identity element for addition

    fn op(a: &Self::T, b: &Self::T) -> Self::T {
        a + b
    }
}

fn main() {
    let values = vec![1, 2, 3, 4, 5];
    let mut seg_tree = SegTree::<SumSpec>::from_vec(&values);

    // Query sum of range [1, 4) -> elements at indices 1, 2, 3
    assert_eq!(seg_tree.query(1, 4), 9); // 2 + 3 + 4

    // Update element at index 2 to 10
    seg_tree.update(2, 10);

    // Query again - sum should reflect the change
    assert_eq!(seg_tree.query(1, 4), 16); // 2 + 10 + 4
    assert_eq!(seg_tree.query(0, 5), 22); // 1 + 2 + 10 + 4 + 5
}

Using Helper Types

For common operations, you can use the provided helper types:

use array_range_query::{SegTreeSum, SegTreeMin, SegTreeMax};

fn main() {
    let values = vec![3, 1, 4, 1, 5, 9, 2, 6];

    // Range sum queries
    let mut sum_tree = SegTreeSum::<i32>::from_vec(&values);
    assert_eq!(sum_tree.query(2, 6), 19); // 4 + 1 + 5 + 9

    // Range minimum queries
    let mut min_tree = SegTreeMin::<i32>::from_vec(&values);
    assert_eq!(min_tree.query(2, 6), 1); // min(4, 1, 5, 9)

    // Range maximum queries
    let mut max_tree = SegTreeMax::<i32>::from_vec(&values);
    assert_eq!(max_tree.query(2, 6), 9); // max(4, 1, 5, 9)
}

Lazy Segment Tree for Range Updates

use array_range_query::{LazySegTree, LazySegTreeSpec};

// Define a spec for range add + range sum
struct RangeAddSum;

impl LazySegTreeSpec for RangeAddSum {
    type T = i64; // Data type (sum values)
    type U = i64; // Update type (add values)
    const ID: Self::T = 0;

    // Combine two data values
    fn op_on_data(d1: &Self::T, d2: &Self::T) -> Self::T {
        d1 + d2
    }

    // Compose two updates
    fn op_on_update(u1: &Self::U, u2: &Self::U) -> Self::U {
        u1 + u2
    }

    // Apply update to data (accounting for range size)
    fn op_update_on_data(u: &Self::U, d: &Self::T, size: usize) -> Self::T {
        d + (u * size as i64)
    }
}

fn main() {
    let values = vec![1, 2, 3, 4, 5];
    let mut lazy_tree = LazySegTree::<RangeAddSum>::from_vec(&values);

    // Initial sum of range [1, 4)
    assert_eq!(lazy_tree.query(1, 4), 9); // 2 + 3 + 4

    // Add 10 to all elements in range [1, 4)
    lazy_tree.update(1, 4, 10);

    // Query the updated range
    assert_eq!(lazy_tree.query(1, 4), 39); // (2+10) + (3+10) + (4+10)

    // Total sum should be updated too
    assert_eq!(lazy_tree.query(0, 5), 45); // 1 + 12 + 13 + 14 + 5
}

Using Lazy Segment Tree Helpers

use array_range_query::{LazySegTreeAddSum, LazySegTreeAddMin, LazySegTreeReplaceSum};

fn main() {
    // Range add + range sum
    let values = vec![1, 2, 3, 4, 5];
    let mut tree = LazySegTreeAddSum::<i64>::from_vec(&values);
    tree.update(1, 3, 5);
    assert_eq!(tree.query(0, 2), 8);  // 1 + (2+5)
    assert_eq!(tree.query(1, 4), 19); // 7 + 8 + 4 = 19

    // Range add + range min
    let values = vec![5, 2, 8, 1, 9, 3];
    let mut min_tree = LazySegTreeAddMin::<i32>::from_vec(&values);
    min_tree.update(1, 4, 2);
    assert_eq!(min_tree.query(0, 6), 3); // min(5, 4, 10, 3, 9, 3)

    // Range assignment (replace) + range sum
    let values = vec![1, 2, 3, 4, 5];
    let mut replace_tree = LazySegTreeReplaceSum::<i32>::from_vec(&values);
    replace_tree.update(1, 4, 10); // Replace [1, 4) with 10
    assert_eq!(replace_tree.query(0, 5), 1 + 10 + 10 + 10 + 5);
}

Advanced Usage

Custom Data Types

You can use segment trees with any type that implements the required traits:

use array_range_query::{SegTree, SegTreeSpec};

#[derive(Clone, PartialEq, Debug)]
struct Point {
    x: i32,
    y: i32,
}

struct PointMaxSpec;

impl SegTreeSpec for PointMaxSpec {
    type T = Point;
    const ID: Self::T = Point { x: i32::MIN, y: i32::MIN };

    fn op(a: &Self::T, b: &Self::T) -> Self::T {
        Point {
            x: a.x.max(b.x),
            y: a.y.max(b.y),
        }
    }
}

fn main() {
    let points = vec![
        Point { x: 1, y: 2 },
        Point { x: 3, y: 1 },
        Point { x: 2, y: 4 },
    ];

    let tree = SegTree::<PointMaxSpec>::from_vec(&points);
    let max_point = tree.query(0, 3);
    assert_eq!(max_point.x, 3);
    assert_eq!(max_point.y, 4);
}

API Reference

SegTree

• SegTree::new(size: usize) - Create empty tree with given size
• SegTree::from_vec(values: &[T]) - Build tree from initial values
• query(&self, left: usize, right: usize) -> T - Query range [left, right)
• update(&mut self, index: usize, value: T) - Update single element

LazySegTree

• LazySegTree::new(size: usize) - Create empty lazy tree
• LazySegTree::from_vec(values: &[T]) - Build from initial values
• query(&self, left: usize, right: usize) -> T - Query range [left, right)
• update(&mut self, left: usize, right: usize, value: U) - Update range [left, right)

Helper Types

Regular Segment Trees:

• SegTreeSum<T> - Range sum queries
• SegTreeMin<T> - Range minimum queries
• SegTreeMax<T> - Range maximum queries

Lazy Segment Trees:

• LazySegTreeAddSum<T> - Range add updates, range sum queries
• LazySegTreeAddMax<T> - Range add updates, range max queries
• LazySegTreeAddMin<T> - Range add updates, range min queries
• LazySegTreeReplaceSum<T> - Range assignment (replace) updates, range sum queries

Performance

All operations have the following time complexities:

• Construction: O(n) for from_vec, O(n) for new
• Point update (SegTree): O(log n)
• Range query: O(log n)
• Range update (LazySegTree): O(log n)

Space complexity: O(n)

Requirements

• Rust 2021 edition or later
• For numeric types: num-traits crate features
• For min/max operations: min_max_traits crate features

License

This project is licensed under the MIT License. See the LICENSE file for details.