CHAPTER 6 DEVELOPMENT OF PARTICLE SWARM OPTIMIZATION BASED ALGORITHM FOR GRAPH PARTITIONING

CHAPTER 6 DEVELOPMENT OF PARTICLE SWARM OPTIMIZATION BASED ALGORITHM FOR GRAPH PARTITIONING 6.1 Introduction From the review, it is studied that the min cut k partitioning problem is a fundamental partitioning problem and is NP hard also. Most of the existing partitioning algorithms are heuristic in nature and they try to find a reasonably good solution. These algorithms falls in move based category in which solution is generated iteratively from an initial solution applying move to the recent solution. Most frequently, these move based approaches are combined with stochastic algorithms In this chapter, we have developed Multilevel Recursive Discrete Particle Swarm Optimization (MRDPSO) technique which integrates a new DPSO based refinement approach and an efficient matching based coarsening scheme for solving GPP. 6.2 Discrete Particle Swarm Optimization The original PSO algorithm optimizes problems in which the elements of solution space are continuous. Since most of the real life applications are based on optimization of discrete valued space, Kennedy et al [163] developed discrete PSO (DPSO) to solve discrete optimization problems. The PSO algorithm regulates the trajectories of population particles through a problem space using

information about the previous best performance of the particle and its neighbor. In PSO, trajectories represent changes in positions of some number of dimensions, whereas in DPSO particles operate on discrete search space and trajectories represents variations in probability which a coordinate takes a value from reasonable discrete values. In this method, each particle specifies the potential solution composed of k elements. The fitness function is used to evaluate the correctness of the solution. Each particle is considered as a position in k dimensional space and each element of particle is restricted to 0 and 1, where 0 represents included and 1 represents not included. Each element can vary from 0 to 1 and vice versa. Additionally, each particle will have k dimensional velocity between the range [- Vmax, Vmax]. Velocities are defined using probabilities that a bit will be in one state or the other. At the initial stage of the algorithm, the number of particles and their velocity vectors are generated randomly. Then in a certain number of iterations, the algorithm aims to find the optimal or close optimal solutions using predefined fitness functions. At each iteration, update the velocity vector using best positions p best,n best and then update the position of particles using velocity vector. p best and n best are k dimensional vectors composed of 0 and 1 which operates as a memory of the algorithm. Since the initial step of algorithm, best position that the particle has visited is p best and the position that the particle and its neighbor has visited is n best. Depending on size of neighborhoods, two distinct PSO algorithms can be developed. If the entire population size of the swarm is considered as the neighbor of particle, then n best is called as global best ( g best ) whereas, if for each particle smaller neighborhood are defined then n best is called

as local best (l best ). Star neighborhood and ring neighborhood are the topologies used by g best and l best respectively. PSO based on g best converges faster than l best based PSO due to its larger particle connectivity, but l best based PSO is less vulnerable to being trapped in local minima. Velocity and position of particle are updated using following equations: ( ) = ( ) + ( ) ( ) + ( ) ( ) (6.1) ( ) = 1, ( ) >, ( ) (6.2) Sigmoid function is given by the relation Sig( ( )) = ( ( )) (6.3) where ( ) - n th element of m th particle in t th iteration of algorithm. ( ) - n th element of the velocity vector of m th particle in t th iteration of algorithm. and and and are positive accelerated constants that controls the influence of on the search process. [0, 1] are random values sampled from a uniform distribution, and [0, 1] is random number. Stopping criteria for DPSO can either be the maximum number of iterations or determining acceptable solution or no further improvement in the number of iterations.

Discrete Particle Swarm Optimization algorithm is Create and initialize k dimensional swarm with N particles Repeat for each particle i = 1, 2,., N if ( ) > then // f represents the fitness function end = ; if > then end = ; end for each particle i = 1, 2,., N do update the velocity vector using equation (6.1) end update the position vector using equation (6.2) until stopping criteria is satisfied. 6.3 DPSO based Mathematical Model of Graph Partitioning Problem Let G = (V, E) be a weighted undirected graph with set of vertices V and E be set of edges, the weight of an edge ( ) and k be natural number greater than 1, then the graph partitioning problem (GPP) is to partition vertex set V into k - blocks,,, such that = and =,. If all blocks have the same weight, then the partition is balanced. For balanced partitioning = (1+ ) for {1,2,, } If = 0, then the partitioning is perfectly balanced. If >, then the block is overloaded.

Objective function for the graph partitioning problem is to minimize the cut of partition cut between two sets and which are subsets of is, = (, ) (6.4) Selection of Discrete PSO for GPP We have developed technique based on discrete particle swarm optimization (DPSO) algorithm to explore good quality approximate solutions of the min cut partitioning problem for graph G = (V, E) with p vertices and q edges. In the problem, each particle of the swarm is considered as partitioning vector where the solution space is p dimensional. Obviously, cut ( ) is the fitness function to be minimized. If the scale of the swarm is N then the position of n th particle will be represented by a p dimensional vector =,,,, {1, 2,, } where each element represents d th dimension of position vector which is restricted to the values zero and one.the velocity of this particle is denoted by another p dimensional vector =,,, in which each particle indicates the probability of the bit possessing the value one. The best formerly visited position of n th particle is represented by the vector =,,,. The best formerly visited position of the swarm is represented by the vector, =,,,, index of best particle in the swarm is i. Let k be iteration number, then the velocity and position used in our DPSO are defined by =. + [0, ] + [0, ] (6.5)

= 1, >, Sigmoid function is Sig( ) = ( ) (6.7) is function following uniform distribution which returns vectors whose positions are randomly selected. represents point wise multiplication of vectors. and are positive constants called as cognitive and social parameter repsectively. [0, 1] generates random number selected from a uniform distribution. The standard discrete PSO is not much effective approach since the space for feasible solutions of min cut partitioning problem is excessively large, particularly when the number of vertices is in the thousands. Hence, we have chosen multilevel partitioning method to combine with discrete PSO and developed, multiple recursive discrete particle swarm optimization algorithm (MRDPSO) for min cut partitioning. As MRDPSO algorithm is applied during the refinement phase of the multilevel method; it improves the quality of level m graph Gm using the boundary refinement policy for partitioning PGm. In addition to this, discrete PSO grouped with local search optimization and hybrid PSO local search is developed which applies the Fiduccia Mattheyses algorithm [55] to each particle for achieving local optimum and maintain the balance constraint of partitioning. This hybrid approach permits the search process to divert from local minima, as well simultaneously reinforces efficiency and achieves a noteworthy speedup for balanced partitioning.

6.4 Core Sort Heavy Edge Matching Sorted heavy edge matching (SHEM) sorts vertices of the graph in ascending order using their degrees to decide the order to visit for matching. We have matched the vertex u with vertex v, if there is an edge of maximum weight from vertex u to an unmatched vertex v. Highly connected groups of vertices are identified and collapsed together in SHEM, this method is discussed in detail in Section 2.5.1. A core number of the vertex is a maximum order of core which contains that vertex [164]. To find core number, the vertices of the graph are to be arranged in ascending degrees, then for each vertex v identified the vertices adjacent to v whose degree is greater than v. Degree of all such vertices is reduced by 1. The process continued until all vertices get core number. The concept of graph core for coarsening power law graphs is introduced in [165]. An algorithm for finding Core Number: Input: Graph G = (V, E) Output: Table representing core number for each vertex Begin Compute degrees of vertices; arrange vertices in ascending order of their degrees; for each in the order do begin [ ] = [ ]; for each h ( ) do begin If [ ] > [ ] then begin [ ] = [ ] 1; end reorder V accordingly; end if end for end for return table with core number for each vertex.

Core sort heavy edge matching algorithm (CSHEM) combines SHEM w ith the concept of core sort. Initially, to decide order to visit for matching by CSHEM, we have sorted vertices of a graph in descending order using core numbers of the vertices. During the process of matching, if tie occurred, according to edge weights, then the vertex with highest core number is selected. An algorithm for Core Sort Heavy Edge Matching: Input: Graph = (, ) Output: Coarsened graph = (, ) Procedure coarsening = (, ) while do Randomly select a vertex Select the (, ) of maximal weight Remove, from Add the new vertex = {, } For each edge (, ) in E do if (, ) then (, ) (, ) + (, ) Remove (, ) from end if end for end while return = (, ) end procedure

6.5 Greedy Graph Growing Partition (GGGP) Graph growing partition algorithm (GGP) introduced by Karypis [17] iteratively generates set A of vertices consisting of half of the vertices such that ( ) = ( ). During implementation of algorithm vertices of graph are divided into three sets A, B, C where B contains vertices that are adjacent to vertices of set A called as border of A, and C contains remaining vertices. Set A is initialized by randomly selecting any vertex of the graph, at each iteration vertices that are adjacent to A are added to A, hence =. The process ends when A will contain set of vertices which represent half the total weight of the graph. Partition generated by GGP is dependent on proper choice of initial vertex. GGP algorithm has been improved by using the concept of cut gain [18] of the vertex and developed a greedy graph growing partition (GGGP). In GGGP also, vertices of graph are divided into three sets A, B, C as in the case of GGP algorithm. A is initialized by randomly selecting any vertex from V and sets B, C are then initialized. For initialization of set A, selected nearest vertex (say u) from the set B to the vertex in set A and then added it to A which is vertex of maximal gain in B. After that, each vertex in set C which is adjacent to u is moved to the set B and calculated its gain. Similarly, recalculate the gain of each vertex in B that is adjacent to u and therefore the next iteration starts. Process continued till the weight of set A reaches to half of the total weight. Algorithm ends when ( ) = ( ). GGGP algorithm generates better result for any choice of the initial vertex to move.

An algorithm for Greedy Graph Growing Partition: Input: Graph = (, ) Output: Set A such that ( ) = ( ) Randomly select a vertex { } { } { h h (, ) } Compute the gains of while ( ) < ( ) do Select the of maximal gain Move from for each edge (, ) in E do if then update the gain of else if then add to compute the gain of end if end for end while return end procedure 6.6 Multilevel Recursive Discrete Particle Swarm Optimization (MRDPSO) Algorithm To develop an algorithm for achieving superior quality approximate solutions of the min cut partitioning problem for graphs, we have used a hybrid approach in which multilevel partitioning method is combined with DPSO.

Our Multilevel recursive discrete particle swarm optimization algorithm (MRDPSO) works in three steps. The first step is an initial portioning phase, in which MRDPSO initializes population on the smallest graph. In the second step i.e, during the refinement phase, it successively projects all the particles back to the next level finer graph. Whereas, in a third step, it recursively partitions bisected graph into k - parts. For the initial partitioning of the given graph G (V, E), we have used GGGP approach. Effective matching based coarsening scheme is applied during the coarsening phase. In this phase, CSHEM algorithm is used on the original graph and SHEM is applied to the coarsened graphs. For graphs with small number vertices, use of core number yields matching same as that of SHEM. Hence, for the graphs with less than fifteen vertices directly SHEM is applied to the original graph. Position vector, velocity vector, and personal best vector on the graph = (, ) for each n th particle are,, and respectively. Initial MDPSO initializes the population on in partitioning phase and projects back successively all the particles,,, to the next level finer graph. In the next stage, internal and external weights for each particle are determined. Internal weight for the n th particle of the vertex xis denoted as and is the sum of weights of the edges incident on the vertex x within the block and external weight for the n th particle of the vertex xis denoted as and is the sum of weights of the edges incident on the vertex x outside the block. = (, ) (, ) (6.8) = (, ) (, ) (6.9)

Boundary vertices for the each particle with positive external weight are stored in a boundary hash table. An initialization phase starts at time t = 0 in which, internal weight, external weight and boundary hash table is computed. The structure of MRDPSO consists of a nested loop. Stopping criteria is decided by the outer loop, either MRDPSO is run for maximum number of cycles or not. Velocity of each particle is to be adjusted by boundary refinement policy in the inner loop of MRDPSO. Enforce maximum value on the velocity, if velocity exceeds the threshold value, then set it. Move vertices between the partitions and update particle s position, this moving may break balance constraint. To maintain the balance constraint of, we prefer to select vertex with highest gain to move from larger block. During these moves, it is very important to inspect and compare boundary vertices by the space which permits to store, recover and update the gains of vertices speedily. In this space, we have used last-in and first-out scheme to achieve an efficient MRDPSO. Internal and external weights of the particle play an important role in computing gain and boundary vertex for the easy implementation of MRDPSO. At each iteration, the weights of all the neighboring vertices of the moved vertex are updated for maintaining the consistency in internal and external weights. The boundary hash table also changes with respect to changes in partitioning. In the third step, we apply recursive algorithm for k partitioning of the bisected graph generated in the first two steps of MRDPSO.

6.7 Result and Discussion To assess the performance of the developed multilevel recursive discrete particle swarm optimization MRDPSO, we carried out experiment on ordinary graphs as well as hypergraphs. For the first one, we have used Walshaw Graph Partitioning Test Bench [82] as characterized in Table 6.1. Graph Size Degree Max Min Avg add20 2395 7462 123 1 6.23 data 2851 15093 17 3 10.59 3elt 4720 13722 9 3 5.81 uk 4824 6837 3 1 2.83 add32 4960 9462 31 1 3.82 bcsstk33 8738 291583 140 19 66.76 whitaker3 9800 28989 8 3 5.92 crack 10240 30380 9 3 5.93 wing-nodal 10937 75488 28 5 13.80 fe_4elt2 11143 32818 12 3 5.89 vibrobox 12328 165250 120 8 26.8 bcsstk29 13992 302748 70 4 43.27 4elt 15606 45878 10 3 5.88 fe_sphere 16386 49152 6 4 5.99 cti 16840 48232 6 3 5.73 memplus 17758 54196 573 1 6.10 cs4 22499 43858 4 2 3.90 bcsstk30 28924 1007284 218 3 69.65 bcsstk31 35588 572914 188 1 32.197 fe_pwt 36519 144794 15 0 7.93 bcsstk32 44609 985046 215 1 44.1636 fe_body 45097 163734 28 0 7.26 t60k 60005 89440 3 2 2.98 wing 62032 121544 4 2 2,57 brack2 62631 366559 32 3 11.71 finan512 74752 261120 54 2 6.99 fe_tooth 78136 452591 39 3 11.58 fe_rotor 99617 662431 125 5 13.30 598a 110971 741934 26 5 13.37 fe_ocean 1433437 409593 6 1 5.71 Table 6.1: WalshawGraph Partitioning Test Benchmark

For hypergraphs ISPD98 benchmark suit [166] is used. The details of graphs are given in Table 6.2. Hypergraphs Vertices Hyperedges ibm01 12752 14111 ibm02 19601 19584 ibm03 23136 27401 ibm04 27507 31970 ibm05 29347 28446 ibm06 32498 34826 ibm07 45926 48117 ibm08 51309 50513 ibm09 53395 60902 ibm10 69429 75196 ibm11 70558 81454 ibm12 71076 77240 ibm13 84199 99666 ibm14 147605 152772 ibm15 161570 186608 ibm16 183484 190048 ibm17 185495 189581 ibm18 210613 201920 Table 6.2: ISPD98 Benchmark Suit 6.7.1 Performance Evaluation for Ordinary Graphs We have compared partitions obtained by our MRDPSO algorithm in short computing time limited to three minutes with state of the-art graph partitioning packages METIS [167], CHACO [168] and multilevel iterated tabu search [169]

Additionally, we have also compared our results with the best partitions ever reported in partitioning archive. Benchmark graphs along with their characteristics are as listed in Table 6. 1. We have used multilevel p METIS algorithm for METIS, multilevel KL algorithm for CHACO with recursive bisection is chosen. For MITS parameters values are: = 0.1, = 0.01, 0.02 and coarsening threshold being 100. METIS and CHACO does not permit randomized repetitions to run the algorithm, we chose to run MRDPSO only once. Furthermore, the optimal set of parameter values chosen for MRDPSO is = 1, = = 0.5, = 4, = 30, = 20. The cutoff limit varies with size of graphs. It is from one second for the graphs with upto 4000 vertices to three minutes for the largest graphs. In the results, METIS, CHACO, tabu search and our approach are labeled as p - METIS, CHACO, MITS and MRDPSO, respectively. Cut value for partition of each graph for different values of k are given in Table 6.3, 6.4, 6.5 and 6.6. If partition is perfectly balanced, then the balance constraint has value = 1. For the situations in which the partition is not perfectly balanced, balance constraint value is also mentioned in parenthesis along with cut value. The number of times each algorithm produces the best partition over the benchmark graphs is given in the last row of each table. CSHEM used for coarsening in MRDPSO immensely reduces the size of graph which helps GGGP for generating better balanced partition. Whereas implementation of DPSO at the most complex and time consuming refinement stage helps a lot to reduce cut value in small cutoff limit also. Hence from the results, we can observe that our MRDPSO approach performs very well in terms of cut value than the other three approaches. From the results, we can also observe that for 32, few partitions by p METIS, MITS and our approach are not perfectly balanced but in our approach this imbalance also occurs very less than other two.

Graph k = 2 k = 4 P - METIS CHACO MITS MRDPSO P - METIS CHACO MITS MRDPSO add20 729 742 708 522 `1292 1329 1224 3 data 218 199 189 183 480 433 427 357 3elt 108 103 90 90 231 234 214 185 uk 23 36 23 15 67 69 47 33 add32 21 11 11 11 42 56 40 37 bcsstk33 10205 10172 10171 10168 23131 23723 22492 20989 whitaker 135 131 127 119 406 425 385 365 3 crack 187 225 184 179 382 445 371 343 wingnodal 1820 1823 1797 1782 4000 4022 3715 3439 fe_4elt2 130 144 130 125 359 402 423 314 vibrobox 12427 11367 11184 9897 21471 21774 19811 18589 bcsstk29 2843 3140 2852 2847 8826 9202 8572 8371 4elt 154 158 139 127 406 433 390 299 fe_sphere 440 424 386 375 872 852 774 697 cti 334 372 366 354 1113 1117 1039 837 memplus 6337 7549 5696 5391 10559 11535 9982 9672 cs4 414 517 377 352 1154 1166 987 991 bcsstk30 6458 6563 9812 6298 17685 17106 22436 15931 bcsstk31 3638 3391 2820 2520 8770 9199 8751 6954 fe_pwt 366 362 360 340 738 911 1249 753 bcsstk32 5672 6137 6936 4421 12205 15704 9864 8674 fe_body 311 1036 271 273 957 1415 728 487 t60k 100 91 86 68 255 235 226 219 wing 950 901 861 657 2086 1982 1770 1599 brack2 738 976 731 719 3250 3462 3291 2878 finan512 162 162 162 162 324 325 405 313 fe_tooth 4297 4642 3827 3803 8577 8430 7460 6334 fe_rotor 2190 2151 2122 1998 8564 8215 7765 6799 598a 2504 2465 2402 2297 8533 8975 8159 8092 fe_ocean 505 499 468 439 2039 2110 2850 1785 Total 3 2 3 27 1 0 1 28 Table 6.3: Comparison of MRDPSO with p - METIS, CHACO, and MITS for k = 2, 4

Graph k = 8 k = 16 P - METIS CHACO MITS MRDPSO P - METIS CHACO MITS MRDPSO add20 1907 1867 1750 1537 2504 2297 2120 1995 data 842 783 679 621 1370 1360 1167 1095 3elt 388 389 352 337 665 660 585 549 uk 101 119 113 76 189 211 163 118 add32 81 115 74 59 128 174 143 99 bcsstk33 40070 39070 34568 34681 59791 61890 55539 52727 whitaker3 719 765 672 639 1237 1218 1120 1103 crack 773 777 720 611 1255 1253 1157 1159 wingnodal 6070 6147 5481 5387 9290 9273 8405 8189 fe_4elt2 654 718 621 625 1152 1135 1039 976 vibrobox 28177 33362 24840 23897 37441 43064 34392 30949 bcsstk29 16555 18158 17014 12887 28151 28029 26055 21356 4elt 635 688 615 497 1056 1083 1005 842 fe_sphere 1330 1302 1243 1099 2030 2037 1855 1942 cti 2110 2102 1838 1763 3181 3083 3033 2908 memplus 13110 14265 12640 12341 14942 16433 14097 12344 cs4 1746 1844 1529 1387 2538 2552 2293 1967 bcsstk30 36357 37406 36373 33741 77293 81069 79265 75333 bcsstk31 16012 15551 15262 15685 27180 28557 25787 22899 fe_pwt 1620 1670 1531 1389 2933 3200 2857 2789 bcsstk32 23601 25719 24435 19987 43371 47829 39902 35742 fe_body 1348 2277 1293 981 2181 2947 2076 2098 t60k 561 524 522 389 998 977 937 765 wing 3205 3174 2686 2476 4666 4671 4188 3747 brack2 7844 8026 7644 7432 12655 13404 12240 10995 finan512 810 648 729 689 1377 1296 1458 1087 fe_tooth 13653 13484 12083 10985 19346 20887 18336 15435 fe_rotor 15712 15244 13558 11037 23863 23936 21687 18767 598a 17276 17530 16270 13887 28922 29674 26565 26189 fe_ocean 4516 5309 4272 3995 9613 9690 8397 6758 Total 0 1 2 27 0 0 2 28 Table 6.4: Comparison of MRDPSO with p - METIS, CHACO, and MITS for k = 8, 16

Graph k = 32 P - METIS CHACO MITS MRDPSO add20 NAN 2684 2524 (1.03) 2243 data 2060 (1.01) 2143 1933 (1.01) 1732 (1.01) 3elt 1093 1106 1053 972 uk 316 (1.01) 343 296 (1.01) 221 add32 288 (1.01) 303 266 (1.01) 198 (1.01) bcsstk33 86008 84613 90438 70535 whitaker3 1891 1895 1758 1572 crack 1890 1962 1741 1632 wing-nodal 13237 13258 12238 10989 fe_4elt2 1787 1796 1688 1672 vibrobox 46112 51006 47048 (1.01) 37847 bcsstk29 41190 42935 38346 32417 4elt 1769 1766 1631 1689 fe_sphere 2913 2920 2701 2731 cti 4605 4532 4479 3989 memplus 17303 17936 NAN 12811 cs4 3579 3588 3137 2639 bcsstk30 131405 128694 117414 108331 bcsstk31 42645 45354 40029 38256 fe_pwt 6029 6036 6596 5138 bcsstk32 70020 73377 64138 58003 fe_body 3424 4194 3290 2491 t60k 1613 1594 1539 1549 wing 6700 6843 6067 5831 brack2 19786 20172 18411 16511 finan512 2592 2592 2592 2592 fe_tooth 29215 29849 26110 22887 fe_rotor 36225 36367 32746 30999 598a 44760 45780 40980 35767 fe_ocean 14613 15059 13358 11789 Total 1 1 4 27 Table 6.5: Comparison of MRDPSO with p - METIS, CHACO, and MITS for k = 32

Graph k = 64 P - METIS CHACO MITS MRDPSO add20 3433 (1.07) 3349 3219 (1.03) 2789 (1.01) data 3116 (1.03) 3145 2924 (1.07) 2645 3elt 1710 1722 1606 (1.03) 1487 uk 495 (1.02) 540 496 (1.03) 395 add32 626 (1.02) 730 571 (1.01) 442 (1.01) bcsstk33 116203 (1.01) 115530 131639 (1.05) 116317 whitaker3 2796 (1.01) 2811 2628 2231 crack 2847 (1.01) 2904 2628 (1.01) 2139 wing-nodal 17899 (1.01) 17783 16258 (1.01) 13669(1.01) fe_4elt2 2765 (1.01) 2781 2590 2591 vibrobox 53764 (1.01) 58392 54503 44665 bcsstk29 62891 (1.01) 63576 59548 (1.01) 54489 (1.01) 4elt 2953 2921 2676 (1.01) 2517 fe_sphere 4191 4151 3776 3789 cti 6461 6334 6181 5441 memplus 19140 (1.01) 18978 NAN 15899 cs4 4791 4817 4286 3972 bcsstk30 191691 191446 175845 (1.02) 168436 bcsstk31 66526 68375 61154 62003 fe_pwt 9310 9231 8487 8054 bcsstk32 106733 108855 96197 88673 fe_body 5843 6326 5097 4613 t60k 2484 2506 2345 1854 wing 9405 9308 8123 7389 brack2 28872 29223 27130 26442 finan512 10842 11962 11077 9782 fe_tooth 40162 40306 35988 32327 fe_rotor 53623 52947 48206 42489 598a 64307 65094 57303 53092 fe_ocean 23317 22692 21212 18985 Total 0 0 3 27 Table 6.6: Comparison of MRDPSO with p - METIS, CHACO, and MITS for k = 64

As it will be noticed from the second experiment that for some graphs more computational time, hence more iterations are required to establish balance partition. Whereas, CHACO generates perfectly balanced partitions for all values of k due to use of recursive bisection. We have also compared the performance of the developed algorithm with best balanced partitions which are reported in the Graph Partitioning archive [82]. Most of these results are generated by the algorithm developed by Schulz et al [170], and their algorithm combines an evolutionary approach with JOSTEL multilevel method. This approach requires large running time (around one week on normal machine for larger graphs), since each run consists of almost 50,000 calls. Other best results are reported by approaches used in [171, 172]. For the second experiment we have increased cutoff limits ranging from one minute for the smallest graph to one hour for largest graph and run MRDPSO algorithm ten times for each value of k for evaluating update in cut value. In table 6.7, 6.8 and 6.9 cut values for the best balance partitions, ever reported in partitioning archive, cut values generated by MRDPSO and standard deviations after ten executions of our algorithm are given. From the results we can be observed that the imbalance is completely removed due increase in cutoff time and the number of executions, but at the same time we can observe that the cut values are same in most of the cases though the time limit is increased. Hence use of DPSO in partitioning gives an optimal solution in less time and less executions also. Cut values generated by MRDPSO are better than best partitions also.

Graph BEST k = 2 k = 4 MRDPSO SD BEST MRDPSO SD (y) (y) (Y add20 596 522 52.3 1154 1089 46 data 189 183 4.24 382 357 17.7 3elt 90 90 0 201 185 11.3 uk 19 15 2.83 41 33 5.66 add32 11 11 0 34 37 2.12 bcsstk33 10171 10168 2.12 21717 20989 515 whitaker3 127 119 5.66 388 365 11.3 crack 184 179 3.54 366 343 16.3 wing-nodal 1707 1782 53 3575 3439 96.2 fe_4elt2 130 125 3.54 349 314 24.7 vibrobox 10343 9897 315 18976 18589 274 bcsstk29 2843 2847 2.83 8035 8371 238 4elt 139 127 8.49 326 299 19.1 fe_sphere 386 375 7.78 768 697 50.2 cti 334 354 14.1 954 837 82.7 memplus 5513 5391 86.3 9448 9672 158 cs4 369 352 12 932 991 29 bcsstk30 6394 6298 67.9 16651 15931 509 bcsstk31 2762 2520 171 7351 6954 281 fe_pwt 340 340 0 705 753 21.2 bcsstk32 4667 4421 174 9314 8674 453 fe_body 262 273 7.78 599 487 79.2 t60k 79 68 7.78 209 219 7.07 wing 789 657 93.3 1623 1599 17 brack2 731 719 8.49 3084 2878 146 finan512 162 162 0 324 313 7.78 fe_tooth 3816 3803 9.19 6889 6334 392 fe_rotor 2098 1998 70.7 7222 6799 299 598a 2398 2297 71.4 8001 8092 64.3 fe_ocean 464 439 17.7 1882 1785 68.6 Total 4 26 5 25 Table 6.7: Comparison of MRDPSO with BEST reported cut for k = 2, 4

Graph BEST k = 8 k = 16 MRDPSO SD BEST MRDPSO SD (y) (y) (Y add20 1686 1537 105 2047 1995 36.77 data 668 621 33.2 1127 1095 22.63 3elt 345 337 5.66 573 549 16.97 uk 84 76 5.66 146 118 19.8 add32 67 59 5.66 118 99 13.44 bcsstk33 34437 34681 173 54680 52727 1381 whitaker3 656 639 12 1088 1103 10.61 crack 679 611 48.1 1088 1159 50.2 wing-nodal 5435 5387 33.9 8334 8189 102.5 fe_4elt2 607 625 12.7 1007 976 21.92 vibrobox 24484 23897 415 31892 30949 666.8 bcsstk29 23986 12887 848 21958 21356 425.7 4elt 545 497 33.9 934 842 65.05 fe_sphere 1156 1099 40.3 1714 1942 121.6 cti 1788 1763 17.7 2793 2908 81.32 memplus 11712 12341 445 12895 12344 389.6 cs4 1440 1387 37.5 2075 1967 76.37 bcsstk30 34846 33741 781 70440 75333 3460 bcsstk31 13285 15685 1697 23869 22899 685.9 fe_pwt 1447 1389 41 2830 2789 28.99 bcsstk32 20070 19987 58.7 36250 35742 359.2 fe_body 1033 981 36.8 1736 2098 256 t60k 456 389 47.4 813 765 33.94 wing 2504 2476 19.8 3876 3747 91.22 brack2 7140 7432 206 11644 10995 458.9 finan512 648 689 29 1296 1087 147.8 fe_tooth 11418 10985 306 17355 15435 1358 fe_rotor 12841 11037 1276 20391 18767 1148 598a 15922 13887 1439 25792 26189 280.7 fe_ocean 4188 3995 136 7713 6758 675.3 Total 6 24 6 24 Table 6.8: Comparison of MRDPSO with BEST reported cut for k = 8, 16

Graph k = 32 k = 64 BEST MRDPSO SD BEST MRDPSO SD add20 2362 2243 84.1 2959 2684 120.208 data 1799 1699 47.4 2839 2645 137.179 3elt 960 972 8.49 1532 1487 31.8198 uk 254 221 23.3 408 395 9.19239 add32 213 191 10.6 485 419 30.4056 bcsstk33 77414 70535 4864 107247 116317 213.46 whitaker3 1668 1572 67.9 2491 2231 183.848 crack 1679 1632 33.2 2541 2139 284.257 wing-nodal 11768 10989 551 15775 13443 1489.17 fe_4elt2 1614 1672 41 2478 2591 37.4767 vibrobox 39477 37847 1153 46653 44665 1405.73 bcsstk29 34968 32417 1804 55521 54376 729.734 4elt 1551 1689 43.8 2566 2517 34.6482 fe_sphere 2490 2731 170 3547 3789 100.409 cti 4049 3989 42.4 5630 5441 133.643 memplus 13953 12811 808 16223 15899 229.103 cs4 2928 2639 204 4027 3972 38.8909 bcsstk30 113443 108331 3615 171279 168436 2010.3 bcsstk31 37158 38256 776 57402 62003 3253.4 fe_pwt 5575 5138 309 8202 8054 104.652 bcsstk32 60038 58003 1439 90895 88673 1571.19 fe_body 2846 2491 251 4801 4613 132.936 t60k 1323 1549 123 2077 1854 157.685 wing 5594 5831 168 7625 7389 166.877 brack2 17387 16511 619 25808 26442 448.306 finan512 2592 2592 0 10560 9782 550.129 fe_tooth 24885 22887 1413 34240 32327 1352.7 fe_rotor 31141 30999 100 45687 42489 2261.33 598a 38581 35767 1990 56097 53092 2124.86 fe_ocean 12684 11789 633 20069 18985 766.504 Total 6 24 3 27 Table 6.9: Comparison of MRDPSO with BEST reported cut for k = 32, 64

Figures 6.1 and 6.2 shows improvement in the partitioning cut obtained by MRDPSO relative to the best cut reported in partitioning archive for k =2, 4, 8 and for k = 16, 32, 64, respectively. Points lying below 1.0 in curves indicate that MRDPSO performs better than other approaches. It can be observed that our approach shows an improvement in 86%, 83%, 80%, 80%, 80% and 90% for k = 2, 4, 8, 16, 32 and 64, respectively. The results show that overall performance of the developed MRDPSO algorithm is noteworthy for generating balanced partition in significantly less time. 1.4 Relative Cut 1.2 1 0.8 0.6 0.4 0.2 0 add20 data 3elt uk add32 bcsstk33 whitaker3 crack wing-nodal fe_4elt2 vibrobox bcsstk29 4elt fe_sphere cti memplus cs4 bcsstk30 bcsstk31 fe_pwt bcsstk32 fe_body t60k wing brack2 finan512 fe_tooth fe_rotor 598a fe_ocean k = 2 k = 4 k = 8 Fig 6.1: Relative improvement in the partitioning cut for k = 2, 4, 8

1.4 Relative Cut 1.2 1 0.8 0.6 0.4 0.2 0 add20 data 3elt uk add32 bcsstk33 whitaker3 crack wing-nodal fe_4elt2 vibrobox bcsstk29 4elt fe_sphere cti memplus cs4 bcsstk30 bcsstk31 fe_pwt bcsstk32 fe_body t60k wing brack2 finan512 fe_tooth fe_rotor 598a fe_ocean k = 16 k = 32 k = 64 Fig 6.2: Relative improvement in the partitioning cut for k = 16, 32, 64 6.7.2 Performance Evaluation for Hypergraphs We have used 18 hypergraphs from ISPD98 benchmark suit with vertices range from 12752 to 210613 and hyperedges range from 14111 to 201920 for performance evaluation of our algorithm. The characteristics of these hypergraphs are listed in Table 6.2. We compared results produced by MRDPSO algorithm with the results derived by METIS recursive bisection (hmetis RB) [173] and METIS k way partitioning (hm ETIS k way) [174]. We have used CSHEM algorithm during the coarsening phase. GGGP algorithm is used to consistently detects smaller edge cuts than the other algorithms during initial

partitioning phase and boundary KL for refinement and then MRDPSO is applied. To guarantee the statistical significance of results, we run the algorithm fifteen times. Furthermore, the optimal set of parameter values is chosen for MRDPSO as = 1, = = 0.5, = 4, = 30, = 20 and balance constraint is = 1.0. Table 6.10, 6.11, and 6.12 gives the number of hyperedges that are cut during the partitioning by MRDPSO algorithm, METIS recursive bisection (hmetis RB) and METIS k way partitioning (hm ETIS k way) for k = 8, 16 and 32 partitions, respectively. In the last row, the total time required by each method for all eighteen hypergraphs is given in minutes. From the results, it can be observed that MRDPSO gives optimal cut value in very less computing time. This is because of the fact that DPSO refinement heuristic is able to perform an excellent job of optimizing the objective function, as it is applied successively to finer coarsen graphs. Furthermore, results produced indicate that, the MRDPSO offers an additional benefit of producing high quality partitioning while enforcing tight balancing constraints. Figures 6.3 and 6.4 shows improvement in the partitioning cut obtained by MRDPSO relative to the METIS recursive bisection (hmetis RB) and METIS k way partitioning (hmetis k way) for k = 8, 16 and 32,respectively. Points below 1.0 indicate that MRDPSO performs better than hmetis RB and hmetis k way. The results illustrate that MRDPSO produces partitions whose cut is far better than the cut obtained by the other two methods.

Hypergraph k =8 hmetis RB hmetis k way MRDPSO ibm01 760 795 382 ibm02 1720 1790 930 ibm03 2503 2553 1363 ibm04 2857 2902 2323 ibm05 4548 4464 2592 ibm06 2452 2397 938 ibm07 3454 3422 2417 ibm08 3696 3544 3893 ibm09 2756 2680 1349 ibm10 4301 4263 2580 ibm11 3592 3713 2873 ibm12 5913 6183 4139 ibm13 3042 2744 1825 ibm14 5501 5244 4125 ibm15 6816 6855 4089 ibm16 6871 6737 5496 ibm17 9341 9420 7753 ibm18 5310 5540 4981 Run - Time 364.53 175.86 113.35 Table 6.10: Comparison of MRDPSO with hmetis RB and hmetis k way for k = 8

Hypergraph k = 16 hmetis RB hmetis k way MRDPSO ibm01 1258 1283 632 ibm02 3150 3210 1732 ibm03 3256 3317 1963 ibm04 3989 3896 3187 ibm05 5465 5612 3943 ibm06 3356 3241 1231 ibm07 4804 4764 3663 ibm08 4916 4718 5187 ibm09 3902 3968 2189 ibm10 6190 6209 4376 ibm11 5260 5371 4063 ibm12 8540 8569 6872 ibm13 5522 5329 2953 ibm14 8362 8293 6761 ibm15 8691 9201 5991 ibm16 10230 10250 8331 ibm17 15088 15206 11445 ibm18 8860 9025 7931 Run - Time 431.902 237.125 192.571 Table 6.11: Comparison of MRDPSO with hmetis RB and hmetis k way for k = 16

Hypergraph k = 32 hmetis RB hmetis k way MRDPSO ibm01 1723 1702 936 ibm02 4412 4380 2342 ibm03 4064 4120 2987 ibm04 5094 5050 4185 ibm05 6211 5948 3341 ibm06 4343 4231 1642 ibm07 6300 6212 4899 ibm08 6489 6154 6503 ibm09 5502 5490 2986 ibm10 8659 8612 6138 ibm11 7514 7534 6841 ibm12 11014 11392 9322 ibm13 7541 7610 3941 ibm14 12681 12838 10532 ibm15 13342 13853 9643 ibm16 15589 15335 11189 ibm17 20175 19812 14767 ibm18 13410 13102 12890 Run - Time 505.42 326.20 274.459 Table 6.12: Comparison of MRDPSO with hmetis RB and hmetis k way for k = 32

Improvement in Cut 1.2 1 0.8 0.6 0.4 0.2 0 k=8 k = 16 k = 32 Fig 6.3: Improvement in the partitioning cut obtained by MRDPSO relative to hmetis RB for k = 8, 16, 32 Improvement in Cut 1.2 1 0.8 0.6 0.4 0.2 0 k=8 k = 16 k = 32 Fig 6.4: Improvement in the partitioning cut obtained by MRDPSO relative to hm ETIS k way for k = 8, 16, 32

Table 6.13 gives improvement percentage in METIS RB and METIS k way corresponding to the results obtained by MRDPSO algorithm, for k = 8, 16 and 32 partitions. The average improvement in METIS RB is 32.14%, 29.41% and 28.76% for k = 8, 16 and 32 respectively. As well as average improvement in METIS k way is 32.14%, 29.41% and 28.76% for k = 8, 16 and 32 respectively. Hypergraph Improvement % in METIS RB Improvement % in METIS k way k = 8 k = 16 k = 32 k = 8 k = 16 k = 32 ibm01 49.73% 49.76% 45.67% 51.94% 50.74% 45.00% ibm02 45.93% 45.01% 46.91% 48.04% 46.04% 46.52% ibm03 45.54% 39.71% 26.50% 46.61% 40.82% 27.5% ibm04 18.69% 20.10% 17.84% 19.95% 18.19% 17.12% ibm05 43.00% 27.85% 46.20% 41.93% 29.73% 43.82% ibm06 61.74% 63.31% 62.19% 60.86% 62.01% 61.19% ibm07 30.02% 23.75% 22.23% 29.36% 23.11% 21.13% ibm08-5.33% -5.51% -0.21% -9.84% -9.94% -5.67% ibm09 51.05% 43.90% 45.72% 49.66% 44.83% 45.61% ibm10 40.01% 29.30% 29.11% 39.47% 29.52% 28.72% ibm11 20.01% 22.75% 8.95% 22.62% 24.35% 9.19% ibm12 30.00% 19.53% 15.36% 33.05% 19.80% 18.17% ibm13 40.00% 46.52% 47.73% 33.49% 44.58% 48.21% ibm14 25.01% 19.14% 16.94% 21.33% 18.47% 17.96% ibm15 40.00% 31.06% 27.72% 40.35% 34.88% 30.39% ibm16 20.01% 18.56% 28.22% 18.42% 18.72% 27.03% ibm17 17.00% 24.14% 26.80% 17.6964 24.7337 25.46% ibm18 6.19% 10.48% 3.87% 10.0903 12.1219 1.61% Average 32.14% 29.41% 28.76% 31.94% 29.59% 28.28% Table 6.11: Improvement percentage in METIS RB and METIS k way corresponding to the results obtained by MRDPSO algorithm, for k = 8, 16 and 32 partitions.

Swarm intelligence based Ant Colony Optimization is used for bisecting graph [20]. Same set up of hypergraphs, we have used to compare our MRDPSO based approach with the bipartitioning result obtained by ACO. Quality measures used for comparison are min cut and average cut. These measures are obtained in fifteen runs with different random seed for everyone. Table 6.12 shows min-cut and average cut obtained by ACO and MRDPSO, and their relative performance along with improvement percentage corresponding to ACO. ACO MRDPSO Relative Performance Improvement % Hypergraph Min - Cut Avg - Cut Min - Cut Avg - Cut Min - Cut Avg - Cut Min - Cut Avg - Cut ibm01 67 95 19 41 0.28358 0.43316 71.64% 56.68% ibm02 171 440 174 300 1.01754 0.68213-1.75% 31.79% ibm03 623 851 544 968 0.87319 1.13648 12.68% 13.65% ibm04 323 504 276 471 0.85449 0.93458 14.55% 6.54% ibm05 1322 1583 640 902 0.48411 0.56987 51.58% 43.01% ibm06 352 997 188 923 0.53409 0.92619 46.59% 7.38% ibm07 375 464 269 345 0.71733 0.74296 28.26% 25.70% ibm08 761 1100 824 1104 1.08279 1.00362-8.27% -0.36% ibm09 364 449 345 368 0.9478 0.8207 5.219% 17.93% ibm10 629 1060 580 820 0.9221 0.77393 7.790% 22.61% ibm11 388 676 375 478 0.96649 0.70759 3.35% 29.24% ibm12 1064 1715 1071 1464 1.00658 0.85365-0.65% 14.63% ibm13 682 1366 548 905 0.80352 0.6625 19.64% 33.75% ibm14 1095 1759 1075 1524 0.98174 0.86627 1.82% 13.37% ibm15 1291 3349 1079 3392 0.83579 1.01295 16.42% -1.30% ibm16 1012 1448 960 1534 0.94862 1.05939 5.13% -5.94% ibm17 1317 1775 1258 1779 0.9552 1.00175 4.47% -0.17% ibm18 1412 1592 1383 1873 0.97946 1.17617 2.05% 17.62% Average 0.84413 0.85355 15.58% 14.65% Table 6.12: Min-cut and average cut obtained by ACO and MRDPSO, relative performance and improvement percentage

Figure 5.6 represents improvement percentage by MRDPSO in Min cut and Average cut corresponding to ACO. Our approach shows an improvement varies from -8.27% and 71.64%with average of 58% in Min Cut and varies from -5.94% and 56.68%with average of 14.65% in Avg Cut corresponding to bisection obtained by ACO based algorithm. Complete evaluation on all 18 graphs using an ACO based algorithm has taken two hours, whereas MRDPSO takes one hour eighteen minutes. 80.00% Improvement % 70.00% 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% -10.00% -20.00% Min - Cut Avg - Cut Fig 6.5: Improvement percentage in Min cut and Average cut for bipartitioning. 6.10 Conclusion In this chapter we have presented MRDPSO, a multilevel recursive discrete PSO approach for balance graph k partitioning. The developed algorithm follows the fundamental concept of multilevel graph partitioning method and integrates most powerful DPSO refinement procedure. We evaluated comprehensively the performance of the algorithm on a collection of the Graph Partitioning Archive

for ordinary graphs as well as hypergraphs for the different values of k, set to = 2, 4, 8, 16, 32, 64. From the results and comparisons, we can observe that the overall performance of developed the MRDPSO algorithm is remarkable for generating balanced partition in significantly less computing time than those produced by METIS, CHACO and Tabu search. The MRDPSO k way partitioning approach substantially outperforms than METIS RB and METIS k way, both for minimizing hyperedge cut as well as minimizing the computation time. Furthermore, our experiment research proves that MRDPSO algorithm is more competent than the ACO based algorithm for bipartitioning.