SESSION GRAPH BASED AND TREE METHODS + RELATED ...

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28 Int'l Conf. Foundations of Computer Science | FCS'12 | 8 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] Z. Kun, W. Heng, and L. Feng-Yu, "Distributed multicast routing for delay and delay variation-bounded Steiner tree using simulated annealing", Computer Communications, vol.28, Issue 11, 5 July 2005, pp. 1356-1370. G. Kulkarni, "A Tabu Search Algorithm for the Steiner Tree Problem", M.Sc. Thesis, North Carolina State uiversity, 2000. C. C. Ribeiro and M. C. De Souza, "Improved Tabu Search for the Steiner Problem in Graphs", Working paper, Catholic University of Rio de Janeiro, Department of Computer Science, 1997. S.M. Wang, "A multiple source algorithm for suboptimum Steiner trees in graphs",Workshop on Graph-Theoretic Concepts, Depatment of Computer Science, 1985, pp. 387- 396. D. E. Drake and S. Hougardy, On Approximation Algorithms for the Terminal Steiner Tree Problem", Information Processing Letters, vol. 89, Number 1, January 2004, pp. 15-18. M. Demange, J. Monnot, and V. Th. Paschos, "Differential Approximation Results for the Steiner Tree Problem", Applied Mathematics Letters, vol. 16, Issue 5, July 2003, pp. 733-739. G. Robins and A. Zelikovsky, "Improved Steiner Tree Approximation in Graphs", In Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms, San Francisco, California, United States, 2000, pp. 770-779. V. V. Vazirani, "Recent results on approximating the Steiner tree problem and its generalizations", Theoretical Computer Science, vol. 235, Issue 1, 17 March 2000, pp. 205- 216. L. Gatani, G. Lo Re, and S. Gaglio, "An efficient distributed algorithm for generating and updating multicast trees", Parallel Computing, vol. 32, Issues 11-12, December 2006, pp. 777-793. URL:http://www.sciencedirect.com. L. M. A. Drummond, E. Uchoa, A. D. Goncalves, J. M.N.Silva, M. C.P. Santos, and M. C. S. de Castro, "A gridenabled distributed branch-and-bound algorithm with application on the Steiner Problem in graphs", Parallel Computing, vol. 32, Issue 9, October 2006, pp. 629-642. URL:http://www.sciencedirect.com/. H. J. Promel and A. Steger, "A New Approximation Algorithm for the Steiner Tree Problem with Performance Ratio 5/3", Journal of Algorithms, vol. 36, Issue 1, July 2000, pp. 89-101. URL:http://www.sciencedirect.com. A. Ghanwani, "Neural and delay based heuristics for the Steiner problem in networks", European Journal of Operational Research, vol. 108, Issue 2, 16 July 1998, pp. 241-265. URL:http://www.sciencedirect.com/ P. Guitart, "A Faster Implementation of Zelikovsky's 11/6- Approximation Algorithm for the Steiner Problem in Graphs", Electronic Notes in Discrete Mathematics, vol. 10, November 2001, pp. 133-136. URL:http://www.sciencedirect.com. S. L. Martins, C. C. Ribeiro, and M. C. Souza, "A Parallel GRASP for the Steiner Problem in Graphs", Lecture Notes In Computer Science; vol. 1457, 1998, pp. 285-297. URL:http://citeseer.ist.psu.edu/martins98parallel.html. H. Esbensen and P. Mazumder, "A Genetic Algorithm for the Steiner Problem in a graph", In Proceedings of the European Design and Test Conference, 1994, pp. 402-406. URL:http://citeseer.ist.psu.edu/185181.html. [16] [17] [18] [19] T. G. Crainic, M. Gendreau, and J. Potvin, "Parallel Tabu Search", Parallel Metaheuristics, E. Alba (Ed.), John Wiley & Sons, 2005. URL:http://www.iro.umontreal.ca/~gendron/Pisa/Reference Meta/Crainic05c.pdf. M. P. Bastos and C. C. Ribeiro, "Reactive Tabu Search with Path Relinking for the Steiner Problem in Graphs", In Proceedings of the Third Metaheuristics International Conference, 1999, pp. 31-36. URL:http://citeseer.ist.psu.edu/bastos99reactive.html. T. G. Crainic and N. Hail, "Parallel Meta-heuristics Applications", Parallel Metaheuristics, E. Alba (Ed.), John Wiley \& Sons, 2005. URL:http://www.iro.umontreal.ca/~gendron/Pisa/Reference Meta/Crainic05b.pdf. H. Takahashi and A.Matsuyama, “An approximate solution for the Steiner problem in Graphs”, Mach. Jap. 24, 1980, pp. 573–577.
Int'l Conf. Foundations of Computer Science | FCS'12 | 29 A Heuristic Algorithm for Solving Steiner Tree Problem on the Graph F. Ghadimi 1 , A. Nourollah 2 , M. Keyvanpour 3 1 Department of Electrical & Computer Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran 2 Department of Electrical & Computer Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran and Department of Electrical & Computer Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran 3 Department of Computer Engineering, Alzahra University, Tehran, Iran Abstract - Steiner tree problem on the graph is an NP- Complete problem and has no exact solution in polynomial time. Since this problem is practically useful, there are more attentions to heuristic and approximation approaches rather than exact ones. By using heuristic algorithms, the near optimum answers are obtained in polynomial time that this is faster than exact approaches. The goal of Steiner Tree problem is to find a minimum cost tree from the main graph that connects a subset of nodes called terminals. In this article, we have proposed a heuristic algorithm that solves Steiner Tree Problem. It has time complexity of �(�(� + � log �)) that � is the number of nodes and � is the number of edges. This algorithm finds the near optimum answer and according to the experimental results and the comparisons, it has an appropriate rate in a reasonable time. Keywords: Steiner Tree on the Graph, NP-Complete problems, Heuristic Algorithms. 1 Introduction The Steiner tree problem (STP) is a well-known issue that is used in many fields, like cabling design, canalization, VLSI design, routing, urban road design and also multicasting in computer networks. According to the different usages of this problem, some various definitions are exist for it: Euclidean STP which finds Steiner tree on the set of points in the plane; Rectilinear STP which finds Steiner tree in Manhattan space; Generalized STP which finds Steiner tree on an undirected, weighted graph; and Directed STP which finds Steiner tree on a directed, weighted graph. In this article, the Steiner tree has been considered on an undirected, weighted graph � = (�, �, �) , which includes of a set of vertices (�), a set of edges (�) and a set of weights of edges (�). The graph � is one of the inputs of STP and it has no negative weight. The other input of this problem is a subset of nodes in �, which called Terminals (�). Steiner tree problem must find a sub graph of � that has the minimum summation of weights and connects all terminals. According to the following proof [1], the Steiner sub graph (�) must be a tree, because there is no negative weight. The proof says that if � isn’t a tree, then it should have at least one cycle. Hence, there are two different paths that connect some terminal nodes to the other. So it is feasible to omit some edges from one of these paths, without losing the connectivity of � . This omission causes a reduction of cost of � and shows that � isn’t the minimum possible answer. In 1972 it was proved that STP is an NP-Complete problem [2], therefore there is no polynomial time approach to compute the exact answer. So far some exponential time approaches have been proposed for this problem that they find the optimum answer and called Exact Algorithms. But these algorithms aren’t suitable for large networks with lots of nodes and edges, because of their low-speed execution. In this reason Heuristic and Approximation algorithms are widely used for solving STP, despite their near optimum answers. These approaches have near optimum answers in polynomial time and their evaluation measure is the ratio between their answer and the optimum answer. In this article, a heuristic algorithm with polynomial time complexity is proposed that can compute Steiner tree on large graphs. This article is organized as follows: In the next Section, pervious works are reviewed. In Section3, our new algorithm and its time complexity are explained. In Section4, the experimental results are presented and finally in Section5 there are the conclusions. 2 Pervious Works All the searching problems that have polynomial time solutions are in the P category, and all the other problems are NP . Because there are some problems that have no polynomial time solutions therefore P ≠ NP [3]. Since STP
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