a markov chain monte carlo approach to the steiner tree problem in water network optimization
DESCRIPTION
A seminar about water network optimization via MCMC I gave in January at AIRO Winter 09TRANSCRIPT
Minimal water networks serving a farm districtA Metropolis algorithm for the Steiner Tree Problem
Carlo Lancia, Alessandro Checco
University of Rome TorVergata
Cortina D’Ampezzo, January 28, 2009
Carlo Lancia, Alessandro Checco Minimal Water Networks
Outline
1 A Model for the Shortest Network ProblemFarm Districts and Water NetworksThe Minimal Steiner Tree Problem
2 Metropolis AlgorithmThe Statistical Mechanics approach to OptimizationMarkov Chain Monte Carlo for the Steiner Tree Problem
3 Numerical Data and ConclusionsNumerical Comparison with Primal-DualConclusions
Carlo Lancia, Alessandro Checco Minimal Water Networks
A Model for the Shortest Network Problem Farm Districts and Water Networks
Introduction
DefinitionA farm district is a set of neighboring farmlands
DefinitionA water network is a set of pipes bringing water to the lands
ConstraintsWe look for the minimal water network such that
Its pipes lie on land boundaries only, so no land bridgingIt does reach at least a corner of each land boundary
Carlo Lancia, Alessandro Checco Minimal Water Networks
A Model for the Shortest Network Problem Farm Districts and Water Networks
Modeling the District
It is very difficult to model the shortest network problem on thegraph induced by the district topology.
Graph induced by the district
Carlo Lancia, Alessandro Checco Minimal Water Networks
A Model for the Shortest Network Problem Farm Districts and Water Networks
Modeling the District
We do prefer to work on a slightly different graph. =Steiner Nodes; =Distribution Node; ◾ =Terminal Nodes
Creation of a new graph G
Carlo Lancia, Alessandro Checco Minimal Water Networks
A Model for the Shortest Network Problem The Minimal Steiner Tree Problem
Water Networks and Steiner Trees
Water networks are equivalent to Steiner Trees on G:
Example
Carlo Lancia, Alessandro Checco Minimal Water Networks
A Model for the Shortest Network Problem The Minimal Steiner Tree Problem
Water Networks and Steiner Trees
Water networks are equivalent to Steiner Trees on G:
Example
Carlo Lancia, Alessandro Checco Minimal Water Networks
A Model for the Shortest Network Problem The Minimal Steiner Tree Problem
The Shortest Network Problem
Some NotationWe call VS the set of Steiner nodes, VT the set of TerminalsA Steiner Tree is any tree T ⊂ G spanning VTThe cost vector c ∶ E → R+ is such that
c(u) = c(v) for fictitious edges u, v situated in the same landfictitious-edge cost is comparable with regular-edge cost
Formulation as Minimal Steiner Tree ProblemOn graph G a water network serving the district is a SteinerTree, so what we are looking for is the shortest Steiner Treehaving root in , with the additional requirement of terminalnodes being leaf nodes (no land crossings allowed)
Carlo Lancia, Alessandro Checco Minimal Water Networks
A Model for the Shortest Network Problem The Minimal Steiner Tree Problem
The Shortest Network Problem
Some NotationWe call VS the set of Steiner nodes, VT the set of TerminalsA Steiner Tree is any tree T ⊂ G spanning VTThe cost vector c ∶ E → R+ is such that
c(u) = c(v) for fictitious edges u, v situated in the same landfictitious-edge cost is comparable with regular-edge cost
Minimal Steiner Tree Problem – NP-hard
minT ⊂G
∑x ∈E(T )
c(x)
s.t. VT ⊂ V (T )deg(v) = 1 ∀ v ∈ VT
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm The Statistical Mechanics Approach
The Statistical Mechanics Approach
Optimization Problem
Problem instance
Cost function, f(ξ)
Optimal solutions
Many Particles System
Particles configuration
Hamiltonian, H(ξ)
Ground states
All we need is a sampling algorithm
When β →∞ Gibbs distributione−βH(ξ)
Zis concentrated on
ground states, that is to say optimal configurations
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm The Statistical Mechanics Approach
The Statistical Mechanics Approach
Optimization Problem
Problem instance
Cost function, f(ξ)
Optimal solutions
Many Particles System
Particles configuration
Hamiltonian, H(ξ)
Ground states
All we need is a sampling algorithm
When β →∞ Gibbs distributione−βH(ξ)
Zis concentrated on
ground states, that is to say optimal configurations
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm The Statistical Mechanics Approach
The Statistical Mechanics Approach
Optimization Problem
Problem instance
Cost function, f(ξ)
Optimal solutions
Many Particles System
Particles configuration
Hamiltonian, H(ξ)
Ground states
All we need is a sampling algorithm
When β →∞ Gibbs distributione−βH(ξ)
Zis concentrated on
ground states, that is to say optimal configurations
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm The Statistical Mechanics Approach
The Statistical Mechanics Approach
Optimization Problem
Problem instance
Cost function, f(ξ)
Optimal solutions
Many Particles System
Particles configuration
Hamiltonian, H(ξ)
Ground states
All we need is a sampling algorithm
When β →∞ Gibbs distributione−βH(ξ)
Zis concentrated on
ground states, that is to say optimal configurations
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm The Statistical Mechanics Approach
The Statistical Mechanics Approach
Optimization Problem
Problem instance
Cost function, f(ξ)
Optimal solutions
Many Particles System
Particles configuration
Hamiltonian, H(ξ)
Ground states
All we need is a sampling algorithm
When β →∞ Gibbs distributione−βH(ξ)
Zis concentrated on
ground states, that is to say optimal configurations
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm The Statistical Mechanics Approach
MCMC Sampling
Let Ω be the set of all Steiner trees over GDefine a Markov chain (Xt) on Ω that is ergodic with limitmeasure equal to Gibbs distribution
Sampling Algorithm1 Follow the evolution of the
chain for a very long time τ2 Return min
t∈[0,τ]Xt
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm The Statistical Mechanics Approach
MCMC Sampling
Let Ω be the set of all Steiner trees over GDefine a Markov chain (Xt) on Ω that is ergodic with limitmeasure equal to Gibbs distribution
Sampling Algorithm1 Follow the evolution of the
chain for a very long time τ2 Return min
t∈[0,τ]Xt
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm MCMC for the Steiner Tree Problem
Cooking Up An Appropriate Markov Chain
Target
We want a chain which limit measure is π(ξ) = e−βH(ξ)
Z
Transition ProbabilitiesMetropolis rule allow us to design a chain that is reversiblewith respect to Gibbs distribution
P (ξ, η) = minπ(η)π(ξ) ,1 = exp−β [H(η) −H(ξ)]+Such design strategy is very general and independent ofthe functional form of Hamiltonian
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm MCMC for the Steiner Tree Problem
Which Hamiltonian?
It is quite natural to choose the Hamiltonian as the cost functionof the problem, that is the length of tree ξ
H(ξ) =∑x∈ξ
c(x)
Upgrading the Hamiltonian
We rather prefer to work with the following Hamiltonian function
H(ξ) =∑x∈ξ
c(x) + h ∑v∈VT
(deg(v) − 1)
where the second sum is extended to all terminal vertices.
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm MCMC for the Steiner Tree Problem
Which Hamiltonian?
Land Bridging Penalties
Employing this Hamiltonian, Steiner trees crossing one or morelands have energy proportional to h. When temperature is lowand h is large such configurations are nearly never sampled.
Upgrading the Hamiltonian
We rather prefer to work with the following Hamiltonian function
H(ξ) =∑x∈ξ
c(x) + h ∑v∈VT
(deg(v) − 1)
where the second sum is extended to all terminal vertices.
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm MCMC for the Steiner Tree Problem
Chain Evolution
At time t denote current tree by Xt. Select an edge x ∈ E(G)u.a.r., then set Xt+1 according to the following rule:
Current tree
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm MCMC for the Steiner Tree Problem
Chain Evolution
Selected edge is the red one. Removing body edges from thetree would result in two separate componentsÔ⇒Xt+1 =Xt
Body edges are not removed from current tree
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm MCMC for the Steiner Tree Problem
Chain Evolution
Adding edges non-adjacent to current tree would result in twoseparate components Ô⇒ Xt+1 =Xt
Separate edges are not added to current tree
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm MCMC for the Steiner Tree Problem
Chain Evolution
When edge x connects a terminal leaf to current tree we setXt+1 =Xt not to compromise solution feasibility
Branch edges connecting terminal leaves are never removed
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm MCMC for the Steiner Tree Problem
Chain Evolution
If x connects a steiner leaf to current tree we cut it out and setXt+1 =Xt ∖ x
Branch edges connecting steiner leaves are always removed
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm MCMC for the Steiner Tree Problem
Chain Evolution
If edge x is adjacent to current tree we may add it. We setXt+1 =Xt ∪ x with probability exp−β c(x)
Edges adjacent to current tree may be added to it
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm MCMC for the Steiner Tree Problem
Chain Evolution
In case adding x to the tree results in a land crossing we setXt+1 =Xt ∪ x with probability exp−β(c(x) + h)
Attention must be paid to land crossing and penalty applied
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm MCMC for the Steiner Tree Problem
Chain Evolution
When adding x to the tree would result in a loop we observethe following procedure which removes the cycle:
Edges introducing a loop may be added by swap procedure
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm MCMC for the Steiner Tree Problem
Chain Evolution
We select uniformly at random an edge y of the loop adjacentto x, then swap x and y with probability exp−β [c(x) − c(y)]+
Edges introducing a loop may be added by swap procedure
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm MCMC for the Steiner Tree Problem
Chain Evolution
We select uniformly at random an edge y of the loop adjacentto x, then swap x and y with probability exp−β [c(x) − c(y)]+
Swap procedure: edge (+) is added and (–) is removed
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm MCMC for the Steiner Tree Problem
Chain Evolution
In case adding x to the tree results in a land crossing loop swapprocedure must be performed more carefully:
Attention must be paid to possible land crossing
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm MCMC for the Steiner Tree Problem
Chain Evolution
Interchanging edges may lead to a land crossing, in such caseswe swap x and y with probability exp−β [c(x) − c(y) + h]+
Swapping must be penalised if it leads to a land bridging
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm MCMC for the Steiner Tree Problem
Chain Evolution
Interchanging edges may lead to a land crossing, in such caseswe swap x and y with probability exp−β [c(x) − c(y) + h]+
Swapping (+) and (–) leads to bridging Ô⇒ penalty imposed
Carlo Lancia, Alessandro Checco Minimal Water Networks
Metropolis Algorithm MCMC for the Steiner Tree Problem
Chain Ergodicity
The chain we have just described is found to beirreducible and aperiodicIt has a unique stationary distribution, namely Gibbs one
limt→∞
P t(ξ, η) = e−βH(η)
Z
Proving irreducibility
Swap procedure grants that the chain can move from a Steinertree to another, stepping through feasible solutions only
Carlo Lancia, Alessandro Checco Minimal Water Networks
Numerical Data and Conclusions Numerical Comparison with Primal-Dual
Numerical Comparison with Primal-Dual
Both Metropolis and Primal-Dual procedure were implementedin Fortran 90 and run on a 1.6 GHz Intel Dual Core Processor
Shortest Network Length (km)
Metropolis Primal-Dual
Test District #1 4,288 5,867Test District #2 6,052 7,123Test District #3 6,588 9,353Test District #4 16,359 20,378Test District #5 5,131 6,660Test District #6 8,123 10,176Test District #7 16,044 22,222
Carlo Lancia, Alessandro Checco Minimal Water Networks
Numerical Data and Conclusions Numerical Comparison with Primal-Dual
Numerical Comparison with Primal-Dual
Algorithm Execution Time (sec)
lands edges Metropolis Primal-Dual
T-D #1 110 909 8 58T-D #2 132 1091 11 62T-D #3 201 1670 16 71T-D #4 458 3858 23 380T-D #5 169 1413 15 66T-D #6 274 2450 20 75T-D #7 605 5117 38 744
Total runtime 131 1456
Carlo Lancia, Alessandro Checco Minimal Water Networks
Numerical Data and Conclusions Numerical Comparison with Primal-Dual
Test District #1 - Primal Dual
Carlo Lancia, Alessandro Checco Minimal Water Networks
Numerical Data and Conclusions Numerical Comparison with Primal-Dual
Test District #1 - Metropolis
Carlo Lancia, Alessandro Checco Minimal Water Networks
Numerical Data and Conclusions Numerical Comparison with Primal-Dual
Test District #3 - Primal Dual
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Carlo Lancia, Alessandro Checco Minimal Water Networks
Numerical Data and Conclusions Numerical Comparison with Primal-Dual
Test District #3 - Metropolis
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Carlo Lancia, Alessandro Checco Minimal Water Networks
Numerical Data and Conclusions Numerical Comparison with Primal-Dual
Test District #7 - Primal Dual
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752
739740
806
747
742
744
755
745
772
759
748
750751
761
754
789
756
768
760
822
763
765
778776
766
767
801
770 771
856
774775
830
793
853
813
779780
839
781
782
849
783784
787786
791
877
788
878
790
871
792
820
794
795
819
797798 799
800
802
821
803
805
831
807
845
808814
827
809 811 812
828
816 817
826
818
865
835
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857
823825824
861
854
832
909
833
837
834
842843
836
901
838
935
846
840841
844
889
847848
850
851852
867
858
855
937
859862
917
860
874
882
898
863864
866
875
868869870
872
873876
888
885
879
883
880 881
932
884
890891
886887
906
894
892
908
893
923
897
933
895
896
900
931
913
899
903
928
902
904905
925
907
910
911
919920
912
918914 915 916
940
929
936
921922
926
967
924
927
930
934
941
938
939
942
979
943
944
945946
947948
950
955
949
954953
951
952
956 957
966
958959
965
960 961
1018
962
1019
964 963
968
1017
970
983
973
971
1029
969
978977
972
974975
980
976
984
1028
986
1037
988
981982985
1031
987
1054
1003
989990
1056
991
1057
992993 994
998
995
1058
996997
1034
999
1061
1000
1001 1002
1042
1004
1064
1005
100610071008100910101011
1079
10121013
1077
101410151016
1036
1045
10201021 10221027 1023 1025
1101
1024
1041
1105
1026
11001098
1032 1030
1074
10331035
1071
1038 1039 1040
1053
1103
1046
1117
1043
1047
1044
1055
1171
1119
1048
1050
1121
1049
1065
1051
1052
1059
1060
10631062
1102
1085
1069
10661067
1068
1092
1070
1086
1072
1115
10731075
1076 1078 1080 1081 1082
1149
1083
1148
1084
1146
1087
1132
1088
10901089
1116
1095
1091
1093
1094 1096 1097
1120
1099
1165
1129
110411061108 11071109
1113
1110111111121114
1118
1131
1122
1123
11241125
1130
1128
1145
11261127
1157
1133 1134 1135
1155
11361137
1172
1138 1141 1139 1140
1167
1169
1142
1143
1144
1147
1150
1152
1151
1153
1154
1176
1156
1158
1159
1160
1161
1162
1163
1164
1166
1168
1170
1173
1177
1174
1175
Carlo Lancia, Alessandro Checco Minimal Water Networks
Numerical Data and Conclusions Numerical Comparison with Primal-Dual
Test District #7 - Metropolis
1
2
3
43
4
5
9
6
7 8
49
10
24
11
38
1213
79
15 14
17
19
102
16
18
22
20
23
21
33
26
111
25
99
27
48
28
52
29
53
30
54
32
57
31
92
34
58
3637
59
35
87
80
39
60
40
62
77
41
71
4244
45
117
46 475051
76
5556
67
61
86
63
64
68
65
163
66
74
113
6970
81
72
108
73
84
123
75
88
83
78
82
91
89
85
97
131
95
112
105
101
90
106
93
100
103
94
9896
193
125124
104
107
166
109
110
115
144
114
126
182
116
120
118
119
127
142
121 122
154
129
171
156
128
130
136
148
132
147
133134135
141
137
138
200
139 140
143
151
229
145 146
152
149
162
150
155
158
197
153
159
157
160161
227
164
173
165
168
230
167
170169
174
172
176
175
181
177
178 179
180
204
183184
198
185186
250
187 188 189
296
190191
326
192
285
194195 196
223
207
249
199
201202
311
206
203
208
205
211
228
362
214
209210 213 212
215
219
237
216
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329
218
225
220
221
233
222
341
224
232
226
242
367
245
231
236
248
342
234 235
282
244
238
239
243
240
246
241
301
259
252
254
247
256
251
277
267
253255
262
257
260258
261
265
360
263
271
453
264
288
270
452
266
275
398
268
269
416
290
397
272273
292
274
304
276
280
300
278
279
297
291
281
331
325
283284
286
302
287
289
371
293
389
294295
305303
298 299
312
380
440
317
306 307308309
368
310
314313
316
321
315
323
438
322
337
318 319320
336
476
327 324328
350
333330
332
334
465
335
338
418
344
357
339340
411
343
345
348346
412
347
425
358
349351
352353
354355356
363
359361
366
401
364
428
365
441
376
386
446449
369
370
372 373374
439
375
385
404
377 378 379
422
381
382383
448
384
393
391
392
387
447
388
485
390
396
480
394395
464
403
409
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430
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414
445
417419
420421 423
427
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431
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436
429 432
433434435
442443
471
466
444
467
463
451 450
454456 455
521
457458
459
488
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501
462
486
532
491
487
468
530
469
500
509
470
472 473474
541
475
477 478
543
479
537
481482483 484
505
499
513
527
489490
492 493
536
494
495 496497
550
498
520
525
508
502503
540
504
529
566
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514
507
557
510
511
563
512
515516
598
517518
524
519
522523
590
526
581
538
587
528
547
531
533
588
535534
582
553
544
539
542
545546
584
609
551548 549
552
570
554
576
555556
559 558
562
594
561
560
578579
577
564
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567 568
569
571
586
572
574573
592
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606
624
580
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671
583
620
591
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604
672
613
616
597
623
602
599600
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622
621
631
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645
610 611
625
612
615614
632
617618619
626
663
629
633
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644
628
745
643
638
680
634 635636
660
637
690
661
670
640
641
647
642
662
646 648
649650
681
654
732
651
723
652653657
655
734
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658 659
666
692
664
689
669
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667 668
695
673
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698
679
742
736
744
682
694
683 684685686 687 688
691
713
708
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722
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700 699
727
704
731
701
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703
729
714
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773
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762
758
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710
794
711712
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715 716
717
740
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808
719
743
720
779
721
724
810
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728
730739733 735737 738
757
748
741
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801
749 750751753 752
761
796
769
759
807
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822
763
766765
764
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781
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857
768
770 771
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830
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833
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777 778
816
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866
782 783
852
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871
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880
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804805 806
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823
813
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850
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892
889 890
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923
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900
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925
906
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997
913
921
914 915 916
1007
917 918
940
919
936
927
928
930
929
931
982
933
934
938
951
942
941
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945946
947948
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965
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1018
961 962 964 963
1014
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1028
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1054
1003
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1056
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1057
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1001
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1058
1059
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1061
1000
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1064
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1010
1081
1011
1079
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1077
1015
1076
1045
10171024
1019 1020
1094
1021 1022 10231026
1101
1031
1041
1025 10291027 10301032
10361033
1035 1034
1071
1037
1109
1038 1039
1053
1040
1103
1046
1042
1043
1044
1049
1047
1119
1048
1050
1123
1052
1126
1055
1060
1070
10631062
1065
1088
10661067
10681069
1086
1087
1115
1072
1073
1112
10741075
1154
1078 1080 1082
1083 1084
1085
1132
10901089
1095
10911092
1093
1122
1096 1097 1098 1099 1100
1165
11051102
1129
110411061108
1160
1107
1161
1113
1110
1162
1111
1163
1114
1156
1164
1127
1116
1117
1118
1120
1121
1143
11241125
1128
1157
1158
1152
1130
1131
1142
1133 1134 1135 1136
1175
1137 1138
1172
1141
1170
1139 1140
11681167
1169
1146
1144
1153
1145
11481147
1149
1151
1150
1155
1176
1159
1166
1171
1173
1174
1177
Carlo Lancia, Alessandro Checco Minimal Water Networks
Numerical Data and Conclusions Conclusions
Conclusions
Final RemarksMetropolis algorithm appears to be more efficient in virtueof better solutions and shorter run-timeIt does not depend on fictitious-edge weightIt needs an initial Steiner tree to take its first move→ a brief high-temperature evolution seems to be a nice fix
Further developmentsMixing time of the chain, possibly via the coupling methodNumerical comparison on a larger data set
Carlo Lancia, Alessandro Checco Minimal Water Networks
Numerical Data and Conclusions Conclusions
Conclusions
Final RemarksMetropolis algorithm appears to be more efficient in virtueof better solutions and shorter run-timeIt does not depend on fictitious-edge weightIt needs an initial Steiner tree to take its first move→ a brief high-temperature evolution seems to be a nice fix
Further developmentsMixing time of the chain, possibly via the coupling methodNumerical comparison on a larger data set
Carlo Lancia, Alessandro Checco Minimal Water Networks