rustworkx_core/connectivity/

biconnected.rs

// Licensed 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 hashbrown::{HashMap, HashSet};
use std::hash::Hash;

use petgraph::{
    visit::{
        EdgeCount, GraphBase, GraphProp, IntoEdges, IntoNodeIdentifiers, NodeIndexable, Time,
        Visitable,
    },
    Undirected,
};

use crate::traversal::{depth_first_search, DfsEvent};

const NULL: usize = usize::MAX;

type Edge<G> = (<G as GraphBase>::NodeId, <G as GraphBase>::NodeId);

#[inline]
fn is_root(parent: &[usize], u: usize) -> bool {
    parent[u] == NULL
}

fn _articulation_points<G>(
    graph: G,
    components: Option<&mut HashMap<Edge<G>, usize>>,
    bridges: Option<&mut HashSet<Edge<G>>>,
) -> HashSet<G::NodeId>
where
    G: GraphProp<EdgeType = Undirected>
        + EdgeCount
        + IntoEdges
        + Visitable
        + NodeIndexable
        + IntoNodeIdentifiers,
    G::NodeId: Eq + Hash,
{
    let num_nodes = graph.node_bound();

    let mut low = vec![NULL; num_nodes];
    let mut disc = vec![NULL; num_nodes];
    let mut parent = vec![NULL; num_nodes];

    let mut root_children: usize = 0;
    let mut points = HashSet::new();

    let mut edge_stack = Vec::new();
    let need_components = components.is_some();
    let mut tmp_components = if need_components {
        HashMap::with_capacity(graph.edge_count())
    } else {
        HashMap::new()
    };
    let need_bridges = bridges.is_some();
    let mut tmp_bridges = if need_bridges {
        HashSet::with_capacity(graph.edge_count())
    } else {
        HashSet::new()
    };
    let mut num_components: usize = 0;

    depth_first_search(graph, graph.node_identifiers(), |event| match event {
        DfsEvent::Discover(u_id, Time(t)) => {
            let u = graph.to_index(u_id);
            low[u] = t;
            disc[u] = t;
        }
        DfsEvent::TreeEdge(u_id, v_id, _) => {
            let u = graph.to_index(u_id);
            let v = graph.to_index(v_id);
            parent[v] = u;
            if is_root(&parent, u) {
                root_children += 1;
            }
            if need_components {
                edge_stack.push((u_id, v_id));
            }
        }
        DfsEvent::BackEdge(u_id, v_id, _) => {
            let u = graph.to_index(u_id);
            let v = graph.to_index(v_id);

            // do *not* consider ``(u, v)`` as a back edge if ``(v, u)`` is a tree edge.
            if v != parent[u] {
                low[u] = low[u].min(disc[v]);
                if need_components {
                    edge_stack.push((u_id, v_id));
                }
            }
        }
        DfsEvent::Finish(u_id, _) => {
            let u = graph.to_index(u_id);
            if is_root(&parent, u) {
                if root_children > 1 {
                    points.insert(u_id);
                }
                // restart ``root_children`` for the remaining connected components
                root_children = 0;
            } else {
                let pu = parent[u];
                let pu_id = graph.from_index(pu);
                low[pu] = low[pu].min(low[u]);

                if !is_root(&parent, pu) && low[u] >= disc[pu] {
                    points.insert(pu_id);
                    // now find a biconnected component that the
                    // current articulation point belongs.
                    if need_components {
                        if let Some(at) = edge_stack.iter().rposition(|&x| x == (pu_id, u_id)) {
                            tmp_components.extend(
                                edge_stack[at..].iter().map(|edge| (*edge, num_components)),
                            );
                            edge_stack.truncate(at);
                            num_components += 1;
                        }
                    }
                    if need_bridges && low[u] != disc[pu] {
                        tmp_bridges.insert((pu_id, u_id));
                    }
                }

                if is_root(&parent, pu) && need_components {
                    if let Some(at) = edge_stack.iter().position(|&x| x == (pu_id, u_id)) {
                        tmp_components
                            .extend(edge_stack[at..].iter().map(|edge| (*edge, num_components)));
                        edge_stack.truncate(at);
                        num_components += 1;
                    }
                }
            }
        }
        _ => (),
    });

    if let Some(x) = components {
        *x = tmp_components;
    }
    if let Some(x) = bridges {
        *x = tmp_bridges;
    }

    points
}

/// Return the articulation points of an undirected graph.
///
/// An articulation point or cut vertex is any node whose removal (along with
/// all its incident edges) increases the number of connected components of
/// a graph. An undirected connected graph without articulation points is
/// biconnected.
///
/// At the same time, you can record the biconnected components in `components`.
///
/// Biconnected components are maximal subgraphs such that the removal
/// of a node (and all edges incident on that node) will not disconnect
/// the subgraph. Note that nodes may be part of more than one biconnected
/// component. Those nodes are articulation points, or cut vertices. The
/// algorithm computes how many biconnected components are in the graph,
/// and assigning each component an integer label.
///
/// # Note
/// The function implicitly assumes that there are no parallel edges
/// or self loops. It may produce incorrect/unexpected results if the
/// input graph has self loops or parallel edges.
///
///
/// # Example:
/// ```rust
/// use std::iter::FromIterator;
/// use hashbrown::{HashMap, HashSet};
///
/// use rustworkx_core::connectivity::articulation_points;
/// use rustworkx_core::petgraph::graph::UnGraph;
/// use rustworkx_core::petgraph::graph::node_index as nx;
///
/// let graph = UnGraph::<(), ()>::from_edges(&[
///    (0, 1), (0, 2), (1, 2), (1, 3),
/// ]);
///
/// let mut bicomp = HashMap::new();
/// let a_points = articulation_points(&graph, Some(&mut bicomp));
///
/// assert_eq!(a_points, HashSet::from_iter([nx(1)]));
/// assert_eq!(bicomp, HashMap::from_iter([
///     ((nx(0), nx(2)), 1), ((nx(2), nx(1)), 1), ((nx(1), nx(0)), 1),
///     ((nx(1), nx(3)), 0)
/// ]));
/// ```
pub fn articulation_points<G>(
    graph: G,
    components: Option<&mut HashMap<Edge<G>, usize>>,
) -> HashSet<G::NodeId>
where
    G: GraphProp<EdgeType = Undirected>
        + EdgeCount
        + IntoEdges
        + Visitable
        + NodeIndexable
        + IntoNodeIdentifiers,
    G::NodeId: Eq + Hash,
{
    _articulation_points(graph, components, None)
}

/// Return the bridges of an undirected graph.
///
/// Bridges are edges that, if removed, would increase the number of
/// connected components of a graph.
///
/// # Note
/// The function implicitly assumes that there are no parallel edges
/// or self loops. It may produce incorrect/unexpected results if the
/// input graph has self loops or parallel edges.
///
///
/// # Example:
/// ```rust
/// use std::iter::FromIterator;
/// use hashbrown::{HashMap, HashSet};
///
/// use rustworkx_core::connectivity::bridges;
/// use rustworkx_core::petgraph::graph::UnGraph;
/// use rustworkx_core::petgraph::graph::node_index as nx;
///
/// let graph = UnGraph::<(), ()>::from_edges(&[
///    (0, 1), (0, 2), (1, 2), (1, 3),
/// ]);
///
/// let bridges = bridges(&graph);
///
/// assert_eq!(bridges, HashSet::from_iter([(nx(1), nx(3))]));
/// ```
pub fn bridges<G>(graph: G) -> HashSet<Edge<G>>
where
    G: GraphProp<EdgeType = Undirected>
        + EdgeCount
        + IntoEdges
        + Visitable
        + NodeIndexable
        + IntoNodeIdentifiers,
    G::NodeId: Eq + Hash,
{
    let mut bridges = HashSet::new();
    _articulation_points(graph, None, Some(&mut bridges));
    bridges
}

#[cfg(test)]
mod tests {
    use crate::connectivity::{articulation_points, bridges};
    use hashbrown::{HashMap, HashSet};
    use petgraph::graph::node_index as nx;
    use petgraph::prelude::*;
    use std::iter::FromIterator;

    #[test]
    fn test_articulation_points_repetitions() {
        let graph = UnGraph::<(), ()>::from_edges([(0, 1), (1, 2), (1, 3)]);

        let a_points = articulation_points(&graph, None);

        assert_eq!(a_points, HashSet::from_iter([nx(1)]));
    }

    #[test]
    fn test_articulation_points_cycle() {
        // create a cycle graph
        let graph = UnGraph::<(), ()>::from_edges([(0, 1), (1, 2), (2, 0), (1, 3), (3, 4), (4, 1)]);

        let a_points = articulation_points(&graph, None);

        assert_eq!(a_points, HashSet::from_iter([nx(1)]));
    }

    #[test]
    fn test_single_bridge() {
        let graph = UnGraph::<(), ()>::from_edges([
            (1, 2),
            (2, 3),
            (3, 4),
            (3, 5),
            (5, 6),
            (6, 7),
            (7, 8),
            (5, 9),
            (9, 10),
            // Nontree edges.
            (1, 3),
            (1, 4),
            (2, 5),
            (5, 10),
            (6, 8),
        ]);

        assert_eq!(bridges(&graph), HashSet::from_iter([(nx(5), nx(6))]));
    }

    #[test]
    // generate test cases for bridges
    fn test_bridges_cycle() {
        let graph = UnGraph::<(), ()>::from_edges([(0, 1), (1, 2), (2, 0), (1, 3), (3, 4), (4, 1)]);
        assert_eq!(bridges(&graph), HashSet::from_iter([]));
    }

    #[test]
    fn test_biconnected_components_cycle() {
        // create a cycle graph
        let graph = UnGraph::<(), ()>::from_edges([(0, 1), (1, 2), (2, 0), (1, 3), (3, 4), (4, 1)]);

        let mut components = HashMap::new();
        let _ = articulation_points(&graph, Some(&mut components));

        assert_eq!(
            components,
            HashMap::from_iter([
                ((nx(1), nx(3)), 0),
                ((nx(3), nx(4)), 0),
                ((nx(4), nx(1)), 0),
                ((nx(1), nx(2)), 1),
                ((nx(2), nx(0)), 1),
                ((nx(0), nx(1)), 1)
            ])
        );
    }

    #[test]
    fn test_biconnected_components1() {
        // example from https://web.archive.org/web/20121229123447/http://www.ibluemojo.com/school/articul_algorithm.html
        let graph = UnGraph::<(), ()>::from_edges([
            (0, 1),
            (0, 5),
            (0, 6),
            (0, 14),
            (1, 5),
            (1, 6),
            (1, 14),
            (2, 4),
            (2, 10),
            (3, 4),
            (3, 15),
            (4, 6),
            (4, 7),
            (4, 10),
            (5, 14),
            (6, 14),
            (7, 9),
            (8, 9),
            (8, 12),
            (8, 13),
            (10, 15),
            (11, 12),
            (11, 13),
            (12, 13),
        ]);

        let mut components = HashMap::new();
        let a_points = articulation_points(&graph, Some(&mut components));

        assert_eq!(
            a_points,
            HashSet::from_iter([nx(4), nx(6), nx(7), nx(8), nx(9)])
        );
        assert_eq!(
            components,
            HashMap::from_iter([
                ((nx(3), nx(4)), 0),
                ((nx(15), nx(3)), 0),
                ((nx(10), nx(15)), 0),
                ((nx(4), nx(10)), 0),
                ((nx(10), nx(2)), 0),
                ((nx(2), nx(4)), 0),
                ((nx(13), nx(12)), 1),
                ((nx(8), nx(13)), 1),
                ((nx(11), nx(13)), 1),
                ((nx(12), nx(11)), 1),
                ((nx(12), nx(8)), 1),
                ((nx(9), nx(8)), 2),
                ((nx(7), nx(9)), 3),
                ((nx(4), nx(7)), 4),
                ((nx(6), nx(4)), 5),
                ((nx(0), nx(14)), 6),
                ((nx(1), nx(5)), 6),
                ((nx(5), nx(0)), 6),
                ((nx(5), nx(14)), 6),
                ((nx(1), nx(14)), 6),
                ((nx(14), nx(6)), 6),
                ((nx(6), nx(0)), 6),
                ((nx(6), nx(1)), 6),
                ((nx(1), nx(0)), 6),
            ])
        )
    }

    #[test]
    fn test_biconnected_components2() {
        let mut graph: Graph<&str, (), Undirected> = Graph::new_undirected();
        let a = graph.add_node("A");
        let b = graph.add_node("B");
        let c = graph.add_node("C");
        let d = graph.add_node("D");
        let e = graph.add_node("E");
        let f = graph.add_node("F");
        let g = graph.add_node("G");
        let h = graph.add_node("H");
        let i = graph.add_node("I");
        let j = graph.add_node("J");

        graph.extend_with_edges([
            (a, b),
            (b, c),
            (c, a),
            (c, d),
            (d, e),
            (e, c),
            (f, i),
            (i, j),
            (j, h),
            (h, g),
            (g, f),
            (g, i),
            (g, j),
            (e, g),
        ]);

        let mut components = HashMap::new();
        let _ = articulation_points(&graph, Some(&mut components));

        assert_eq!(
            components,
            HashMap::from_iter([
                ((f, g), 0),
                ((i, f), 0),
                ((i, g), 0),
                ((j, i), 0),
                ((g, j), 0),
                ((j, h), 0),
                ((h, g), 0),
                ((e, g), 1),
                ((c, d), 2),
                ((d, e), 2),
                ((e, c), 2),
                ((c, a), 3),
                ((a, b), 3),
                ((b, c), 3),
            ])
        )
    }

    #[test]
    fn test_null_graph() {
        let graph: Graph<(), (), Undirected> = Graph::new_undirected();

        let mut components = HashMap::new();
        let a_points = articulation_points(&graph, Some(&mut components));
        let bridges = bridges(&graph);

        assert_eq!(a_points, HashSet::new());
        assert_eq!(bridges, HashSet::new());
        assert_eq!(components, HashMap::new());
    }
}