2010 Second International Conference on Communication Software and Networks A Novel Particle Swarm Optimization – based Algorithm for the Optimal Communication Spanning Tree Problem Anh Tuan Hoang, Vinh Trong Le Nhu Gia Nguyen Department of Mathematics – Mechanics - Informatics Hanoi University of Science, Vietnam {hoangtuananh, letrongvinh}@hus.edu.vn Faculty of Information Technology Duy Tan University, Vietnam nguyengianhu@duytan.edu.vn Abstract — In this paper, we propose a novel approach for the optimal communication spanning tree (OCST) problem Our algorithm is based on the Particle Swarm Optimization (PSO) technique and take account into node biased encoding (NBE) scheme to find nearly optimal solution The new algorithm can achieve a result that is better than known heuristic algorithms do, as verified by a set of public benchmark problem instances Figure The distant matrix of Palmer’s 6-nodes instance Keywords — Optimal Communication Spanning Tree, Node Biased Encoding, Particle Swarm Optimization I INTRODUCTION The problem of finding optimal communication spanning tree was introduced by Hu in 1974 [1] Its NP- hard property was shown by Johnson et al [2] The formal definition of this problem is below Figure The requirement matrix of Palmer’s 6-nodes instance Let G = (V, E) is an undirected weighted graph, in which V is set of n vertices; E is set of m edges Moreover, let D = (dij)nxn and R= (rij)nxn be the distant matrix and requirement matrix of the graph G, respectively For any pair of nodes i and j in V, dij is the weight of the edge (i,j) and is infinite if not; rij is the communication requirement between them Both R and D are all symmetric and their elements are nonnegative In this context, each spanning tree T of G has a cost w(T) which is total of the weighted of paths over all pairs of vertices in T In general, w(T) is computed by following equation: Z 7 The cost of the tree T in Fig3.a is compute as below: w(T) = r12.d12 + r13(d12+ d23) + r14(d12+ d24) + + r15(d12+ d25+ d45) + r16(d12 + d24 + d46) + + … + r56(d54+d46) (5) = 817626 (1) T where pij denotes the distant between node i and node j on the tree T Let {i=v1, v2, …, vk=j} be the path between node i and j in the tree T, then pijT is calculated by the following equation: pijT = (2) (a) The function f is defined as the product of two variables: f(rij,pijT) = rij.pijT Figure Two communication spanning tree (a) of Palmer’s 6-nodes (3) For this instance, the optimal solution is given in Fig3.b with the minimum cost is 693180 The objective function is then: minimizeT w(T) It is important to note that the OCST problem is far more different from the minimum spanning tree problem, which is already solved by several polynomial algorithms (such as Prim [5] and Kruskal) Moreover, there are several variations of this problem when the form of the distant matrix D or the requirement matrix R changes More information can be found in [6] (4) where is the set of nn-2 labeled spanning tree of G [3] We take the Palmer’s 6-nodes instance from [4] as an example to clear out the definition Example 1: The Palmer’s 6-nodes instance The distant matrix and requirement matrix are given below: 978-0-7695-3961-4/10 $26.00 © 2010 IEEE DOI 10.1109/ICCSN.2010.111 (b) 232 A Previous work Due to its large application, there are lots of research on OCST problem, from both exact and heuristic approaches Authors in [1] defined the problem and discussed two simplest cases The NP- hard property of the problem was showed [2] and Reshef showed that the problem is even a MAX SNP – hard one, which cannot be solved by a polynomial approximation algorithm unless P = NP [6] Therefore, heuristic optimization methods have been chosen by researchers to find optimal or nearly optimal solutions for the OCST problems One of the first heuristic approaches was presented by Ahuja and Murty, which is closes to local search methods Palmer, in his PhD thesis [8], proposed a genetic algorithm (GA) using the node and link biased encoding that will be clearly descript in section II From the same approach, Li and Bouchebaba proposed another GA which works on a tree chromosome without intermediate encoding and decoding [9] This GA uses a Prim – based algorithm to randomly generate the initial population and performs better than the former one Some other GAs using variants of Prufer numbers were proposed, such as Picciotto [10], Julstrom [11, 12] However, Jens Gottlieb stated that Prufer numbers based chromosome presentations are not good for constrained spanning tree optimization problems [13] After all, from statistical view, Rothlauf argued that good solution for the OCST are biased towards minimum spanning trees [14] to move away from its neighbors if they are too close Second, each particle steers towards the average heading of its neighbors And the third, each particle tries to go towards the average position of its neighbors Kennedy and Eberhart generalized these properties to be an optimization technique as below Consider the optimization problem P First, we randomly initiate a set of feasible solutions; each of single solution is a “bird” in search space and called “particle” All of particles have fitness values which are evaluated by the fitness function to be optimized, and have velocities which direct the flying of the particles The particles fly through the problem space by following the current optimum particles The better solutions are found by updating particle’s position In every iteration, each particle is updated by following two "best" values The first one is the best solution (fitness) it has achieved so far (The fitness value is also stored.) This value is called pbest Another "best" value that is tracked by the particle swarm optimizer is the best value, obtained so far by any particle in the population This best value is a global best and called gbest When a particle takes part of the population as its topological neighbors, the best value is a local best and is called lbest After finding the two best values, the particle updates its velocity and positions with following equation: (6.a) (which use global best gbest) or (6.b) (which use local best lbest) and (7) B This work Our main contributions are as follows: We present a new method of finding optimal solution for OCST problem In our methods, PSO technique [15] is employed to reduce search space mentioned above but still converge to a global good solution; the NBE schema is used to represent candidate solutions Our approach outperforms Palmer’s and Li and Bouchebaba’s GAs in not only final solution but also computation time v[ ] = v[ ] + c1* rand() * (pbest[ ] - present[ ]) + + c2 * rand() * (gbest[ ] - present[ ]) v[ ] = v[ ] + c1* rand() * (pbest[ ] - present[ ]) + + c2 * rand() * (lbest[ ] - present[ ]) present[ ] = persent[ ] + v[ ] The rest of the paper is organized as follows The main concepts of PSO are presented in section II.A The NBE schema is described in section II.B We present our method in section II.C Section III presents some experimental results We make conclusion and recommend some future works in section IV (6.a) (6.b) (7) In those above equation, rand() is a random number between and 1; c1 and c2 are cognitive parameter and social parameter restectively PSO Algorithm {For each particle Initialize particle EndFor Do For each particle Calculate fitness value If the fitness value is better than the best fitness value (pBest) in history Set current value as the new pBest EndIf EndFor Choose the particle with the best fitness value of all the particles as the gBest (or Choose the particle with the best fitness value of all the neighbors particles as the lBest) For each particle Calculate particle velocity according to (6.a) (or (6.b)) II PARTICLE SWARM OPTIMIZATION FOR SOLVING GRAPH COLORING PROBLEM A Particle swarm optimization Particle swarm optimization is a stochastic optimization technique developed by Dr Eberhart and Dr Kennedy [15], inspired by social behavior of bird flocking or fish schooling It shares many similarities with other evolutionary computation techniques such as genetic algorithms (GA) The algorithm is initialized with a population of random solutions and searches for optima by updating generations However, unlike the GA, the PSO algorithm has no evolution operators such as the crossover and the mutation operator In the PSO algorithm, the potential solutions, called particles, fly through the problem space by following the current optimum particle By observing bird flocking or fish schooling, we found that their searching progress has three important properties First, each particle tries 233 Update particle position according to (7) EndFor While (stop condition is true)} The stop condition mentioned in the above algorithm can be the maximum number of interaction is not reached or the minimum error criteria are not attained Figure The modified distant matrix of temple graph in example B Node Biased Encoding The node biased encodings is among the class of weighted linear chromosome representation for spanning tree optimization problems Its original version was developed by Palmer [8] In this scheme, each spanning tree is encoded by assigning every nodes and link on graph to a corresponding weight The details of encode and decode phases are given below Figure Spanning tree computed in example 1) Represent and decode a particle: We use NBE scheme to represent particles Each particle is a vector with n components; every component is a real number corresponds to one in n vertices To decode a particle we exactly steps described in previous section 2) Initiate population: As mentioned in section I.A, good solution for the OCST problem are biased towards minimum spanning trees, we initiate a particle corespond to the mininmum spanning tree one This particle is the vector whose components are all zero Others n specially initiated particles are of the form {1, 1, ,1, 0, 1, 1, ,1} Those particle corespond to star – shaped trees This is motivated by the fact that a star –shaped tree is the optimal solution for a special case [1] All others P-(n+1) are vectors whose components are real numbers between and which are randomly generated (P is a designed parameter) 3) Fitness function: If the jth particle results a solution cost cj, then the fitness value of this paritcle is computed with following equation: (9) Fj = 1/cj 1) Encode phase: Each spanning tree T of graph G = (V, E) is represented by a n – components real vector b = (b1, b2, …,bn) Remind that n is the cardinant of V b is now called the weighted vector 2) Decode phase: Take two following steps to calculate spanning tree T correspond to the weighted vector b a) Step 1: Construct the temple graph G’ = (V, E) with modified distant matrix D’ = (dij)nxn computed as below equation: d’ij = dij + P2(bi+bj) dmax (8) where dmax = maxij(dij) and P2 is the node biased parameter which controls the node weights on the tree construction b) Step 2: Calculate T as the minimum spanning tree of temple graph G’ by using Prim algorithm Note that, by running Prim algorithm, nodes i with low bi will probably be interior nodes, and on the otherhands, nodes j with high bj will probably be leaf nodes Therefore, the higher the weight bi, the higher is the probability that node i will be a leaf node We take the example below to illustrate the functionality of the NBE scheme Example 2: NBE scheme Let G is a 5-nodes complete graph with distant matrix D given in Fig.4 4) Stop condition: The stop condition we used in this paper is defined as the maximum number of interaction Ngen (Ngen is also a designed parameter) D The complexity In this section, we compute the complexity of the above algorithm To initiate the population size P, we have to randomly generate a matrix with P rows and n columns So, the complexity of initiate phase is O(P.n) It’s clear that the population updating has also the complexity of O(P.n) In particle decoding, the complexity of Step is O(n2), and in Step 2, we use Prim algorithm, so the complexity of this step is also O(n.log(m)) since we use heap and adjacent list in implementation Therefore, the total complexity of our algorithm with P particles and Ngen generations is O(Ngen.P.max{n2, nlog(m)}) = O(n2) Figure The distant matrix of graph in example Then the modified distant D’ correspond to vector b = (0.7, 0.5, 0.8, 0, 1) and parameter P2 = is given in Fig.5 and by running the Prim algorithm we get the tree in Fig.6 C Solving the OCST problem base on PSO In this section, we present application of PSO technique for the OCST problem Without loss of generality, we can assume that G is undirected complete graph Our new algorithm is described as follows III EXPERIMENT RESULTS In order to measure the performance, we use a set of standard benchmark instances They are nodes (Palmer6), 12nodes (Palmer12) and 24 nodes (Palmer24) networks from 234 Palmer; 10 nodes (Raidl10), 20 nodes (Raidl20), 50 nodes (Raidl50, Raidl50gen) networks from Raidl and nodes (Berry6) networks from Berrry These instances can be found in [4] The node biased parameter is set to and the parameters for the PSO are set as below: In the future, we will test our algorithm on more problem instances, both from literature and real world We will also try find out a better encoding scheme and adapt our method to other problems ACKNOWLEDGMENT x Population size P = 1000 x Maximum number of interaction Ngen = 500 x Cognitive parameter c1 = x Social parameter c2 = x Update population according to (6.b) and (7) with number of neighbor is K = This research is partly conducted as a program for the QT09-03 and TN-09-03 in Funds for Science and Technology by the Vietnam National University REFERENCES [1] Hu, T.C, Optimum Communication Spanning Trees, SIAM Journal of Computing, 3(3), 188–195 (1974) Note that: the author of [16] recommended that the local best PSO is better than a global one and this is verified by us [2] Garey M R., Johnson D S., Computers Intractability: A Guide to the Theory of Completeness, W.H Freeman, 1979 We run our algorithm on those above instances and compare to the results produced by Palmer’s GA and Li and Bouchebaba’s GA in Table All three algorithms were implemented in C++ and run on a workstation with AMD Athlon™ 64X2 Dulal Core – 2.30 GHz processor and 2GB RAM TABLE I [3] A Cayley, A theorem on trees, Quart J Math 23 (1889), 376–378 [4] Franz Rothlauf, Representations for Genetic and Evolutionary Algorithms, Springer, Heidelberg New York, 2nd Edition, 2006 [5] Thomas H Cormen, Charles E Leiserson, Ronald L RESULTS FOR STANDARD BENCHMARK INSTANCE Palmer' GA Li & Bouchebaba's GA and NP- Rivest, Clifford Stein Introduction to Algorithms, Second Edition MIT Press and McGraw-Hill, 2001 PSO [6] Bang Ye Wu Kun-Mao Chao, Spanning Trees and Instance Optimization Problems, Chapman & Hall/CRC, 2004 Cost Time Cost Time Cost Time Palmer6 709770 6.88 708090 693180* 4.1 Palmer12 3876488 35.53 3457952 24.7 3428509* 22 Palmer24 1959790 229.67 10866568* 162 1138360 143 communication cost spanning, trees and related problems Master’s thesis, Feinberg Graduate School of the Weizmann Institute of Science, Rehovot 76100, Israel 14 [8] Palmer C C, An approach to problem in network design * * Raidl10 58352 22.21 53674 15.93 53674 Raidl20 165788 138.5 157570* 97.3 157570* 87.3 Raidl50 911987 1883.09 134931 1350 107746 1178 Raidl50gen 123013 1883.86 964140 1351 826499 1177 Berry6 534 * 6.8 534 * 5.74 534 * [7] Reshef, E (1999, April) Approximating minimum using genetic algorithms, PhD Thesis, Polytechnic University, Computer Science Department, Brooklyn, New York, 1994 [9] Li, Y., & Bouchebaba, Y, A new genetic algorithm for the 4.6 optimal communication spanning tree problem, Proceedings of Artificial Evolution: Fifth European Conference (pp 162–173) Berlin, Springer, 1999 (An entry marked by a star (*) denotes a optimal solution) [10] Picciotto, S, How to encode a tree, Ph D thesis, For Palmer series, we have the two optimal solutions (Palmer6 and Palmer12) For Raild series, we have the same two optimal solutions (Raild10 and Raild20) with Li & Bouchebaba’s In other instances (Raidl50 and Raidl50gen), we also have better solution than Palmer’s and Li & Bouchebaba’s The last instance (Berry6), all three algorithms result optimal solution However, ours still the fastest IV University of California, San Diego, USA, 1999 [11] Julstrom, B A The blob code: A better string coding of spanning trees for evolutionary search In A S Wu (Ed.), Proceedings of the 2001 Genetic and Evolutionary Computaton Conference Workshop Program, San Francisco, California, USA, pp 256–261, 2001 [12] Julstrom, B A The blob code is competitive with edge- CONCLUSION AND FUTURE WORK sets in genetic algorithms for the minimum routing cost spanning tree problem In Beyer, Hans-Georg et al (Ed.), Proceedings of the Genetic and Evolutionary Computation Conference 2005, New York, pp 585–590, ACM Press, 2005 In this paper, we present a novel PSO-based algorithm to solve the OCST problem in which the each particle is encode using NBE scheme The experiments results show that the proposed algorithm is effective to get optimal and near optimal solution in term of final solution cost and running time [13] Gottlieb, J., B A Julstrom, G R Raidl, and F Rothlauf, Prăufer numbers: A poor representation of spanning trees 235 [15] J Kennedy and R Eberhart, Swarm Intelligence, Morgan for evolutionary search,Proceedings of the Genetic and Kaufmann Publisher Inc, 2001 Evolutionary Computation Conference (GECCO2001): 343–350 2001 [16] M Clerc, A method to improve Standard PSO ,Technical Report, http://clerc.maurice.free.fr/pso/ [14] Rothlauf, F., J Gerstacker, and A Heinzl (2003) On the optimal communication spanning tree problem Technical Report 15/2003, Department of Information Systems, University of Mannheim 236 ... fitness value of all the neighbors particles as the lBest) For each particle Calculate particle velocity according to (6 .a) (or (6.b)) II PARTICLE SWARM OPTIMIZATION FOR SOLVING GRAPH COLORING PROBLEM. .. motivated by the fact that a star –shaped tree is the optimal solution for a special case [1] All others P-(n+1) are vectors whose components are real numbers between and which are randomly generated... numbers based chromosome presentations are not good for constrained spanning tree optimization problems [13] After all, from statistical view, Rothlauf argued that good solution for the OCST are biased