Source file nonnegative.ml
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
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
module Imperative
(G: Sig.IM)
(W: Sig.WEIGHT with type edge = G.E.t) = struct
module S = Set.Make(G.V)
module M = Map.Make(G.V)
module V = G.V
module E = G.E
type t = G.t * S.t ref * (G.E.t option * W.t) M.t M.t ref
type edge = G.edge
type vertex = G.vertex
let sov v = string_of_int (Obj.magic (V.label v))
let dump_cycle cycle =
let v0 = G.E.src (List.hd cycle) in
print_string ("(" ^ (sov v0) ^ ")");
let v1 = List.fold_left (fun v e ->
assert ((G.V.compare v (G.E.src e)) = 0);
let v = G.E.dst e in
print_string ("-(" ^ (sov v) ^ ")");
v) v0 cycle in
assert (G.V.equal v0 v1);
print_string "\n"
let dump_set = S.iter (fun x -> print_string ((sov x) ^ ", "))
let dump (src, dist) =
print_string "====================\nS: ";
dump_set !src;
print_string "\nMap:";
M.iter (fun k v ->
print_string ("\n " ^ (sov k) ^ ": ");
M.iter (fun k (origin, dist) ->
print_string (
"(" ^ (sov k) ^ ">>" ^
(match origin with
| None -> "---"
| Some e -> (sov (G.E.src e)) ^ ">"
^ (sov (G.E.dst e))) ^ ":" ^
(string_of_int (Obj.magic dist)) ^ ") ")) v) !dist;
print_string "\n"
exception Negative_cycle of G.E.t list
let create ?size () =
let g = match size with
| Some size -> G.create ~size ()
| None -> G.create () in
(g, ref S.empty, ref M.empty)
let copy (g, src, dist) =
(G.copy g, ref (!src), ref (!dist))
let clear (g, src, dist) =
G.clear g;
src := S.empty;
dist := M.empty
let add_vertex (g, src, dist) v =
if not (G.mem_vertex g v) then begin
G.add_vertex g v;
src := S.add v !src;
dist := M.add v (M.add v (None, W.zero) M.empty) !dist;
dump (src, dist)
end
let rec propagate (g, src, dist) q start =
if Queue.is_empty q then (g, src, dist)
else begin
let (v1, v1src) = Queue.pop q in
let v1dist = M.find v1 dist in
let dist = G.fold_succ_e (fun e dist ->
let v2 = G.E.dst e in
let v2dist = if M.mem v2 dist then M.find v2 dist else M.empty in
let (v2dist, nextSrc) = S.fold (fun x (v2dist, nextSrc) ->
let _, dev1 = M.find x v1dist in
let ndev2 = W.add dev1 (W.weight e) in
let improvement =
try
let _, dev2 = M.find x v2dist in
W.compare ndev2 dev2 < 0
with Not_found -> true in
if improvement then
let v2dist = M.add x (Some e, ndev2) v2dist in
let nextSrc = S.add x nextSrc in
(v2dist, nextSrc)
else
(v2dist, nextSrc)
) v1src (v2dist, S.empty) in
if S.is_empty nextSrc then
dist
else if G.V.equal start v2 then
let dist = M.add v2 v2dist dist in
let cycle = S.fold (fun s x ->
let rec build_cycle x ret =
match M.find s (M.find x dist) with
| Some e, _ ->
let y = G.E.src e in
let cycle = e :: ret in
if G.V.equal start y then Some cycle
else build_cycle y cycle
| _ -> None in
match x with
| None -> build_cycle v2 []
| Some _ -> x) nextSrc None in
let cycle = match cycle with
| Some x -> x | None -> assert false in
dump_cycle cycle;
raise (Negative_cycle cycle)
else
begin
Queue.push (v2, nextSrc) q;
M.add v2 v2dist dist
end
) g v1 dist in
propagate (g, src, dist) q start
end
let m_cardinal m = M.fold (fun _ _ acc -> acc+1) m 0
let set_of_map m = M.fold (fun k _ acc -> S.add k acc) m S.empty
let add_edge_internal (g, src, dist) v1 v2 =
let dv1 = M.find v1 dist in
let q = Queue.create () in
if m_cardinal dv1 = 1 && M.mem v2 dv1 then (
Queue.add (v1, (S.add v2 S.empty)) q;
propagate (g, src, dist) q v1
) else (
let (src, dist, dv1) =
if not (S.mem v2 src) then
(src, dist, dv1)
else
let src = S.remove v2 src in
let dist = M.map (M.remove v2) dist in
let dv1 = M.find v1 dist in
(src, dist, dv1) in
Queue.add (v1, set_of_map dv1) q;
propagate (g, src, dist) q v1)
let add_edge_e (g, src, dist) e =
if not (G.mem_edge_e g e) then begin
let v1 = G.E.src e in
let v2 = G.E.dst e in
List.iter (add_vertex (g, src, dist)) [v1 ; v2];
begin try
G.add_edge_e g e;
let (_, src', dist') = add_edge_internal (g, !src, !dist) v1 v2 in
src := src'; dist := dist'
with exp ->
G.remove_edge_e g e;
raise exp
end;
dump (src, dist)
end
let add_edge (g, src, dist) v1 v2 =
if not (G.mem_edge g v1 v2) then begin
List.iter (add_vertex (g, src, dist)) [v1 ; v2];
begin try
G.add_edge g v1 v2;
let (_, src', dist') = add_edge_internal (g, !src, !dist) v1 v2 in
src := src'; dist := dist'
with exp ->
G.remove_edge g v1 v2;
raise exp
end;
dump (src, dist)
end
let remove_edge_internal (g, src) v2 =
let q = Queue.create () in
print_string ("dump: ");
dump_set src;
let dist = S.fold (fun x dist ->
print_string ("source: " ^ (sov x) ^ "\n");
Queue.add (x, (S.add x S.empty)) q;
M.add x (M.add x (None, W.zero) M.empty) dist) src M.empty in
let g, src, dist = propagate (g, src, dist) q (S.choose src) in
if M.mem v2 dist then
(g, src, dist)
else (
Queue.add (v2, (S.add v2 S.empty)) q;
let src = S.add v2 src in
let dist = M.add v2 (M.add v2 (None, W.zero) M.empty) dist in
propagate (g, src, dist) q v2)
let remove_edge_e (g, src, dist) e =
if G.mem_edge_e g e then begin
G.remove_edge_e g e;
let v2 = G.E.dst e in
let (_, src', dist') = remove_edge_internal (g, !src) v2 in
src := src';
dist := dist';
dump (src, dist)
end
let remove_edge (g, src, dist) v1 v2 =
if G.mem_edge g v1 v2 then begin
G.remove_edge g v1 v2;
let (_, src', dist') = remove_edge_internal (g, !src) v2 in
src := src';
dist := dist';
dump (src, dist)
end
let remove_vertex (g, src, dist) v =
if G.mem_vertex g v then begin
G.iter_succ_e (fun e -> remove_edge_e (g, src, dist) e) g v;
G.iter_pred_e (fun e -> remove_edge_e (g, src, dist) e) g v;
G.remove_vertex g v;
src := S.remove v !src;
dist := M.remove v (M.map (M.remove v) !dist);
dump (src, dist)
end
let map_vertex f (g, src, dist) =
let map_map update m =
M.fold (fun v m acc -> M.add (f v) (update m) acc) m M.empty
in
let (g, src, dist) = (G.map_vertex f g,
S.fold (fun v acc -> S.add (f v) acc) !src S.empty,
let update = function
| None, _ as v -> v
| Some e, w ->
Some (E.create (f (E.src e)) (E.label e) (f (E.dst e))), w
in
map_map (map_map update) !dist) in
(g, ref src, ref dist)
let fold_pred_e f (g, _, _) = G.fold_pred_e f g
let iter_pred_e f (g, _, _) = G.iter_pred_e f g
let fold_succ_e f (g, _, _) = G.fold_succ_e f g
let iter_succ_e f (g, _, _) = G.iter_succ_e f g
let fold_pred f (g, _, _) = G.fold_pred f g
let fold_succ f (g, _, _) = G.fold_succ f g
let iter_pred f (g, _, _) = G.iter_pred f g
let iter_succ f (g, _, _) = G.iter_succ f g
let fold_edges_e f (g, _, _) = G.fold_edges_e f g
let iter_edges_e f (g, _, _) = G.iter_edges_e f g
let fold_edges f (g, _, _) = G.fold_edges f g
let iter_edges f (g, _, _) = G.iter_edges f g
let fold_vertex f (g, _, _) = G.fold_vertex f g
let iter_vertex f (g, _, _) = G.iter_vertex f g
let pred_e (g, _, _) = G.pred_e g
let succ_e (g, _, _) = G.succ_e g
let pred (g, _, _) = G.pred g
let succ (g, _, _) = G.succ g
let find_all_edges (g, _, _) = G.find_all_edges g
let find_edge (g, _, _) = G.find_edge g
let mem_edge_e (g, _, _) = G.mem_edge_e g
let mem_edge (g, _, _) = G.mem_edge g
let mem_vertex (g, _, _) = G.mem_vertex g
let in_degree (g, _, _) = G.in_degree g
let out_degree (g, _, _) = G.out_degree g
let nb_edges (g, _, _) = G.nb_edges g
let nb_vertex (g, _, _) = G.nb_vertex g
let is_empty (g, _, _) = G.is_empty g
let is_directed = G.is_directed
module Mark = struct
type graph = t
type vertex = G.vertex
let clear g = let (g, _, _) = g in G.Mark.clear g
let get = G.Mark.get
let set = G.Mark.set
end
end
module Persistent
(G: Sig.P)
(W: Sig.WEIGHT with type edge = G.E.t) = struct
module S = Set.Make(G.V)
module M = Map.Make(G.V)
module E = G.E
module V = G.V
type t = G.t * S.t * (G.E.t option * W.t) M.t M.t
type edge = G.edge
type vertex = G.vertex
exception Negative_cycle of G.E.t list
let empty : t =
let g = G.empty in
let src = S.empty in
let dist = M.empty in
(g, src, dist)
let add_vertex (g, src, dist) v =
if G.mem_vertex g v then
(g, src, dist)
else
(G.add_vertex g v),
(S.add v src),
(M.add v (M.add v (None, W.zero) M.empty) dist)
let rec propagate (g, src, dist) q start =
if Queue.is_empty q then (g, src, dist)
else begin
let (v1, v1src) = Queue.pop q in
let v1dist = M.find v1 dist in
let dist = G.fold_succ_e (fun e dist ->
let v2 = G.E.dst e in
let v2dist = M.find v2 dist in
let (v2dist, nextSrc) = S.fold (fun x (v2dist, nextSrc) ->
let _, dev1 = M.find x v1dist in
let ndev2 = W.add dev1 (W.weight e) in
let improvement =
try
let _, dev2 = M.find x v2dist in
W.compare ndev2 dev2 < 0
with Not_found -> true in
if improvement then
let v2dist = M.add x (Some e, ndev2) v2dist in
let nextSrc = S.add x nextSrc in
(v2dist, nextSrc)
else
(v2dist, nextSrc)
) v1src (v2dist, S.empty) in
if S.is_empty nextSrc then
dist
else if G.V.equal start v2 then
let s = S.choose nextSrc in
let rec build_cycle x ret =
match M.find s (M.find x dist) with
| Some e, _ ->
let y = G.E.src e in
let cycle = e :: ret in
if G.V.equal start y then cycle
else build_cycle y cycle
| _ -> assert false in
raise (Negative_cycle (build_cycle v2 []))
else
begin
Queue.push (v2, nextSrc) q;
M.add v2 v2dist dist
end
) g v1 dist in
propagate (g, src, dist) q start
end
let m_cardinal m = M.fold (fun _ _ acc -> acc+1) m 0
let set_of_map m = M.fold (fun k _ acc -> S.add k acc) m S.empty
let add_edge_internal (g, src, dist) v1 v2 =
let dv1 = M.find v1 dist in
let q = Queue.create () in
if m_cardinal dv1 = 1 && M.mem v2 dv1 then (
Queue.add (v1, (S.add v2 S.empty)) q;
propagate (g, src, dist) q v1
) else (
let (src, dist, dv1) =
if not (S.mem v2 src) then
(src, dist, dv1)
else
((S.remove v2 src),
(M.map (M.remove v2) dist),
(M.find v1 dist)) in
Queue.add (v1, set_of_map dv1) q;
propagate (g, src, dist) q v1)
let add_edge_e (g, src, dist) e =
if G.mem_edge_e g e then
(g, src, dist)
else begin
let v1 = G.E.src e in
let v2 = G.E.dst e in
let (g, src, dist) = List.fold_left
add_vertex (g, src, dist) [v1 ; v2] in
let g = G.add_edge_e g e in
add_edge_internal (g, src, dist) v1 v2
end
let add_edge (g, src, dist) v1 v2 =
if G.mem_edge g v1 v2 then
(g, src, dist)
else begin
let (g, src, dist) = List.fold_left
add_vertex (g, src, dist) [v1 ; v2] in
let g = G.add_edge g v1 v2 in
add_edge_internal (g, src, dist) v1 v2
end
let remove_edge_internal (g, src) v2 =
let q = Queue.create () in
let dist = S.fold (fun x dist ->
Queue.add (x, (S.add x S.empty)) q;
M.add x (M.add x (None, W.zero) M.empty) dist) src M.empty in
let g, src, dist = propagate (g, src, dist) q (S.choose src) in
if M.mem v2 dist then
(g, src, dist)
else (
Queue.add (v2, (S.add v2 S.empty)) q;
let src = S.add v2 src in
let dist = M.add v2 (M.add v2 (None, W.zero) M.empty) dist in
propagate (g, src, dist) q v2)
let remove_edge_e (g, src, dist) e =
if not (G.mem_edge_e g e) then
(g, src, dist)
else begin
let g = G.remove_edge_e g e in
let v2 = G.E.dst e in
remove_edge_internal (g, src) v2
end
let remove_edge (g, src, dist) v1 v2 =
if not (G.mem_edge g v1 v2) then
(g, src, dist)
else begin
let g = G.remove_edge g v1 v2 in
remove_edge_internal (g, src) v2
end
let remove_vertex t v =
let (g, _, _) = t in
let t = G.fold_succ_e (fun e t -> remove_edge_e t e) g v t in
let (g, _, _) = t in
let t = G.fold_pred_e (fun e t -> remove_edge_e t e) g v t in
let (g, src, dist) = t in
(G.remove_vertex g v,
(S.remove v src),
(M.map (M.remove v) dist))
let map_vertex f (g, src, dist) =
let map_map update m =
M.fold (fun v m acc -> M.add (f v) (update m) acc) m M.empty
in
(G.map_vertex f g,
S.fold (fun v acc -> S.add (f v) acc) src S.empty,
let update = function
| None, _ as v -> v
| Some e, w ->
Some (E.create (f (E.src e)) (E.label e) (f (E.dst e))), w
in
map_map (map_map update) dist)
let fold_pred_e f (g, _, _) = G.fold_pred_e f g
let iter_pred_e f (g, _, _) = G.iter_pred_e f g
let fold_succ_e f (g, _, _) = G.fold_succ_e f g
let iter_succ_e f (g, _, _) = G.iter_succ_e f g
let fold_pred f (g, _, _) = G.fold_pred f g
let fold_succ f (g, _, _) = G.fold_succ f g
let iter_pred f (g, _, _) = G.iter_pred f g
let iter_succ f (g, _, _) = G.iter_succ f g
let fold_edges_e f (g, _, _) = G.fold_edges_e f g
let iter_edges_e f (g, _, _) = G.iter_edges_e f g
let fold_edges f (g, _, _) = G.fold_edges f g
let iter_edges f (g, _, _) = G.iter_edges f g
let fold_vertex f (g, _, _) = G.fold_vertex f g
let iter_vertex f (g, _, _) = G.iter_vertex f g
let pred_e (g, _, _) = G.pred_e g
let succ_e (g, _, _) = G.succ_e g
let pred (g, _, _) = G.pred g
let succ (g, _, _) = G.succ g
let find_all_edges (g, _, _) = G.find_all_edges g
let find_edge (g, _, _) = G.find_edge g
let mem_edge_e (g, _, _) = G.mem_edge_e g
let mem_edge (g, _, _) = G.mem_edge g
let mem_vertex (g, _, _) = G.mem_vertex g
let in_degree (g, _, _) = G.in_degree g
let out_degree (g, _, _) = G.out_degree g
let nb_edges (g, _, _) = G.nb_edges g
let nb_vertex (g, _, _) = G.nb_vertex g
let is_empty (g, _, _) = G.is_empty g
let is_directed = G.is_directed
end