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
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
//! Graph algorithms.
//!
//! It is a goal to gradually migrate the algorithms to be based on graph traits
//! so that they are generally applicable. For now, some of these still require
//! the `Graph` type.

pub mod dominators;

use std::collections::BinaryHeap;
use std::cmp::min;

use prelude::*;

use super::{
    EdgeType,
};
use scored::MinScored;
use super::visit::{
    GraphRef,
    GraphBase,
    Visitable,
    VisitMap,
    IntoNeighbors,
    IntoNeighborsDirected,
    IntoNodeIdentifiers,
    NodeCount,
    NodeIndexable,
    NodeCompactIndexable,
    IntoEdgeReferences,
    IntoEdges,
    Reversed,
};
use super::unionfind::UnionFind;
use super::graph::{
    IndexType,
};
use visit::{Data, NodeRef, IntoNodeReferences};
use data::{
    Element,
};

pub use super::isomorphism::{
    is_isomorphic,
    is_isomorphic_matching,
};
pub use super::dijkstra::dijkstra;
pub use super::astar::astar;

/// [Generic] Return the number of connected components of the graph.
///
/// For a directed graph, this is the *weakly* connected components.
pub fn connected_components<G>(g: G) -> usize
    where G: NodeCompactIndexable + IntoEdgeReferences,
{
    let mut vertex_sets = UnionFind::new(g.node_bound());
    for edge in g.edge_references() {
        let (a, b) = (edge.source(), edge.target());

        // union the two vertices of the edge
        vertex_sets.union(g.to_index(a), g.to_index(b));
    }
    let mut labels = vertex_sets.into_labeling();
    labels.sort();
    labels.dedup();
    labels.len()
}


/// [Generic] Return `true` if the input graph contains a cycle.
///
/// Always treats the input graph as if undirected.
pub fn is_cyclic_undirected<G>(g: G) -> bool
    where G: NodeIndexable + IntoEdgeReferences
{
    let mut edge_sets = UnionFind::new(g.node_bound());
    for edge in g.edge_references() {
        let (a, b) = (edge.source(), edge.target());

        // union the two vertices of the edge
        //  -- if they were already the same, then we have a cycle
        if !edge_sets.union(g.to_index(a), g.to_index(b)) {
            return true
        }
    }
    false
}


/// [Generic] Perform a topological sort of a directed graph.
///
/// If the graph was acyclic, return a vector of nodes in topological order:
/// each node is ordered before its successors.
/// Otherwise, it will return a `Cycle` error. Self loops are also cycles.
///
/// To handle graphs with cycles, use the scc algorithms or `DfsPostOrder`
/// instead of this function.
///
/// If `space` is not `None`, it is used instead of creating a new workspace for
/// graph traversal. The implementation is iterative.
pub fn toposort<G>(g: G, space: Option<&mut DfsSpace<G::NodeId, G::Map>>)
    -> Result<Vec<G::NodeId>, Cycle<G::NodeId>>
    where G: IntoNeighborsDirected + IntoNodeIdentifiers + Visitable,
{
    // based on kosaraju scc
    with_dfs(g, space, |dfs| {
        dfs.reset(g);
        let mut finished = g.visit_map();

        let mut finish_stack = Vec::new();
        for i in g.node_identifiers() {
            if dfs.discovered.is_visited(&i) {
                continue;
            }
            dfs.stack.push(i);
            while let Some(&nx) = dfs.stack.last() {
                if dfs.discovered.visit(nx) {
                    // First time visiting `nx`: Push neighbors, don't pop `nx`
                    for succ in g.neighbors(nx) {
                        if succ == nx {
                            // self cycle
                            return Err(Cycle(nx));
                        }
                        if !dfs.discovered.is_visited(&succ) {
                            dfs.stack.push(succ);
                        } 
                    }
                } else {
                    dfs.stack.pop();
                    if finished.visit(nx) {
                        // Second time: All reachable nodes must have been finished
                        finish_stack.push(nx);
                    }
                }
            }
        }
        finish_stack.reverse();

        dfs.reset(g);
        for &i in &finish_stack {
            dfs.move_to(i);
            let mut cycle = false;
            while let Some(j) = dfs.next(Reversed(g)) {
                if cycle {
                    return Err(Cycle(j));
                }
                cycle = true;
            }
        }

        Ok(finish_stack)
    })
}

/// [Generic] Return `true` if the input directed graph contains a cycle.
///
/// This implementation is recursive; use `toposort` if an alternative is
/// needed.
pub fn is_cyclic_directed<G>(g: G) -> bool
    where G: IntoNodeIdentifiers + IntoNeighbors + Visitable,
{
    use visit::{depth_first_search, DfsEvent};

    depth_first_search(g, g.node_identifiers(), |event| {
        match event {
            DfsEvent::BackEdge(_, _) => Err(()),
            _ => Ok(()),
        }
    }).is_err()
}

type DfsSpaceType<G> = DfsSpace<<G as GraphBase>::NodeId, <G as Visitable>::Map>;

/// Workspace for a graph traversal.
#[derive(Clone, Debug)]
pub struct DfsSpace<N, VM> {
    dfs: Dfs<N, VM>,
}

impl<N, VM> DfsSpace<N, VM>
    where N: Copy + PartialEq,
          VM: VisitMap<N>,
{
    pub fn new<G>(g: G) -> Self
        where G: GraphRef + Visitable<NodeId=N, Map=VM>,
    {
        DfsSpace {
            dfs: Dfs::empty(g)
        }
    }
}

impl<N, VM> Default for DfsSpace<N, VM>
    where VM: VisitMap<N> + Default,
{
    fn default() -> Self {
        DfsSpace {
            dfs: Dfs {
                stack: <_>::default(),
                discovered: <_>::default(),
            }
        }
    }
}

/// Create a Dfs if it's needed
fn with_dfs<G, F, R>(g: G, space: Option<&mut DfsSpaceType<G>>, f: F) -> R
    where G: GraphRef + Visitable,
          F: FnOnce(&mut Dfs<G::NodeId, G::Map>) -> R
{
    let mut local_visitor;
    let dfs = if let Some(v) = space { &mut v.dfs } else {
        local_visitor = Dfs::empty(g);
        &mut local_visitor
    };
    f(dfs)
}

/// [Generic] Check if there exists a path starting at `from` and reaching `to`.
///
/// If `from` and `to` are equal, this function returns true.
///
/// If `space` is not `None`, it is used instead of creating a new workspace for
/// graph traversal.
pub fn has_path_connecting<G>(g: G, from: G::NodeId, to: G::NodeId,
                              space: Option<&mut DfsSpace<G::NodeId, G::Map>>)
    -> bool
    where G: IntoNeighbors + Visitable,
{
    with_dfs(g, space, |dfs| {
        dfs.reset(g);
        dfs.move_to(from);
        while let Some(x) = dfs.next(g) {
            if x == to {
                return true;
            }
        }
        false
    })
}

/// Renamed to `kosaraju_scc`.
#[deprecated(note = "renamed to kosaraju_scc")]
pub fn scc<G>(g: G) -> Vec<Vec<G::NodeId>>
    where G: IntoNeighborsDirected + Visitable + IntoNodeIdentifiers,
{
    kosaraju_scc(g)
}

/// [Generic] Compute the *strongly connected components* using [Kosaraju's algorithm][1].
///
/// [1]: https://en.wikipedia.org/wiki/Kosaraju%27s_algorithm
///
/// Return a vector where each element is a strongly connected component (scc).
/// The order of node ids within each scc is arbitrary, but the order of
/// the sccs is their postorder (reverse topological sort).
///
/// For an undirected graph, the sccs are simply the connected components.
///
/// This implementation is iterative and does two passes over the nodes.
pub fn kosaraju_scc<G>(g: G) -> Vec<Vec<G::NodeId>>
    where G: IntoNeighborsDirected + Visitable + IntoNodeIdentifiers,
{
    let mut dfs = DfsPostOrder::empty(g);

    // First phase, reverse dfs pass, compute finishing times.
    // http://stackoverflow.com/a/26780899/161659
    let mut finish_order = Vec::with_capacity(0);
    for i in g.node_identifiers() {
        if dfs.discovered.is_visited(&i) {
            continue
        }

        dfs.move_to(i);
        while let Some(nx) = dfs.next(Reversed(g)) {
            finish_order.push(nx);
        }
    }

    let mut dfs = Dfs::from_parts(dfs.stack, dfs.discovered);
    dfs.reset(g);
    let mut sccs = Vec::new();

    // Second phase
    // Process in decreasing finishing time order
    for i in finish_order.into_iter().rev() {
        if dfs.discovered.is_visited(&i) {
            continue;
        }
        // Move to the leader node `i`.
        dfs.move_to(i);
        let mut scc = Vec::new();
        while let Some(nx) = dfs.next(g) {
            scc.push(nx);
        }
        sccs.push(scc);
    }
    sccs
}

/// [Generic] Compute the *strongly connected components* using [Tarjan's algorithm][1].
///
/// [1]: https://en.wikipedia.org/wiki/Tarjan%27s_strongly_connected_components_algorithm
///
/// Return a vector where each element is a strongly connected component (scc).
/// The order of node ids within each scc is arbitrary, but the order of
/// the sccs is their postorder (reverse topological sort).
///
/// For an undirected graph, the sccs are simply the connected components.
///
/// This implementation is recursive and does one pass over the nodes.
pub fn tarjan_scc<G>(g: G) -> Vec<Vec<G::NodeId>>
    where G: IntoNodeIdentifiers + IntoNeighbors + NodeIndexable
{
    #[derive(Copy, Clone)]
    #[derive(Debug)]
    struct NodeData {
        index: Option<usize>,
        lowlink: usize,
        on_stack: bool,
    }

    #[derive(Debug)]
    struct Data<'a, G>
        where G: NodeIndexable, 
          G::NodeId: 'a
    {
        index: usize,
        nodes: Vec<NodeData>,
        stack: Vec<G::NodeId>,
        sccs: &'a mut Vec<Vec<G::NodeId>>,
    }

    fn scc_visit<G>(v: G::NodeId, g: G, data: &mut Data<G>) 
        where G: IntoNeighbors + NodeIndexable
    {
        macro_rules! node {
            ($node:expr) => (data.nodes[g.to_index($node)])
        }

        if node![v].index.is_some() {
            // already visited
            return;
        }

        let v_index = data.index;
        node![v].index = Some(v_index);
        node![v].lowlink = v_index;
        node![v].on_stack = true;
        data.stack.push(v);
        data.index += 1;

        for w in g.neighbors(v) {
            match node![w].index {
                None => {
                    scc_visit(w, g, data);
                    node![v].lowlink = min(node![v].lowlink, node![w].lowlink);
                }
                Some(w_index) => {
                    if node![w].on_stack {
                        // Successor w is in stack S and hence in the current SCC
                        let v_lowlink = &mut node![v].lowlink;
                        *v_lowlink = min(*v_lowlink, w_index);
                    }
                }
            }
        }

        // If v is a root node, pop the stack and generate an SCC
        if let Some(v_index) = node![v].index {
            if node![v].lowlink == v_index {
                let mut cur_scc = Vec::new();
                loop {
                    let w = data.stack.pop().unwrap();
                    node![w].on_stack = false;
                    cur_scc.push(w);
                    if g.to_index(w) == g.to_index(v) { break; }
                }
                data.sccs.push(cur_scc);
            }
        }
    }

    let mut sccs = Vec::new();
    {
        let map = vec![NodeData { index: None, lowlink: !0, on_stack: false }; g.node_bound()];

        let mut data = Data {
            index: 0,
            nodes: map,
            stack: Vec::new(),
            sccs: &mut sccs,
        };

        for n in g.node_identifiers() {
            scc_visit(n, g, &mut data);
        }
    }
    sccs
}

/// [Graph] Condense every strongly connected component into a single node and return the result.
///
/// If `make_acyclic` is true, self-loops and multi edges are ignored, guaranteeing that
/// the output is acyclic.
pub fn condensation<N, E, Ty, Ix>(g: Graph<N, E, Ty, Ix>, make_acyclic: bool) -> Graph<Vec<N>, E, Ty, Ix>
    where Ty: EdgeType,
          Ix: IndexType,
{
    let sccs = kosaraju_scc(&g);
    let mut condensed: Graph<Vec<N>, E, Ty, Ix> = Graph::with_capacity(sccs.len(), g.edge_count());

    // Build a map from old indices to new ones.
    let mut node_map = vec![NodeIndex::end(); g.node_count()];
    for comp in sccs {
        let new_nix = condensed.add_node(Vec::new());
        for nix in comp {
            node_map[nix.index()] = new_nix;
        }
    }

    // Consume nodes and edges of the old graph and insert them into the new one.
    let (nodes, edges) = g.into_nodes_edges();
    for (nix, node) in nodes.into_iter().enumerate() {
        condensed[node_map[nix]].push(node.weight);
    }
    for edge in edges {
        let source = node_map[edge.source().index()];
        let target = node_map[edge.target().index()];
        if make_acyclic {
            if source != target {
                condensed.update_edge(source, target, edge.weight);
            }
        } else {
            condensed.add_edge(source, target, edge.weight);
        }
    }
    condensed
}

/// [Generic] Compute a *minimum spanning tree* of a graph.
///
/// The input graph is treated as if undirected.
///
/// Using Kruskal's algorithm with runtime **O(|E| log |E|)**. We actually
/// return a minimum spanning forest, i.e. a minimum spanning tree for each connected
/// component of the graph.
///
/// The resulting graph has all the vertices of the input graph (with identical node indices),
/// and **|V| - c** edges, where **c** is the number of connected components in `g`.
///
/// Use `from_elements` to create a graph from the resulting iterator.
pub fn min_spanning_tree<G>(g: G) -> MinSpanningTree<G>
    where G::NodeWeight: Clone,
          G::EdgeWeight: Clone + PartialOrd,
          G: IntoNodeReferences + IntoEdgeReferences + NodeIndexable,
{

    // Initially each vertex is its own disjoint subgraph, track the connectedness
    // of the pre-MST with a union & find datastructure.
    let subgraphs = UnionFind::new(g.node_bound());

    let edges = g.edge_references();
    let mut sort_edges = BinaryHeap::with_capacity(edges.size_hint().0);
    for edge in edges {
        sort_edges.push(MinScored(edge.weight().clone(), (edge.source(), edge.target())));
    }

    MinSpanningTree {
        graph: g,
        node_ids: Some(g.node_references()),
        subgraphs: subgraphs,
        sort_edges: sort_edges,
    }

}

/// An iterator producing a minimum spanning forest of a graph.
pub struct MinSpanningTree<G>
    where G: Data + IntoNodeReferences,
{
    graph: G,
    node_ids: Option<G::NodeReferences>,
    subgraphs: UnionFind<usize>,
    sort_edges: BinaryHeap<MinScored<G::EdgeWeight, (G::NodeId, G::NodeId)>>,
}


impl<G> Iterator for MinSpanningTree<G>
    where G: IntoNodeReferences + NodeIndexable,
          G::NodeWeight: Clone,
          G::EdgeWeight: PartialOrd,
{
    type Item = Element<G::NodeWeight, G::EdgeWeight>;

    fn next(&mut self) -> Option<Self::Item> {
        if let Some(ref mut iter) = self.node_ids {
            if let Some(node) = iter.next() {
                return Some(Element::Node { weight: node.weight().clone() });
            }
        }
        self.node_ids = None;

        // Kruskal's algorithm.
        // Algorithm is this:
        //
        // 1. Create a pre-MST with all the vertices and no edges.
        // 2. Repeat:
        //
        //  a. Remove the shortest edge from the original graph.
        //  b. If the edge connects two disjoint trees in the pre-MST,
        //     add the edge.
        while let Some(MinScored(score, (a, b))) = self.sort_edges.pop() {
            let g = self.graph;
            // check if the edge would connect two disjoint parts
            if self.subgraphs.union(g.to_index(a), g.to_index(b)) {
                return Some(Element::Edge {
                    source: g.to_index(a),
                    target: g.to_index(b),
                    weight: score,
                });
            }
        }
        None
    }
}

/// An algorithm error: a cycle was found in the graph.
#[derive(Clone, Debug, PartialEq)]
pub struct Cycle<N>(N);

impl<N> Cycle<N> {
    /// Return a node id that participates in the cycle
    pub fn node_id(&self) -> N
        where N: Copy
    {
        self.0
    }
}
/// An algorithm error: a cycle of negative weights was found in the graph.
#[derive(Clone, Debug, PartialEq)]
pub struct NegativeCycle(());

/// [Generic] Compute shortest paths from node `source` to all other.
///
/// Using the [Bellman–Ford algorithm][bf]; negative edge costs are
/// permitted, but the graph must not have a cycle of negative weights
/// (in that case it will return an error).
///
/// On success, return one vec with path costs, and another one which points
/// out the predecessor of a node along a shortest path. The vectors
/// are indexed by the graph's node indices.
///
/// [bf]: https://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithm
pub fn bellman_ford<G>(g: G, source: G::NodeId)
    -> Result<(Vec<G::EdgeWeight>, Vec<Option<G::NodeId>>), NegativeCycle>
    where G: NodeCount + IntoNodeIdentifiers + IntoEdges + NodeIndexable,
          G::EdgeWeight: FloatMeasure,
{
    let mut predecessor = vec![None; g.node_bound()];
    let mut distance = vec![<_>::infinite(); g.node_bound()];

    let ix = |i| g.to_index(i);

    distance[ix(source)] = <_>::zero();
    // scan up to |V| - 1 times.
    for _ in 1..g.node_count() {
        let mut did_update = false;
        for i in g.node_identifiers() {
            for edge in g.edges(i) {
                let i = edge.source();
                let j = edge.target();
                let w = *edge.weight();
                if distance[ix(i)] + w < distance[ix(j)] {
                    distance[ix(j)] = distance[ix(i)] + w;
                    predecessor[ix(j)] = Some(i);
                    did_update = true;
                }
            }
        }
        if !did_update {
            break;
        }
    }

    // check for negative weight cycle
    for i in g.node_identifiers() {
        for edge in g.edges(i) {
            let j = edge.target();
            let w = *edge.weight();
            if distance[ix(i)] + w < distance[ix(j)] {
                //println!("neg cycle, detected from {} to {}, weight={}", i, j, w);
                return Err(NegativeCycle(()));
            }
        }
    }

    Ok((distance, predecessor))
}

use std::ops::Add;
use std::fmt::Debug;

/// Associated data that can be used for measures (such as length).
pub trait Measure : Debug + PartialOrd + Add<Self, Output=Self> + Default + Clone {
}

impl<M> Measure for M
    where M: Debug + PartialOrd + Add<M, Output=M> + Default + Clone,
{ }

/// A floating-point measure.
pub trait FloatMeasure : Measure + Copy {
    fn zero() -> Self;
    fn infinite() -> Self;
}

impl FloatMeasure for f32 {
    fn zero() -> Self { 0. }
    fn infinite() -> Self { 1./0. }
}

impl FloatMeasure for f64 {
    fn zero() -> Self { 0. }
    fn infinite() -> Self { 1./0. }
}