1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
use std::collections::VecDeque;
use std::hash::Hash;

use crate::visit::{
    EdgeRef, GraphBase, IntoEdges, IntoNeighbors, IntoNodeIdentifiers, NodeCount, NodeIndexable,
    VisitMap, Visitable,
};

/// Computed
/// [*matching*](https://en.wikipedia.org/wiki/Matching_(graph_theory)#Definitions)
/// of the graph.
pub struct Matching<G: GraphBase> {
    graph: G,
    mate: Vec<Option<G::NodeId>>,
    n_edges: usize,
}

impl<G> Matching<G>
where
    G: GraphBase,
{
    fn new(graph: G, mate: Vec<Option<G::NodeId>>, n_edges: usize) -> Self {
        Self {
            graph,
            mate,
            n_edges,
        }
    }
}

impl<G> Matching<G>
where
    G: NodeIndexable,
{
    /// Gets the matched counterpart of given node, if there is any.
    ///
    /// Returns `None` if the node is not matched or does not exist.
    pub fn mate(&self, node: G::NodeId) -> Option<G::NodeId> {
        self.mate.get(self.graph.to_index(node)).and_then(|&id| id)
    }

    /// Iterates over all edges from the matching.
    ///
    /// An edge is represented by its endpoints. The graph is considered
    /// undirected and every pair of matched nodes is reported only once.
    pub fn edges(&self) -> MatchedEdges<'_, G> {
        MatchedEdges {
            graph: &self.graph,
            mate: self.mate.as_slice(),
            current: 0,
        }
    }

    /// Iterates over all nodes from the matching.
    pub fn nodes(&self) -> MatchedNodes<'_, G> {
        MatchedNodes {
            graph: &self.graph,
            mate: self.mate.as_slice(),
            current: 0,
        }
    }

    /// Returns `true` if given edge is in the matching, or `false` otherwise.
    ///
    /// If any of the nodes does not exist, `false` is returned.
    pub fn contains_edge(&self, a: G::NodeId, b: G::NodeId) -> bool {
        match self.mate(a) {
            Some(mate) => mate == b,
            None => false,
        }
    }

    /// Returns `true` if given node is in the matching, or `false` otherwise.
    ///
    /// If the node does not exist, `false` is returned.
    pub fn contains_node(&self, node: G::NodeId) -> bool {
        self.mate(node).is_some()
    }

    /// Gets the number of matched **edges**.
    pub fn len(&self) -> usize {
        self.n_edges
    }

    /// Returns `true` if the number of matched **edges** is 0.
    pub fn is_empty(&self) -> bool {
        self.len() == 0
    }
}

impl<G> Matching<G>
where
    G: NodeCount,
{
    /// Returns `true` if the matching is perfect.
    ///
    /// A matching is
    /// [*perfect*](https://en.wikipedia.org/wiki/Matching_(graph_theory)#Definitions)
    /// if every node in the graph is incident to an edge from the matching.
    pub fn is_perfect(&self) -> bool {
        let n_nodes = self.graph.node_count();
        n_nodes % 2 == 0 && self.n_edges == n_nodes / 2
    }
}

trait WithDummy: NodeIndexable {
    fn dummy_idx(&self) -> usize;
    fn node_bound_with_dummy(&self) -> usize;
    /// Convert `i` to a node index, returns None for the dummy node
    fn try_from_index(&self, i: usize) -> Option<Self::NodeId>;
}

impl<G: NodeIndexable> WithDummy for G {
    fn dummy_idx(&self) -> usize {
        // Gabow numbers the vertices from 1 to n, and uses 0 as the dummy
        // vertex. Our vertex indices are zero-based and so we use the node
        // bound as the dummy node.
        self.node_bound()
    }

    fn node_bound_with_dummy(&self) -> usize {
        self.node_bound() + 1
    }

    fn try_from_index(&self, i: usize) -> Option<Self::NodeId> {
        if i != self.dummy_idx() {
            Some(self.from_index(i))
        } else {
            None
        }
    }
}

pub struct MatchedNodes<'a, G: GraphBase> {
    graph: &'a G,
    mate: &'a [Option<G::NodeId>],
    current: usize,
}

impl<G> Iterator for MatchedNodes<'_, G>
where
    G: NodeIndexable,
{
    type Item = G::NodeId;

    fn next(&mut self) -> Option<Self::Item> {
        while self.current != self.mate.len() {
            let current = self.current;
            self.current += 1;

            if self.mate[current].is_some() {
                return Some(self.graph.from_index(current));
            }
        }

        None
    }
}

pub struct MatchedEdges<'a, G: GraphBase> {
    graph: &'a G,
    mate: &'a [Option<G::NodeId>],
    current: usize,
}

impl<G> Iterator for MatchedEdges<'_, G>
where
    G: NodeIndexable,
{
    type Item = (G::NodeId, G::NodeId);

    fn next(&mut self) -> Option<Self::Item> {
        while self.current != self.mate.len() {
            let current = self.current;
            self.current += 1;

            if let Some(mate) = self.mate[current] {
                // Check if the mate is a node after the current one. If not, then
                // do not report that edge since it has been already reported (the
                // graph is considered undirected).
                if self.graph.to_index(mate) > current {
                    let this = self.graph.from_index(current);
                    return Some((this, mate));
                }
            }
        }

        None
    }
}

/// \[Generic\] Compute a
/// [*matching*](https://en.wikipedia.org/wiki/Matching_(graph_theory)) using a
/// greedy heuristic.
///
/// The input graph is treated as if undirected. The underlying heuristic is
/// unspecified, but is guaranteed to be bounded by *O(|V| + |E|)*. No
/// guarantees about the output are given other than that it is a valid
/// matching.
///
/// If you require a maximum matching, use [`maximum_matching`][1] function
/// instead.
///
/// [1]: fn.maximum_matching.html
pub fn greedy_matching<G>(graph: G) -> Matching<G>
where
    G: Visitable + IntoNodeIdentifiers + NodeIndexable + IntoNeighbors,
    G::NodeId: Eq + Hash,
    G::EdgeId: Eq + Hash,
{
    let (mates, n_edges) = greedy_matching_inner(&graph);
    Matching::new(graph, mates, n_edges)
}

#[inline]
fn greedy_matching_inner<G>(graph: &G) -> (Vec<Option<G::NodeId>>, usize)
where
    G: Visitable + IntoNodeIdentifiers + NodeIndexable + IntoNeighbors,
{
    let mut mate = vec![None; graph.node_bound()];
    let mut n_edges = 0;
    let visited = &mut graph.visit_map();

    for start in graph.node_identifiers() {
        let mut last = Some(start);

        // Function non_backtracking_dfs does not expand the node if it has been
        // already visited.
        non_backtracking_dfs(graph, start, visited, |next| {
            // Alternate matched and unmatched edges.
            if let Some(pred) = last.take() {
                mate[graph.to_index(pred)] = Some(next);
                mate[graph.to_index(next)] = Some(pred);
                n_edges += 1;
            } else {
                last = Some(next);
            }
        });
    }

    (mate, n_edges)
}

fn non_backtracking_dfs<G, F>(graph: &G, source: G::NodeId, visited: &mut G::Map, mut visitor: F)
where
    G: Visitable + IntoNeighbors,
    F: FnMut(G::NodeId),
{
    if visited.visit(source) {
        for target in graph.neighbors(source) {
            if !visited.is_visited(&target) {
                visitor(target);
                non_backtracking_dfs(graph, target, visited, visitor);

                // Non-backtracking traversal, stop iterating over the
                // neighbors.
                break;
            }
        }
    }
}

#[derive(Clone, Copy)]
enum Label<G: GraphBase> {
    None,
    Start,
    // If node v is outer node, then label(v) = w is another outer node on path
    // from v to start u.
    Vertex(G::NodeId),
    // If node v is outer node, then label(v) = (r, s) are two outer vertices
    // (connected by an edge)
    Edge(G::EdgeId, [G::NodeId; 2]),
    // Flag is a special label used in searching for the join vertex of two
    // paths.
    Flag(G::EdgeId),
}

impl<G: GraphBase> Label<G> {
    fn is_outer(&self) -> bool {
        self != &Label::None
            && !match self {
                Label::Flag(_) => true,
                _ => false,
            }
    }

    fn is_inner(&self) -> bool {
        !self.is_outer()
    }

    fn to_vertex(&self) -> Option<G::NodeId> {
        match *self {
            Label::Vertex(v) => Some(v),
            _ => None,
        }
    }

    fn is_flagged(&self, edge: G::EdgeId) -> bool {
        match self {
            Label::Flag(flag) if flag == &edge => true,
            _ => false,
        }
    }
}

impl<G: GraphBase> Default for Label<G> {
    fn default() -> Self {
        Label::None
    }
}

impl<G: GraphBase> PartialEq for Label<G> {
    fn eq(&self, other: &Self) -> bool {
        match (self, other) {
            (Label::None, Label::None) => true,
            (Label::Start, Label::Start) => true,
            (Label::Vertex(v1), Label::Vertex(v2)) => v1 == v2,
            (Label::Edge(e1, _), Label::Edge(e2, _)) => e1 == e2,
            (Label::Flag(e1), Label::Flag(e2)) => e1 == e2,
            _ => false,
        }
    }
}

/// \[Generic\] Compute the [*maximum
/// matching*](https://en.wikipedia.org/wiki/Matching_(graph_theory)) using
/// [Gabow's algorithm][1].
///
/// [1]: https://dl.acm.org/doi/10.1145/321941.321942
///
/// The input graph is treated as if undirected. The algorithm runs in
/// *O(|V|³)*. An algorithm with a better time complexity might be used in the
/// future.
///
/// **Panics** if `g.node_bound()` is `std::usize::MAX`.
///
/// # Examples
///
/// ```
/// use petgraph::prelude::*;
/// use petgraph::algo::maximum_matching;
///
/// // The example graph:
/// //
/// //    +-- b ---- d ---- f
/// //   /    |      |
/// //  a     |      |
/// //   \    |      |
/// //    +-- c ---- e
/// //
/// // Maximum matching: { (a, b), (c, e), (d, f) }
///
/// let mut graph: UnGraph<(), ()> = UnGraph::new_undirected();
/// let a = graph.add_node(());
/// let b = graph.add_node(());
/// let c = graph.add_node(());
/// let d = graph.add_node(());
/// let e = graph.add_node(());
/// let f = graph.add_node(());
/// graph.extend_with_edges(&[(a, b), (a, c), (b, c), (b, d), (c, e), (d, e), (d, f)]);
///
/// let matching = maximum_matching(&graph);
/// assert!(matching.contains_edge(a, b));
/// assert!(matching.contains_edge(c, e));
/// assert_eq!(matching.mate(d), Some(f));
/// assert_eq!(matching.mate(f), Some(d));
/// ```
pub fn maximum_matching<G>(graph: G) -> Matching<G>
where
    G: Visitable + NodeIndexable + IntoNodeIdentifiers + IntoEdges,
{
    // The dummy identifier needs an unused index
    assert_ne!(
        graph.node_bound(),
        std::usize::MAX,
        "The input graph capacity should be strictly less than std::usize::MAX."
    );

    // Greedy algorithm should create a fairly good initial matching. The hope
    // is that it speeds up the computation by doing les work in the complex
    // algorithm.
    let (mut mate, mut n_edges) = greedy_matching_inner(&graph);

    // Gabow's algorithm uses a dummy node in the mate array.
    mate.push(None);
    let len = graph.node_bound() + 1;
    debug_assert_eq!(mate.len(), len);

    let mut label: Vec<Label<G>> = vec![Label::None; len];
    let mut first_inner = vec![std::usize::MAX; len];
    let visited = &mut graph.visit_map();

    for start in 0..graph.node_bound() {
        if mate[start].is_some() {
            // The vertex is already matched. A start must be a free vertex.
            continue;
        }

        // Begin search from the node.
        label[start] = Label::Start;
        first_inner[start] = graph.dummy_idx();
        graph.reset_map(visited);

        // start is never a dummy index
        let start = graph.from_index(start);

        // Queue will contain outer vertices that should be processed next. The
        // start vertex is considered an outer vertex.
        let mut queue = VecDeque::new();
        queue.push_back(start);
        // Mark the start vertex so it is not processed repeatedly.
        visited.visit(start);

        'search: while let Some(outer_vertex) = queue.pop_front() {
            for edge in graph.edges(outer_vertex) {
                if edge.source() == edge.target() {
                    // Ignore self-loops.
                    continue;
                }

                let other_vertex = edge.target();
                let other_idx = graph.to_index(other_vertex);

                if mate[other_idx].is_none() && other_vertex != start {
                    // An augmenting path was found. Augment the matching. If
                    // `other` is actually the start node, then the augmentation
                    // must not be performed, because the start vertex would be
                    // incident to two edges, which violates the matching
                    // property.
                    mate[other_idx] = Some(outer_vertex);
                    augment_path(&graph, outer_vertex, other_vertex, &mut mate, &label);
                    n_edges += 1;

                    // The path is augmented, so the start is no longer free
                    // vertex. We need to begin with a new start.
                    break 'search;
                } else if label[other_idx].is_outer() {
                    // The `other` is an outer vertex (a label has been set to
                    // it). An odd cycle (blossom) was found. Assign this edge
                    // as a label to all inner vertices in paths P(outer) and
                    // P(other).
                    find_join(
                        &graph,
                        edge,
                        &mate,
                        &mut label,
                        &mut first_inner,
                        |labeled| {
                            if visited.visit(labeled) {
                                queue.push_back(labeled);
                            }
                        },
                    );
                } else {
                    let mate_vertex = mate[other_idx];
                    let mate_idx = mate_vertex.map_or(graph.dummy_idx(), |id| graph.to_index(id));

                    if label[mate_idx].is_inner() {
                        // Mate of `other` vertex is inner (no label has been
                        // set to it so far). But it actually is an outer vertex
                        // (it is on a path to the start vertex that begins with
                        // a matched edge, since it is a mate of `other`).
                        // Assign the label of this mate to the `outer` vertex,
                        // so the path for it can be reconstructed using `mate`
                        // and this label.
                        label[mate_idx] = Label::Vertex(outer_vertex);
                        first_inner[mate_idx] = other_idx;
                    }

                    // Add the vertex to the queue only if it's not the dummy and this is its first
                    // discovery.
                    if let Some(mate_vertex) = mate_vertex {
                        if visited.visit(mate_vertex) {
                            queue.push_back(mate_vertex);
                        }
                    }
                }
            }
        }

        // Reset the labels. All vertices are inner for the next search.
        for lbl in label.iter_mut() {
            *lbl = Label::None;
        }
    }

    // Discard the dummy node.
    mate.pop();

    Matching::new(graph, mate, n_edges)
}

fn find_join<G, F>(
    graph: &G,
    edge: G::EdgeRef,
    mate: &[Option<G::NodeId>],
    label: &mut [Label<G>],
    first_inner: &mut [usize],
    mut visitor: F,
) where
    G: IntoEdges + NodeIndexable + Visitable,
    F: FnMut(G::NodeId),
{
    // Simultaneously traverse the inner vertices on paths P(source) and
    // P(target) to find a join vertex - an inner vertex that is shared by these
    // paths.
    let source = graph.to_index(edge.source());
    let target = graph.to_index(edge.target());

    let mut left = first_inner[source];
    let mut right = first_inner[target];

    if left == right {
        // No vertices can be labeled, since both paths already refer to a
        // common vertex - the join.
        return;
    }

    // Flag the (first) inner vertices. This ensures that they are assigned the
    // join as their first inner vertex.
    let flag = Label::Flag(edge.id());
    label[left] = flag;
    label[right] = flag;

    // Find the join.
    let join = loop {
        // Swap the sides. Do not swap if the right side is already finished.
        if right != graph.dummy_idx() {
            std::mem::swap(&mut left, &mut right);
        }

        // Set left to the next inner vertex in P(source) or P(target).
        // The unwraps are safe because left is not the dummy node.
        let left_mate = graph.to_index(mate[left].unwrap());
        let next_inner = label[left_mate].to_vertex().unwrap();
        left = first_inner[graph.to_index(next_inner)];

        if !label[left].is_flagged(edge.id()) {
            // The inner vertex is not flagged yet, so flag it.
            label[left] = flag;
        } else {
            // The inner vertex is already flagged. It means that the other side
            // had to visit it already. Therefore it is the join vertex.
            break left;
        }
    };

    // Label all inner vertices on P(source) and P(target) with the found join.
    for endpoint in [source, target].iter().copied() {
        let mut inner = first_inner[endpoint];
        while inner != join {
            // Notify the caller about labeling a vertex.
            if let Some(ix) = graph.try_from_index(inner) {
                visitor(ix);
            }

            label[inner] = Label::Edge(edge.id(), [edge.source(), edge.target()]);
            first_inner[inner] = join;
            let inner_mate = graph.to_index(mate[inner].unwrap());
            let next_inner = label[inner_mate].to_vertex().unwrap();
            inner = first_inner[graph.to_index(next_inner)];
        }
    }

    for (vertex_idx, vertex_label) in label.iter().enumerate() {
        // To all outer vertices that are on paths P(source) and P(target) until
        // the join, se the join as their first inner vertex.
        if vertex_idx != graph.dummy_idx()
            && vertex_label.is_outer()
            && label[first_inner[vertex_idx]].is_outer()
        {
            first_inner[vertex_idx] = join;
        }
    }
}

fn augment_path<G>(
    graph: &G,
    outer: G::NodeId,
    other: G::NodeId,
    mate: &mut [Option<G::NodeId>],
    label: &[Label<G>],
) where
    G: NodeIndexable,
{
    let outer_idx = graph.to_index(outer);

    let temp = mate[outer_idx];
    let temp_idx = temp.map_or(graph.dummy_idx(), |id| graph.to_index(id));
    mate[outer_idx] = Some(other);

    if mate[temp_idx] != Some(outer) {
        // We are at the end of the path and so the entire path is completely
        // rematched/augmented.
    } else if let Label::Vertex(vertex) = label[outer_idx] {
        // The outer vertex has a vertex label which refers to another outer
        // vertex on the path. So we set this another outer node as the mate for
        // the previous mate of the outer node.
        mate[temp_idx] = Some(vertex);
        if let Some(temp) = temp {
            augment_path(graph, vertex, temp, mate, label);
        }
    } else if let Label::Edge(_, [source, target]) = label[outer_idx] {
        // The outer vertex has an edge label which refers to an edge in a
        // blossom. We need to augment both directions along the blossom.
        augment_path(graph, source, target, mate, label);
        augment_path(graph, target, source, mate, label);
    } else {
        panic!("Unexpected label when augmenting path");
    }
}