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//! Algorithm to reduce combinatorial structures modulo isomorphism.
//!
//! This can typically be used to to test if two graphs are isomorphic.
//!
//! The algorithm manipulates its input as a black box by
//! the action of permutations
//! and by testing equallity with element of its orbit,
//! plus some user-defined functions
//! that help to break symmetries.
//!
//!```
//!use canonical_form::Canonize;
//!
//!// Simple Graph implementation as adjacency lists
//!#[derive(Ord, PartialOrd, PartialEq, Eq, Clone, Debug)]
//!struct Graph {
//! adj: Vec<Vec<usize>>,
//!}
//!
//!
//!impl Graph {
//! fn new(n: usize, edges: &[(usize, usize)]) -> Self {
//! let mut adj = vec![Vec::new(); n];
//! for &(u, v) in edges {
//! adj[u].push(v);
//! adj[v].push(u);
//! }
//! for list in &mut adj {
//! list.sort() // Necessary to make the derived `==` correct
//! }
//! Graph { adj }
//! }
//!}
//!
//!// The Canonize trait allows to use the canonial form algorithms
//!impl Canonize for Graph {
//! fn size(&self) -> usize {
//! self.adj.len()
//! }
//! fn apply_morphism(&self, perm: &[usize]) -> Self {
//! let mut adj = vec![Vec::new(); self.size()];
//! for (i, nbrs) in self.adj.iter().enumerate() {
//! adj[perm[i]] = nbrs.iter().map(|&u| perm[u]).collect();
//! adj[perm[i]].sort();
//! }
//! Graph { adj }
//! }
//! fn invariant_neighborhood(&self, u: usize) -> Vec<Vec<usize>> {
//! vec![self.adj[u].clone()]
//! }
//!}
//!
//! // Usage of library functions
//! // Two isomorphic graphs
//! let c5 = Graph::new(5, &[(0, 1), (1, 2), (2, 3), (3, 4), (4, 0)]);
//! let other_c5 = Graph::new(5, &[(0, 2), (2, 1), (1, 4), (4, 3), (3, 0)]);
//! assert_eq!(c5.canonical(), other_c5.canonical());
//!
//! // Non-isomorphic graphs
//! let p5 = Graph::new(5, &[(0, 1), (1, 2), (2, 3), (3, 4)]);
//! assert!(c5.canonical() != p5.canonical());
//!
//! // Recovering the permutation that gives the canonical form
//! let p = c5.morphism_to_canonical();
//! assert_eq!(c5.apply_morphism(&p), c5.canonical());
//!
//! // Enumerating automorphisms
//! assert_eq!(c5.canonical().automorphisms().count(), 10)
//!```
#![warn(
missing_docs,
missing_debug_implementations,
missing_copy_implementations,
trivial_casts,
trivial_numeric_casts,
unsafe_code,
unstable_features,
unused_import_braces,
unused_qualifications,
unused_labels,
unused_results
)]
mod refine;
use crate::refine::Partition;
use std::collections::btree_map::Entry::{Occupied, Vacant};
use std::collections::BTreeMap;
use std::rc::Rc;
pub mod example;
/// Objects that can be reduced modulo the actions of a permutation group.
///
/// An object that implement this trait has a set of elements assimilated to
/// {0,...,n-1} on which the group of permutations can act.
/// The purpose of the trait is to compute a normal form of
/// the object modulo the permutation of its elements.
pub trait Canonize
where
Self: Sized + Ord + Clone,
{
/// Return the number of vertices.
///
/// The elements of `self` are assimilated to the number of `0..self.size()`.
fn size(&self) -> usize;
/// Return the result of the action of a permuation `p` on the object.
///
/// The input `p` is guarenteed to be a permutation represented
/// as a slice of size `self.size()`
/// where `p[u]` is the image of `u` by the permutation.
///
/// The action of the permutation set on `Self` is assumed to be a group action.
/// ```
/// use canonical_form::Canonize;
/// use canonical_form::example::Graph;
///
/// let g = Graph::new(4, &[(1, 2), (2, 3)]);
///
/// let p = &[1, 2, 0, 3];
/// assert_eq!(g.apply_morphism(p), Graph::new(4, &[(2, 0), (0, 3)]));
///
/// let identity = &[0, 1, 2, 3];
/// assert_eq!(g.apply_morphism(identity), g);
///
/// let q = &[1, 0, 3, 2];
/// let p_circle_q = &[2, 1, 3, 0];
/// assert_eq!(
/// g.apply_morphism(q).apply_morphism(p),
/// g.apply_morphism(p_circle_q)
/// )
/// ```
fn apply_morphism(&self, p: &[usize]) -> Self;
/// Optionally returns a value for each node that is invariant by isomorphism.
///
/// If defined, the returned vector `c` must have size `self.len()`, where `c[u]`
/// is the value associated to the element `u`. It must satisfy the property that if
/// `c[u]` and `c[v]` are different then no automorphism of `self`
/// maps `u` to `v`.
fn invariant_coloring(&self) -> Option<Vec<u64>> {
None
}
/// Return lists of vertices that are invariant isomorphism.
///
/// This function helps the algorithm to be efficient.
/// The output `invar` is such that each `invar[i]` is a vector
/// of vertices `[v1, ..., vk]`
/// (so the `vi`s are elements of `0..self.size()`) such that
/// for every permutation `p`,
/// `self.apply_morphism(p).invariant_neighborhood(p[u])[i]`
/// is equal to `[p[v1], ..., p[vk]]` up to reordering.
///
/// The length of the output (the number of lists) has to be independent from `u`.
fn invariant_neighborhood(&self, _u: usize) -> Vec<Vec<usize>> {
Vec::new()
}
/// Computes a canonical form of a combinatorial object.
///
/// This is the main function provided by this trait.
/// A canonical form is a function that assigns to an object `g` (e.g. a graph)
/// another object of sane type `g.canonical()` that is isomorphic to `g`
/// with the property that `g1` and `g2` are isomorphic if and only if
/// `g1.canocial() == g2.canonical()`.
/// ```
/// use canonical_form::Canonize;
/// use canonical_form::example::Graph;
///
/// let p5 = Graph::new(5, &[(0, 1), (1, 2), (2, 3), (3, 4)]);
/// let other_p5 = Graph::new(5, &[(3, 4), (0, 4), (0, 2), (1, 2)]);
/// assert_eq!(p5.canonical(), other_p5.canonical());
///
/// let not_p5 = Graph::new(5, &[(0, 1), (1, 2), (2, 0), (3, 4)]);
/// assert!(p5.canonical() != not_p5.canonical());
/// ```
fn canonical(&self) -> Self {
self.canonical_typed(0)
}
/// The "typed" objects refers to the case where only
/// the action of permutations that are constant
/// on `0..sigma` are considered.
///
/// So `g.canonical_typed(sigma)` returns a normal form of `g`
/// modulo the permutations that stabilize the `sigma` first vertices.
/// ```
/// use canonical_form::Canonize;
/// use canonical_form::example::Graph;
///
/// // p5 with the edge (0, 1) at an end
/// let p5 = Graph::new(5, &[(0, 1), (1, 2), (2, 3), (3, 4)]);
/// // p5 with the edge (0, 1) is in the middle
/// let p5_bis = Graph::new(5, &[(3, 4), (4, 1), (1, 0), (0, 2)]);
/// // If we fix the vertices 0 and 1, these two (rooted) graphs are different
/// assert!(p5.canonical_typed(2) != p5_bis.canonical_typed(2));
///
/// let p5_ter = Graph::new(5, &[(0, 1), (1, 3), (3, 4), (4, 2)]);
/// assert_eq!(p5.canonical_typed(2), p5_ter.canonical_typed(2));
/// ```
fn canonical_typed(&self, sigma: usize) -> Self {
let partition = Partition::with_singletons(self.size(), sigma);
canonical_constraint(self, partition)
}
#[inline]
/// Return a permutation `p` such that `self.apply_morphism(&p) = self.canonical()`.
/// ```
/// use canonical_form::Canonize;
/// use canonical_form::example::Graph;
///
/// let g = Graph::new(6, &[(0, 1), (1, 2), (2, 3), (3, 4), (3, 5)]);
/// let p = g.morphism_to_canonical();
/// assert_eq!(g.apply_morphism(&p), g.canonical());
/// ```
fn morphism_to_canonical(&self) -> Vec<usize> {
self.morphism_to_canonical_typed(0)
}
/// Return a permutation `phi` such that
/// `g.apply_morphism(&phi) = canonical_typed(&g, sigma)`.
/// ```
/// use canonical_form::Canonize;
/// use canonical_form::example::Graph;
///
/// let g = Graph::new(5, &[(0, 1), (1, 2), (2, 3), (3, 4)]);
/// let p = g.morphism_to_canonical_typed(2);
/// assert_eq!(g.apply_morphism(&p), g.canonical_typed(2));
/// ```
fn morphism_to_canonical_typed(&self, sigma: usize) -> Vec<usize> {
assert!(sigma <= self.size());
let partition = Partition::with_singletons(self.size(), sigma);
morphism_to_canonical_constraint(self, partition)
}
/// Iterator on the automorphism group of `g`.
///
/// The input `g` must be in normal form.
/// ```
/// use canonical_form::Canonize;
/// use canonical_form::example::Graph;
///
/// let c6 = Graph::new(6, &[(0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 0)]).canonical();
///
/// let mut count = 0;
/// for p in c6.automorphisms() {
/// assert_eq!(c6.apply_morphism(&p), c6);
/// count += 1;
/// }
/// assert_eq!(count, 12);
/// ```
#[inline]
fn automorphisms(&self) -> AutomorphismIterator<Self> {
self.stabilizer(0)
}
/// Iterator on the automorphisms of `g`
/// that fix the `sigma` first vertices.
///
/// The input `g` must be in normal form computed with `canonical_typed`.
/// ```
/// use canonical_form::Canonize;
/// use canonical_form::example::Graph;
///
/// // Cube graph with one fixed vertex
/// let cube = Graph::new(8, &[(0, 1), (1, 2), (2, 3), (3, 0),
/// (4, 5), (5, 6), (6, 7), (7, 4),
/// (0, 4), (1, 5), (2, 6), (3, 7)]).canonical_typed(1);
///
/// let mut count = 0;
/// for p in cube.stabilizer(1) {
/// assert_eq!(cube.apply_morphism(&p), cube);
/// assert_eq!(p[0], 0);
/// count += 1;
/// }
/// assert_eq!(count, 6);
/// ```
#[inline]
fn stabilizer(&self, sigma: usize) -> AutomorphismIterator<Self> {
let mut partition = Partition::simple(self.size());
for i in 0..sigma {
let _ = partition.individualize(i);
}
AutomorphismIterator::new(self, partition)
}
}
/// Return the next part to be refined.
/// This part is chosen as a smallest part with at least 2 elements.
/// Return None is the partition is discrete.
fn target_selector(part: &Partition) -> Option<usize> {
let mut min = usize::max_value();
let mut arg_min = None;
for i in part.parts() {
let length = part.part(i).len();
if 2 <= length && (length < min) {
min = length;
arg_min = Some(i);
}
}
arg_min
}
fn precompute_invariant<F>(g: &F) -> Vec<Vec<Vec<usize>>>
where
F: Canonize,
{
let n = g.size();
let mut res = Vec::with_capacity(n);
for i in 0..n {
res.push(g.invariant_neighborhood(i))
}
res
}
/// Compute the coarsest refinement of `partition` with part undistinguishable
/// by the invarriants.
/// If `new_part` is `Some(p)`, assumes that the partition is up-to-date up to the creation
/// of the part `p`.
fn refine(partition: &mut Partition, invariants: &[Vec<Vec<usize>>], new_part: Option<usize>) {
if !partition.is_discrete() {
let n = partition.num_elems();
assert!(n >= 2);
let invariant_size = invariants[0].len();
debug_assert!(invariants.iter().all(|v| v.len() == invariant_size));
// Stack contains the new created partitions
let mut stack: Vec<_> = match new_part {
Some(p) => vec![p],
None => partition.parts().collect(),
};
// base
let max_step = ((n + 1 - partition.num_parts()) as u64).pow(invariant_size as u32);
let threshold = u64::max_value() / max_step; //
let mut part_buffer = Vec::new();
while !stack.is_empty() && !partition.is_discrete() {
let mut weight = 1; // multiplicator to make the values in the sieve unique
while let Some(part) = stack.pop() {
part_buffer.clear();
part_buffer.extend_from_slice(partition.part(part));
let factor = (part_buffer.len() + 1) as u64;
for i in 0..invariant_size {
weight *= factor;
// Compute sieve
for &u in &part_buffer {
for &v in &invariants[u][i] {
partition.sieve(v, weight)
}
}
}
if weight > threshold {
break;
};
}
partition.split(|new| {
stack.push(new);
})
}
}
}
/// Return the first index on which `u` and `v` differ.
fn fca(u: &[usize], v: &[usize]) -> usize {
let mut i = 0;
while i < u.len() && i < v.len() && u[i] == v[i] {
i += 1;
}
i
}
/// Node of the tree of the normalization process
#[derive(Clone, Debug)]
struct IsoTreeNode {
nparts: usize,
children: Vec<usize>,
inv: Rc<Vec<Vec<Vec<usize>>>>,
}
impl IsoTreeNode {
fn root<F: Canonize>(partition: &mut Partition, g: &F) -> Self {
let inv = Rc::new(precompute_invariant(g));
if let Some(coloring) = g.invariant_coloring() {
partition.refine_by_value(&coloring, |_| {})
}
Self::new(partition, inv, None)
}
fn new(
partition: &mut Partition,
inv: Rc<Vec<Vec<Vec<usize>>>>,
new_part: Option<usize>,
) -> Self {
refine(partition, &inv, new_part);
Self {
children: match target_selector(partition) {
Some(set) => partition.part(set).to_vec(),
None => Vec::new(),
},
nparts: partition.num_parts(),
inv,
}
}
fn explore(&self, v: usize, pi: &mut Partition) -> Self {
debug_assert!(self.is_restored(pi));
let new_part = pi.individualize(v);
Self::new(pi, self.inv.clone(), new_part)
}
// Should never be used
fn dummy() -> Self {
Self {
children: Vec::new(),
nparts: 1,
inv: Rc::new(Vec::new()),
}
}
fn restore(&self, partition: &mut Partition) {
partition.undo(self.nparts)
}
fn is_restored(&self, partition: &mut Partition) -> bool {
partition.num_parts() == self.nparts
}
}
/// Normal form of `g` under the action of isomorphisms that
/// stabilize the parts of `partition`.
fn canonical_constraint<F>(g: &F, mut partition: Partition) -> F
where
F: Canonize,
{
// contains the images of `g` already computed associated to the path to the corresponding leaf
let mut zeta: BTreeMap<F, Vec<usize>> = BTreeMap::new();
let mut tree = Vec::new(); // A stack of IsoTreeNode
let mut path = Vec::new(); // Current path as a vector of chosen vertices
let mut node = IsoTreeNode::root(&mut partition, g);
loop {
// If we have a leaf, treat it
if let Some(phi) = partition.as_bijection() {
match zeta.entry(g.apply_morphism(phi)) {
Occupied(entry) =>
// We are in a branch isomorphic to a branch we explored
{
let k = fca(entry.get(), &path) + 1;
tree.truncate(k);
path.truncate(k);
}
Vacant(entry) => {
let _ = entry.insert(path.clone());
}
}
};
// If there is a child, explore it
if let Some(u) = node.children.pop() {
let new_node = node.explore(u, &mut partition);
tree.push(node);
path.push(u);
node = new_node;
} else {
match tree.pop() {
Some(n) => {
node = n;
let _ = path.pop();
node.restore(&mut partition); // backtrack the partition
}
None => break,
}
};
}
let (g_max, _) = zeta.into_iter().next_back().unwrap(); // return the largest image found
g_max
}
/// Iterator on the automorphisms of a combinatorial structure.
#[derive(Clone, Debug)]
pub struct AutomorphismIterator<F> {
tree: Vec<IsoTreeNode>,
node: IsoTreeNode,
partition: Partition,
g: F,
}
impl<F: Canonize> AutomorphismIterator<F> {
/// Iterator on the automorphisms of `g` that preserve `partition`.
fn new(g: &F, mut partition: Partition) -> Self {
debug_assert!(g == &canonical_constraint(g, partition.clone()));
Self {
tree: vec![IsoTreeNode::root(&mut partition, g)],
partition,
node: IsoTreeNode::dummy(), // Dummy node that will be unstacked at the first iteration
g: g.clone(),
}
}
}
impl<F: Canonize> Iterator for AutomorphismIterator<F> {
type Item = Vec<usize>;
#[inline]
fn next(&mut self) -> Option<Self::Item> {
loop {
if let Some(u) = self.node.children.pop() {
let new_node = self.node.explore(u, &mut self.partition);
let old_node = std::mem::replace(&mut self.node, new_node);
self.tree.push(old_node)
} else {
match self.tree.pop() {
Some(n) => {
n.restore(&mut self.partition);
self.node = n
}
None => return None,
}
}
if let Some(phi) = self.partition.as_bijection() {
if self.g.apply_morphism(phi) == self.g {
return Some(phi.to_vec());
}
};
}
}
}
/// Return a morphism `phi`
/// such that `g.apply_morphism(phi) = canonical_constraint(g, partition)`.
fn morphism_to_canonical_constraint<F>(g: &F, mut partition: Partition) -> Vec<usize>
where
F: Canonize,
{
// initialisation
let mut tree = Vec::new();
let mut node = IsoTreeNode::root(&mut partition, g);
let mut max = None;
let mut phimax = Vec::new();
loop {
if let Some(phi) = partition.as_bijection() {
// If node is a leaf
let phi_g = Some(g.apply_morphism(phi));
if phi_g > max {
max = phi_g;
phimax = phi.to_vec();
}
};
if let Some(u) = node.children.pop() {
let new_node = node.explore(u, &mut partition);
tree.push(node);
node = new_node;
} else {
match tree.pop() {
Some(n) => {
n.restore(&mut partition);
node = n
}
None => break,
}
}
}
phimax
}
/// Tests
#[cfg(test)]
mod tests {
use super::*;
#[derive(Ord, PartialOrd, PartialEq, Eq, Clone, Debug)]
struct Graph {
adj: Vec<Vec<usize>>,
}
impl Graph {
fn new(n: usize, edges: &[(usize, usize)]) -> Self {
let mut adj = vec![Vec::new(); n];
for &(u, v) in edges {
adj[u].push(v);
adj[v].push(u);
}
Graph { adj }
}
}
impl Canonize for Graph {
fn size(&self) -> usize {
self.adj.len()
}
fn apply_morphism(&self, perm: &[usize]) -> Self {
let mut adj = vec![Vec::new(); self.size()];
for (i, nbrs) in self.adj.iter().enumerate() {
adj[perm[i]] = nbrs.iter().map(|&u| perm[u]).collect();
adj[perm[i]].sort();
}
Graph { adj }
}
fn invariant_neighborhood(&self, u: usize) -> Vec<Vec<usize>> {
vec![self.adj[u].clone()]
}
}
#[test]
fn graph() {
let c5 = Graph::new(5, &[(0, 1), (1, 2), (2, 3), (3, 4), (4, 0)]);
let other_c5 = Graph::new(5, &[(0, 2), (2, 1), (1, 4), (4, 3), (3, 0)]);
assert_eq!(c5.canonical(), other_c5.canonical());
let p5 = Graph::new(5, &[(0, 1), (1, 2), (2, 3), (3, 4)]);
assert!(c5.canonical() != p5.canonical());
let p = c5.morphism_to_canonical();
assert_eq!(c5.apply_morphism(&p), c5.canonical());
}
#[test]
fn empty_graphs() {
let empty = Graph::new(0, &[]);
assert_eq!(empty, empty.canonical());
assert_eq!(empty, empty.canonical_typed(0));
assert_eq!(empty.automorphisms().count(), 1);
}
#[test]
fn automorphisms_iterator() {
let c4 = Graph::new(4, &[(0, 1), (1, 2), (2, 3), (3, 0)]).canonical();
let mut count = 0;
for phi in c4.automorphisms() {
assert_eq!(c4.apply_morphism(&phi), c4);
count += 1;
}
assert_eq!(count, 8)
}
}