feat: implement Bron-Kerbosch algorithm

to find maximal cliques in a graph.

Author: gzsombor

src/lib.rs

@@ -33,6 +33,7 @@
 //! - [connected components](undirected/connected_components/index.html): find disjoint connected sets of vertices ([⇒ Wikipedia][Connected components])
 //! - [Kruskal](undirected/kruskal/index.html): find a minimum-spanning-tree ([⇒ Wikipedia][Kruskal])
 //! - [Prim](undirected/prim/index.html): find a minimum-spanning-tree ([⇒ Wikipedia][Prim])
+//! - [cliques]: find maximum cliques in a graph ([= Wikipedia][BronKerbosch])
 //!
 //! ### Matching
 //!
@@ -80,6 +81,7 @@
 //! [A*]: https://en.wikipedia.org/wiki/A*_search_algorithm
 //! [BFS]: https://en.wikipedia.org/wiki/Breadth-first_search
 //! [Brent]: https://en.wikipedia.org/wiki/Cycle_detection#Brent's_algorithm
+//! [BronKerbosch]: https://en.wikipedia.org/wiki/Bron%E2%80%93Kerbosch_algorithm
 //! [Connected components]: https://en.wikipedia.org/wiki/Connected_component_(graph_theory)
 //! [DFS]: https://en.wikipedia.org/wiki/Depth-first_search
 //! [Dijkstra]: https://en.wikipedia.org/wiki/Dijkstra's_algorithm
@@ -131,6 +133,7 @@ pub mod prelude {
     pub use crate::grid::*;
     pub use crate::kuhn_munkres::*;
     pub use crate::matrix::*;
+    pub use crate::undirected::cliques::*;
     pub use crate::undirected::connected_components::*;
     pub use crate::undirected::kruskal::*;
     pub use crate::utils::*;

src/undirected/cliques.rs

@@ -0,0 +1,108 @@
+//! Find cliques in an undirected graph.
+
+use std::collections::HashSet;
+use std::hash::Hash;
+
+/// Algorithm for finding all maximal cliques in an undirected graph.
+/// That is, it lists all subsets of vertices with the two properties that each pair of vertices in
+/// one of the listed subsets is connected by an edge, and no listed subset can have
+/// any additional vertices added to it while preserving its complete connectivity.
+///  [Bron-Kerbosch algorithm](https://en.wikipedia.org/wiki/Bron%E2%80%93Kerbosch_algorithm).
+///
+///
+/// - `vertices` is the list of all nodes.
+/// - `connected` returns true if the two given node is connected.
+/// - return a list of cliques.
+pub fn maximal_cliques_collect<N, FN, IN>(vertices: IN, connected: &mut FN) -> Vec<HashSet<N>>
+where
+    N: Eq + Hash + Clone,
+    FN: FnMut(&N, &N) -> bool,
+    IN: IntoIterator<Item = N>,
+{
+    let mut result = Vec::new();
+    let mut consumer = |n: &HashSet<N>| result.push(n.to_owned());
+    let mut remaining_nodes: HashSet<N> = vertices.into_iter().collect::<HashSet<_>>();
+    bron_kerbosch(
+        connected,
+        &HashSet::new(),
+        &mut remaining_nodes,
+        &mut HashSet::new(),
+        &mut consumer,
+    );
+    result
+}
+
+/// Algorithm for finding all maximal cliques in an undirected graph.
+/// That is, it lists all subsets of vertices with the two properties that each pair of vertices in
+/// one of the listed subsets is connected by an edge, and no listed subset can have
+/// any additional vertices added to it while preserving its complete connectivity.
+///  [Bron-Kerbosch algorithm](https://en.wikipedia.org/wiki/Bron%E2%80%93Kerbosch_algorithm).
+///
+///
+/// - `vertices` is the list of all nodes.
+/// - `connected` returns true if the two given node is connected.
+/// - 'consumer' function which called for each clique.
+///
+pub fn maximal_cliques<N, FN, IN, CO>(vertices: IN, connected: &mut FN, consumer: &mut CO)
+where
+    N: Eq + Hash + Clone,
+    FN: FnMut(&N, &N) -> bool,
+    IN: IntoIterator<Item = N>,
+    CO: FnMut(&HashSet<N>),
+{
+    let mut remaining_nodes: HashSet<N> = vertices.into_iter().collect();
+    bron_kerbosch(
+        connected,
+        &HashSet::new(),
+        &mut remaining_nodes,
+        &mut HashSet::new(),
+        consumer,
+    );
+}
+
+fn bron_kerbosch<N, FN, CO>(
+    connected: &mut FN,
+    potential_clique: &HashSet<N>,
+    remaining_nodes: &mut HashSet<N>,
+    skip_nodes: &mut HashSet<N>,
+    consumer: &mut CO,
+) where
+    N: Eq + Hash + Clone,
+    FN: FnMut(&N, &N) -> bool,
+    CO: FnMut(&HashSet<N>),
+{
+    if remaining_nodes.is_empty() && skip_nodes.is_empty() {
+        consumer(potential_clique);
+        return;
+    }
+    let nodes_to_check = remaining_nodes.clone();
+    for node in &nodes_to_check {
+        let mut new_potential_clique = potential_clique.clone();
+        new_potential_clique.insert(node.to_owned());
+
+        let mut new_remaining_nodes: HashSet<N> = remaining_nodes
+            .iter()
+            .filter(|n| *n != node && connected(node, n))
+            .cloned()
+            .collect();
+
+        let mut new_skip_list: HashSet<N> = skip_nodes
+            .iter()
+            .filter(|n| *n != node && connected(node, n))
+            .cloned()
+            .collect();
+        bron_kerbosch(
+            connected,
+            &new_potential_clique,
+            &mut new_remaining_nodes,
+            &mut new_skip_list,
+            consumer,
+        );
+
+        // We're done considering this node. If there was a way to form a clique with it, we
+        // already discovered its maximal clique in the recursive call above.  So, go ahead
+        // and remove it from the list of remaining nodes and add it to the skip list.
+        remaining_nodes.remove(node);
+        skip_nodes.insert(node.to_owned());
+    }
+}

src/undirected/mod.rs

@@ -1,5 +1,6 @@
 //! Algorithms for undirected graphs.
 
+pub mod cliques;
 pub mod connected_components;
 pub mod kruskal;
 pub mod prim;

tests/cliques.rs

@@ -0,0 +1,49 @@
+use std::collections::HashSet;
+
+use itertools::Itertools;
+use pathfinding::prelude::*;
+
+#[test]
+fn find_cliques() {
+    let vertices: Vec<i32> = (1..10).collect_vec();
+    let cliques = maximal_cliques_collect(&vertices, &mut |a, b| (*a - *b) % 3 == 0);
+    let cliques_as_vectors: Vec<Vec<i32>> = sort(&cliques);
+
+    assert_eq!(
+        vec![vec![1, 4, 7], vec![2, 5, 8], vec![3, 6, 9]],
+        cliques_as_vectors
+    );
+}
+
+#[test]
+fn test_same_node_appears_in_multiple_clique() {
+    let vertices: Vec<i32> = (1..10).collect_vec();
+    let cliques = maximal_cliques_collect(&vertices, &mut |a, b| {
+        (*a % 3 == 0) && (*b % 3 == 0) || ((*a - *b) % 4 == 0)
+    });
+    let cliques_as_vectors: Vec<Vec<i32>> = sort(&cliques);
+
+    assert_eq!(
+        vec![
+            vec![1, 5, 9],
+            vec![2, 6],
+            vec![3, 6, 9],
+            vec![3, 7],
+            vec![4, 8]
+        ],
+        cliques_as_vectors
+    );
+}
+
+fn sort(cliques: &[HashSet<&i32>]) -> Vec<Vec<i32>> {
+    let mut cliques_as_vectors: Vec<Vec<i32>> = cliques
+        .iter()
+        .map(|cliq| {
+            let mut s = cliq.iter().map(|&x| *x).collect_vec();
+            s.sort_unstable();
+            s
+        })
+        .collect();
+    cliques_as_vectors.sort();
+    cliques_as_vectors
+}