Genetic algorithms for solving bounded diameter minimum spanning tree problem

Genetic algorithms for solving bounded diameter minimum spanning tree problem Genetic algorithms for solving bounded diameter minimum spanning tree problem Genetic algorithms for solving bounded diameter minimum spanning tree problem luận văn tốt nghiệp,luận văn thạc sĩ, luận văn cao học, luận văn đại học, luận án tiến sĩ, đồ án tốt nghiệp luận văn tốt nghiệp,luận văn thạc sĩ, luận văn cao học, luận văn đại học, luận án tiến sĩ, đồ án tốt nghiệp

MINISTRY OF EDUCATION AND TRAINING HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY GENETIC ALGORITHMS FOR SOLVING BOUNDED DIAMETER MINIMUM SPANNING TREE PROBLEM By Huynh Thi Thanh Binh Supervisor: Associate Professor Nguyen Duc Nghia A Dissertation submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in Engineering HaNoi, 2011 Table of Contents Table of Contents i Abstract Acknowledgements Introduction 1.1 Motivation 1.2 Methodologies 1.3 Scope of research 14 1.4 Contributions 14 1.5 Outline 16 Bounded Diameter Minimum Spanning Tree and Related Works 18 2.1 Problem formulation 18 2.2 Related Optimization and Decision Problems 20 2.3 Related works 22 2.3.1 Exact approaches 23 2.3.2 Heuristic Methods 23 2.3.2.1 One Time Tree Construction Algorithm 24 2.3.2.2 Center-Based Tree Construction Algorithm 25 2.3.2.3 Randomized Greedy Heuristic Algorithm 25 ii 2.3.2.4 Improved Greedy Heurisitics (RGH − I and CBT C − I) 27 2.3.2.5 Hierarchical clustering heuristic algorithm - HCH 28 2.3.2.6 Comments 29 2.3.3 Metaheuristic algorithms 30 2.3.4 Conclusion 39 Center-Based Recursive Clustering Heuristic Algorithm 41 3.1 The new greedy heuristic - Center-Based Recursive Clustering (CBRC) 41 3.2 The improvement of Centre-Based Recursive Clustering - CBRC − I 44 3.3 Experiments 45 3.3.1 Problem instances 45 3.3.2 Experiment setup 46 3.3.3 Result 46 3.4 Discussion 55 3.5 Conclusion 56 Genetic algorithm with multi-parent recombination operator 57 4.1 Individual representation and genetic operators 58 4.2 Experiments 61 4.2.1 Problem instances 61 4.2.2 Experiment setup 62 4.2.3 System setting 62 4.2.4 Results and discussion 63 Conclusion 77 4.3 Multi-population Genetic Algorithm 79 5.1 Structure of the genetic algorithm 80 5.2 Experiments 83 iii 5.3 5.2.1 Problem instances 83 5.2.2 Experiment setup 83 5.2.3 System setting 84 5.2.4 Result 85 5.2.5 Discussion 85 Conclusion 95 Steady-state genetic algorithm 6.1 97 Steady state genetic algorithm structure 97 6.1.1 Individual representation and initial population 97 6.1.2 Crossover 98 6.1.3 Mutation 98 6.1.4 Selection 99 6.2 Replacement policy 99 6.3 Experiments 100 6.4 6.3.1 Problem instances 100 6.3.2 Experiment setup 101 6.3.3 Parameter 101 6.3.4 Result 101 Conclusion 106 Conclusion 108 Bibliography 111 iv List of Figures 1.1 Scheme of genetic algorithm 2.1 The BDST with 19 vertices and bounded diameter D=4, v is the center of the tree 2.2 27 The best BDST found by RGH algorithm on the Euclidean problem instance with n = 100, D = 10 2.7 27 The best BDST found by CBT C − I algorithm on the Euclidean problem instance with n = 100, D = 10 2.6 24 The best BDST found by CBT C algorithm on the Euclidean problem instance with n = 100, D = 10 2.5 20 The best BDST found by OT T C algorithm on the Euclidean problem instance with n = 100, D = 2.4 20 The BDST with 19 vertices and bounded diameter D=5, v1 , v2 are the centers of the tree 2.3 11 28 The best BDST found by RGH − I algorithm on the Euclidean problem instance with n = 100, D = 10 28 2.8 A spanning tree on twelve nodes and an its edge-set representation 31 2.9 A spanning tree on eleven nodes and an its permutation-code representation 33 2.10 Center Move Mutation 34 2.11 Edge Delete Mutation 34 2.12 Subtree-Optimize Mutation 35 v 2.13 The best BDST found by JR−ESEA algorithm on the Euclidean problem instance with n = 250, D = 15 39 2.14 The best BDST found by JR − P EA algorithm on the Euclidean problem instance with n = 250, D = 15 39 2.15 The best BDST found by P EA − I algorithm on the Euclidean problem instance with n = 250, D = 15 39 3.1 A star-like structure of a typical solution to the BDM ST problem 42 3.2 Greedy Edge Delete Local search 45 3.3 The best BDST found by CBRC heuristic on the Euclidean problem instance with n = 100, D = 10 3.4 The best BDST found by CBRC − I heuristic on the Euclidean problem instance with n = 100, D = 10 4.1 69 Comparison between the sum of the best solutions found by EA − xgk algorithm on all the problem instances (x = b, r, l; k = 2, 5, 7, 9) 4.7 68 Comparison between the sum of the best solutions found by EA − xdk algorithm on all the problem instances (x = b, r, l; k = 2, 5, 7, 9) 4.6 68 Comparison between the sum of the best solutions found by EA − xy9 algorithm on all the problem instances 4.5 68 Comparison between the sum of the best solutions found by EA − xy7 algorithm on all the problem instances 4.4 68 Comparison between the sum of the best solutions found by EA − xy5 algorithm on all the problem instances 4.3 45 Comparison between the sum of the best solutions found by EA − xy2 algorithm on all the problem instances 4.2 45 69 Comparison between the sum of the best solutions found by EA − xmk algorithm on all the problem instances (x = b, r, l; k = 2, 5, 7, 9) vi 69 4.8 Comparison between the sum of the average solutions found by EA − xy2 algorithm on all the problem instances (x = b, r, l) 4.9 69 Comparison between the sum of the average solutions found by EA − xy5 algorithm on all the problem instances 70 4.10 Comparison between the sum of the average solutions found by EA − xy7 algorithm on all the problem instances 70 4.11 Comparison between the sum of the average solutions found by EA − xy9 algorithm on all the problem instances (x = b, r, l) 70 4.12 Comparison between the sum of the average solutions found by EA − xdk algorithm on all the problem instances (x = b, r, l; k = 2, 5, 7, 9) 70 4.13 Comparison between the sum of the average solutions found by EA − xgk algorithm on all the problem instances (x = b, r, l; k = 2, 5, 7, 9) 71 4.14 Comparison between the sum of the average solutions found by EA − xmk algorithm on all the problem instances (x = b, r, l; k = 2, 5, 7, 9) 71 4.15 Comparision between the best solution found by GA1 , GA2 , GA3 , GA4 , GA5 , GA6 on all the problem instance 71 4.16 Comparision between the standard deviation of the solution found by GA1 , GA2 , GA3 , GA4 , GA5 , GA6 on all the problem instance 71 5.1 Multi-population model 80 5.2 The comparision between the best results found by GA11 , GA12 , GA13 , GA14 and HGA on the instance with n = 250, D = 15, instance 5.3 86 The comparision between the mean results found by GA11 , GA12 , GA13 , GA14 and HGA on the instance with n = 250, D = 15, instance 91 5.4 The number of individuals from GA11 , GA12 , GA13 , GA14 migrate to GAf inal 91 6.1 P EA − I algorithm vii 99 List of Tables 3.1 Diameter Bound 3.2 Results of OT T C, CBT C, RGH, CBRC, CBRC − I, RGH − I on the 46 Euclidean instances of the BDM ST problem with n = 100 and D = 5, 7, 9, 11, 13, 15 3.3 47 Results of OT T C, CBT C, RGH, CBRC, CBRC − I, RGH − I on the Euclidean instances of the BDM ST problem with n = 250 and D = 5, 10, 13, 15, 17, 20, 25 3.4 48 Results of OT T C, CBT C, RGH, CBRC, CBRC − I, RGH − I on the Euclidean instances of the BDM ST problem with n = 500 and D = 10, 15, 18, 20, 22, 25, 30 3.5 49 Results of OT T C, CBT C, RGH, CBRC, CBRC − I, RGH − I on the Euclidean instances of the BDM ST problem with n = 1000 and D = 15, 20, 23, 25, 27, 30, 35 3.6 50 Results of OT T C, CBT C, RGH, CBRC, CBRC − I, RGH − I on the Non-Euclidean instances of the BDM ST problem with n = 100 and D = 5, 7, 9, 11, 13, 15 3.7 51 Results of OT T C, CBT C, RGH, CBRC, CBRC − I, RGH − I on the Non-Euclidean instances of the BDM ST problem with n = 250 and D = 5, 10, 13, 15, 17, 20, 25 viii 52 3.8 Results of OT T C, CBT C, RGH, CBRC, CBRC − I, RGH − I on the Non-Euclidean instances of the BDM ST problem with n = 500 and D = 10, 15, 18, 20, 22, 25, 30 3.9 53 Results of OT T C, CBT C, RGH, CBRC, CBRC − I, RGH − I on the Non-Euclidean instances of the BDM ST problem with n = 1000 and D = 15, 20, 23, 25, 27, 30, 35 4.1 The rate of the heuristic algorithms use for initialization of the population in each experiment genetic algorithm 4.2 86 87 Comparision between the result found by RJ − ESEA, P EA − I, HGA, M HGA on the 20 Non-Euclidean problem instances 5.5 85 Comparision between the result found by RJ − ESEA, P EA − I, HGA, M HGA on the 20 Euclidean problem instances 5.4 67 Comparision between the result with different crossover probabily on the Euclidean problem instance with number of vertices are 250, D=15 5.3 66 Comparision between the result with different crossover probabily on the Euclidean problem instance with number of vertices are 250, D=15 5.2 65 Comparision between the result found by EA − xy9; x = d, g, m; y = l, r, b on the 20 Euclidean problem instances 5.1 64 Comparision between the result found by EA − xy7; x = d, g, m; y = l, r, b on the 20 Euclidean problem instances 4.5 Comparision between the result found by EA − xy5; x = d, g, m; y = l, r, b on the 20 Euclidean problem instances 4.4 63 Comparision between the result found by EA − xy2; x = d, g, m; y = l, r, b on the 20 Euclidean problem instances 4.3 54 88 Result of GA11 , GA12 , GA13 , GA14 and HGA on 20 Euclidean BDM ST problem instances ix 89 5.6 Result of GA11 , GA12 , GA13 , GA14 and HGA on 20 Non-Euclidean BDM ST problem instances 5.7 Result of GA21 , GA22 , GA23 , GA24 and M HGA on 20 Euclidean BDM ST problem instances 5.8 92 Result of GA21 , GA22 , GA23 , GA24 and M HGA on 20 Euclidean BDM ST problem instances 6.1 90 93 Results of P EA − RGH, P EA − RGHI, P EA − CBRC, P EA − CBRCI, P EA − I on the 20 Euclidean instances of the BDM ST problem of size 100, 250, 500 and 1,000 102 6.2 Average number of iterations required by P EA − RGH, P EA − RGHI, P EA − CBRC, P EA − CBRCI, P EA − I to reach the best solution on the 20 Euclidean instances of BDM ST problem of size 100, 250, 500 and 1,000 103 6.3 Results of P EA − RGH, P EA − RGHI, P EA − CBRC, P EA − CBRCI, P EA − I on the 20 Non-Euclidean instances of the BDM ST problem of size 100, 250, 500 and 1,000 6.4 104 Average number of iterations required by P EA − RGH, P EA − RGHI, P EA − CBRC, P EA − CBRCI, P EA − I to reach the best solution on the 20 Non-Euclidean instances of BDM ST problem of size 100, 250, 500 and 1,000 105 x Table 6.1: Results of P EA − RGH, P EA − RGHI, P EA − CBRC, P EA − CBRCI, P EA − I on the 20 Euclidean instances of the BDM ST problem of size 100, 250, 500 and 1,000 Instance PEA-RGH PEA-RGHI PEA-CBRC PEA-CBRCI PEA-I n k No Best Avg SD Best Avg SD Best Avg SD Best Avg SD Best Avg SD 100 10 7.76 7.82 0.03 7.76 7.81 0.02 7.78 7.84 0.03 7.76 7.81 0.02 7.76 7.82 0.03 100 10 7.85 7.88 0.04 7.85 7.86 0.03 7.85 7.88 0.03 7.77 7.83 0.03 7.85 7.89 0.04 100 10 7.92 7.96 0.04 7.91 7.96 0.03 7.92 7.99 0.04 7.91 7.95 0.02 7.90 7.97 0.04 100 10 7.98 8.04 0.03 7.98 8.01 0.02 7.98 8.03 0.03 7.73 7.85 0.02 7.98 8.04 0.03 100 10 8.16 8.20 0.03 8.16 8.17 0.02 8.16 8.20 0.03 8.16 8.18 0.02 8.16 8.21 0.03 250 15 12.43 12.43 0.04 12.25 12.32 0.04 12.25 12.38 0.05 12.21 12.26 0.04 12.24 12.36 0.05 250 15 12.08 12.17 0.05 12.05 12.05 0.00 12.08 12.17 0.05 12.10 12.15 0.04 12.04 12.13 0.04 250 15 12.05 12.09 0.03 12.00 12.03 0.02 12.05 12.12 0.04 11.98 12.03 0.03 12.03 12.11 0.05 250 15 12.47 12.58 0.05 12.47 12.58 0.05 12.48 12.59 0.06 12.44 12.51 0.04 12.42 12.57 0.05 250 15 12.26 12.38 0.05 12.34 12.34 0.00 12.28 12.38 0.05 12.22 12.33 0.05 12.28 12.39 0.05 500 20 16.97 17.10 0.07 16.85 16.94 0.04 16.97 17.10 0.06 16.30 16.41 0.07 16.96 17.13 0.06 500 20 16.86 16.96 0.05 16.75 16.82 0.05 16.84 16.98 0.07 16.74 16.82 0.05 16.81 16.99 0.07 500 20 16.91 17.03 0.06 16.81 16.90 0.04 16.90 17.02 0.06 16.82 16.82 0.05 16.89 17.04 0.06 500 20 16.98 17.10 0.07 16.90 16.96 0.04 16.98 17.06 0.05 16.87 16.97 0.06 16.96 17.10 0.06 500 20 16.60 16.72 0.06 16.52 16.56 0.04 16.58 16.69 0.07 16.51 16.56 0.06 16.58 16.72 0.06 1000 25 23.83 23.91 0.07 23.81 23.86 0.05 24.16 24.22 0.05 23.76 23.89 0.08 23.97 24.19 0.10 1000 25 23.69 23.74 0.05 23.51 23.58 0.07 23.86 23.92 0.06 23.54 23.66 0.07 23.70 23.98 0.13 1000 25 23.65 23.72 0.04 23.62 23.71 0.09 24.03 24.13 0.05 23.26 23.37 0.07 23.61 23.76 0.08 1000 25 23.99 24.13 0.11 23.73 23.86 0.09 24.03 24.13 0.05 23.71 23.88 0.09 24.04 24.16 0.07 1000 25 23.76 23.80 0.04 23.43 23.56 0.07 23.83 23.94 0.09 23.53 23.59 0.04 23.75 23.90 0.07 n: number of vetices k: diameter bound No: number of instance Best: Min weight of the best trees obtained by the algorithm over the runs Avg: Mean weight of the best tree obtained by the algorithm over the runs SD: Standard deviation of Best 102 Table 6.2: Average number of iterations required by P EA−RGH, P EA−RGHI, P EA− CBRC, P EA−CBRCI, P EA−I to reach the best solution on the 20 Euclidean instances of BDM ST problem of size 100, 250, 500 and 1,000 Instance PEA-RGH PEA-RGHI PEA-CBRC PEA-CBRCI PEA-I n k No Avg Itr Avg Itr Avg Itr Avg Itr Avg Itr 100 10 178,000 135,000 209,014 123,350 103,262 100 10 150,800 115,883 157,564 114,220 83,394 100 10 213,089 168,569 216,345 163,434 136,975 100 10 195,100 131,688 198,851 130,141 113,883 100 10 194,438 120,818 198,888 121,673 114,407 250 15 348,828 348,828 527,999 413,849 386,197 250 15 371,872 371,872 512,245 343,841 367,366 250 15 479,542 321,843 364,769 364,769 406,414 250 15 517,925 511,324 504,705 514,201 425,633 250 15 490,473 509,346 472,317 462,911 351,804 500 20 1,064,689 701,120 1,100,429 766,998 818,202 500 20 1,067,481 776,586 998,823 708,815 833,715 500 20 1,103,919 617,849 937,661 704,520 892,252 500 20 679,311 617,849 1,069,642 679,311 872,195 500 20 1,069,055 632,847 1,138,248 677,045 848,124 1000 25 1,275,042 1,307,843 1,781,575 1,228,143 1,768,532 1000 25 1,737,109 1,055,733 1,872,158 1,177,029 1,527,331 1000 25 1,926,683 1,092,697 1,709,077 1,211,080 1,783,477 1000 25 1,853,140 1,058,943 1,787,181 1,111,753 1,730,925 1000 25 2,171,615 1,333,279 1,763,359 1,197,522 1,681,034 n: number of vetices k: diameter bound No: number of instance Avg.Itr: Average iteration 103 Table 6.3: Results of P EA − RGH, P EA − RGHI, P EA − CBRC, P EA − CBRCI, P EA − I on the 20 Non-Euclidean instances of the BDM ST problem of size 100, 250, 500 and 1,000 Instance PEA-RGH PEA-RGHI PEA-CBRC PEA-CBRCI PEA-PEA-I n k No Best Avg SD Best Avg SD Best Avg SD Best Avg SD Best Avg SD 100 10 2.34 2.40 0.04 2.33 2.35 0.02 2.33 2.40 0.04 2.21 2.23 0.01 2.32 2.38 0.03 100 10 2.18 2.24 0.04 2.18 2.22 0.02 2.18 2.24 0.04 2.08 2.09 0.01 2.20 2.22 0.02 100 10 2.40 2.53 0.06 2.40 2.41 0.02 2.40 2.52 0.05 2.34 2.35 0.00 2.40 2.43 0.04 100 10 2.17 2.23 0.03 2.18 2.22 0.02 2.18 2.22 0.03 2.05 2.07 0.01 2.18 2.23 0.02 100 10 2.36 2.43 0.04 2.35 2.40 0.03 2.39 2.44 0.03 2.25 2.26 0.01 2.35 2.42 0.04 250 15 3.74 3.79 0.03 3.71 3.73 0.02 3.73 3.78 0.03 3.71 3.73 0.02 3.73 3.79 0.03 250 15 3.79 3.85 0.04 3.71 3.73 0.02 3.77 3.86 0.06 3.77 3.79 0.02 3.79 3.83 0.03 250 15 3.70 3.78 0.03 3.70 3.74 0.02 3.72 3.79 0.04 3.68 3.71 0.02 3.69 3.76 0.03 250 15 3.78 3.84 0.03 3.75 3.77 0.02 3.72 3.79 0.04 3.86 3.89 0.01 3.76 3.82 0.03 250 15 3.90 3.98 0.04 3.86 3.89 0.02 3.79 3.84 0.04 3.75 3.78 0.01 3.88 3.95 0.04 500 20 6.26 6.31 0.02 6.28 6.31 0.02 6.27 6.31 0.03 6.19 6.21 0.01 6.24 6.29 0.03 500 20 6.35 6.39 0.03 6.27 6.28 0.01 6.35 6.39 0.03 6.23 6.25 0.01 6.30 6.36 0.03 500 20 6.21 6.25 0.02 6.12 6.14 0.01 6.22 6.26 0.02 6.12 6.13 0.01 6.16 6.22 0.03 500 20 6.28 6.33 0.03 6.22 6.23 0.01 6.22 6.23 0.01 6.22 6.24 0.01 6.25 6.32 0.03 500 20 6.21 6.25 0.02 6.20 6.22 0.01 6.27 6.33 0.05 6.15 6.24 0.01 6.25 6.29 0.04 1000 25 11.31 11.36 0.02 11.21 11.22 0.01 11.35 11.39 0.03 11.20 11.22 0.01 11.26 11.31 0.03 1000 25 11.35 11.41 0.03 11.23 11.25 0.01 11.37 11.40 0.03 11.23 11.25 0.02 11.30 11.34 0.03 1000 25 11.38 11.41 0.03 11.23 11.26 0.01 11.35 11.42 0.05 11.25 11.26 0.01 11.30 11.35 0.03 1000 25 11.28 11.31 0.02 11.22 11.29 0.01 11.28 11.32 0.02 11.15 11.16 0.01 11.22 11.26 0.03 1000 25 11.44 11.49 0.03 11.30 11.32 0.01 11.42 11.49 0.04 11.31 11.32 0.01 11.39 11.42 0.03 n: number of vetices k: diameter bound No: number of instance Best: Min weight of the best trees obtained by the algorithm over the runs Avg: Mean weight of the best tree obtained by the algorithm over the runs SD: Standard deviation of Best 104 Table 6.4: Average number of iterations required by P EA−RGH, P EA−RGHI, P EA− CBRC, P EA − CBRCI, P EA − I to reach the best solution on the 20 Non-Euclidean instances of BDM ST problem of size 100, 250, 500 and 1,000 Instance PEA-RGH PEA-RGHI PEA-CBRC PEA-CBRCI PEA-I n k No Avg Itr Avg Itr Avg Itr Avg Itr Avg Itr 100 10 124,199 77,629 129,206 93,791 73,408 100 10 141,255 88,840 132,538 82,002 72,978 100 10 115,386 59,367 122,435 76,089 60,274 100 10 128,560 75,084 130,567 66,234 62,525 100 10 164,190 77,433 131,823 68,457 66,128 250 15 455,276 241,988 485,771 225,146 257,584 250 15 491,791 236,129 455,726 263,574 272,453 250 15 469,407 218,206 422,591 237,056 268,446 250 15 395,140 228,775 462,874 266,387 226,044 250 15 478,605 244,394 394,520 233,514 235,091 500 20 951,705 408,078 942,897 418,173 474,220 500 20 1,062,778 481,510 885,779 399,670 493,360 500 20 940,920 464,312 1,117,531 427,504 520,809 500 20 987,256 506,365 879,541 431,634 447,458 500 20 1,002,389 424,270 927,313 405,393 479,958 1000 25 1,105,176 768,487 934,756 740,707 939,944 1000 25 1,127,522 670,230 1,181,498 690,462 966,064 1000 25 1,038,656 752,934 1,080,855 715,921 898,320 1000 25 1,075,065 840,684 1,100,168 684,416 977,801 1000 25 1,159,085 702,703 1,095,189 684,111 948,178 n: number of vetices k: diameter bound No: number of instance Avg.Itr: Average iteration 105 by P EA − RGH, P EA − RGHI, P EA − CBRC, P EA − CBRCI, P EA − I are not different but it is clearly with large instances Table 6.2 shows the average number of iterations required by P EA−RGH, P EA−RGHI, P EA − CBRC, P EA − CBRCI, P EA − I to reach the best solution on the 20 Euclidean instances of BDM ST problem of size 100, 250, 500 and 1,000 On almost instances, the average iterations required by P EA − CBRCI to reach the best solution are smallest and the one required P EA − RGH are biggest Especially, on mall instances (n = 100), the average iterations required by P EA − I are smallest Table 6.3 shows the results of P EA − RGH, P EA − RGHI, P EA − CBRC, P EA − CBRCI, P EA − I on the 20 Non-Euclidean instances of the BDM ST problem of size 100, 250, 500 and 1,000 On almost instances, the best results found by P EA − RGH, P EA − RGHI, P EA − CBRC, P EA − CBRCI, P EA − I are not different The standard deviation values found by P EA − CBRCI are smallest Table 6.4 shows the average number of iterations required by P EA − RGH, P EA − RGHI, P EA − CBRC, P EA − CBRCI, P EA − I to reach the best solution on the 20 Non-Euclidean instances of BDM ST problem of size 100, 250, 500 and 1,000 On almost instances, the average iterations required by P EA − CBRCI to reach the best solution are smallest and the one required P EA − RGH are biggest Especially, on small instances (n = 100), the average iterations required by P EA − I are smallest 6.4 Conclusion In this chapter, we modify the decoder and the replacement policy used in P EA−I We use four decoders by different well-known heuristic algorithms: RGH, RGH − I, CBRC, CBRC −I and compare the results found by them The experimental results show that, on almost the instances, the results found by P EA−RGHI, P EA−CBRC, P EA− CBRCI are the best and the standard deviation found by P EA − CBRCI is smallest 106 The reason is that we use better heuristic in the decoding step in genetic algorithm for solving BDM ST 107 Conclusion The Bounded Diameter Minimum Spanning Tree (BDM ST ) problem is a hard combinatorial optimization problem that arises in many applications such as design of wire-based communication networks under quality of service requirements; in linear lightwave networks, where it can minimize interference in the network by limiting the traffic in the network lines Another practical application requiring a BDM ST arises in data compression, where some algorithms compress a file utilizing a tree data-structure, and decompress a path in the tree to access a record in ad-hoc wireless networks distributed mutual exclusion algorithms New approaches for solving BDM ST problem are presented in this thesis: Center-Based Recursive Clustering (CBRC) heuristic algorithm, multi-parent recombination operator, multi-population genetic algorithm, steady-state genetic algorithm for solving BDM ST These algorithms demonstrated their effectiveness in comparison to state-of-the-art approaches from the literature CBRC is based on RGH (and CBT C) and it extend the concept of center to each level of the partially constructed spanning tree The algorithm can be seen as recursively clustering the vertices of the graph: every in-node of the spanning tree is the center of the sub-graph of nodes in the subtree rooted at this node This thesis also survey the constraint between the weight of tree and bounded diameter The experiments and comparision between the result of CBRC and others - RGH, RGH − I, CBT C, OT T C, CBT C − I - on the Euclidean and Non-Euclidean instances up to 1000 vertices are presented The results show 108 the effectiveness of proposed algorithms Three multi-parent recombination operators in genetic algorithm for solving BDM ST problem are introduced The proposed multi-parent recombination operator can use more than two parents to create offspring Three different methods for choosing parents are considered: the first one is based on Levenshtein distance between the parents, the second one uses the best individual in the population and the last one uses randomly chosen individual in the population This thesis also experiment each method of choosing parents with three ways for adding edges from the parents into the offspring: choose the edge randomly, choose the edge which have minimum weight, choose the edge which have minimum weight in maximum sharing edge from the parents This thesis experiment on the Euclidean instances up to 1000 vertices and concentrate on analyzing the recombination operator in genetic algorithms The comparision between the results of proposed algorithms using, respectively, three mentioned multi-parent recombination operators with another genetic algorithm using two-parent recombination operator on the same problem are presented This thesis also present a new hybrid genetic algorithm for solving BDM ST problem The new genetic algorithm use a multi-population, where each population is initialized with a different well known heuristic The individuals in each population will subsequently compete for positions in a selection population, using a simulated annealing mechanism based on proportionate selection; in the selection population, they will combine and evolve toward the optimum Therefore, our research approach is to employ different initial biases by using different heuristics for initialization, and to hybridize the individuals from these populations to promote the exploratory capacity of the GA The comparision between our results with current best known genetic algorithms, namely, the genetic algorithm in [40] of Raidl and Julstrom (called RJ − ESEA), the genetic algorithm of Alok and Gupta in [46] (called P EA − I) and the genetic algorithm in each population on the Euclidean and Non-Euclidean instances up to 1000 vertices are presented The results show the effetiveness of proposed algorithm 109 Last but not least, steady-state genetic algorithms which use differents heuristic algorithms for decoding are introduced The modification of the decoder and the replacement policy used in P EA − I so as to improve its performance is presented Four decoders by different well-known heuristic algorithms: RGH, RGH − I, CBRC, CBRC − I are used The experiment on the Euclidean and Non-Euclidean instances up to 1000 vertices and the result show the outperform of proposed algorithm than the others Altogether, these are some interesting research challenges for the near future 110 Bibliography [1] Abdalla, A Computing a Diameter-constrained Minimum Spanning Tree PhD thesis, The School of Electrical Engineering and Computer 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Suppose T = (V, ET ) be a spanning tree of G Problem 1: Bounded Weighted Diameter Minimum Spanning Tree problem (BW DM ST ) Among all spanning trees of G whose weight of diameters not exceed a given... bound D, find the spanning tree with the minimal cost Problem 2: Minimum Weighted Diameter Bounded Spanning Tree problem (M W DBST ) Among all spanning trees of G whose weight of tree not exceed... S, find the spanning tree with the minimal weighted radius Problem 5: Bounded Weighted Diameter Bounded Spanning Tree problem (BW DBST ) Among all spanning trees of G whose weight of diameters