Improved Minimum Spanning Tree Heuristics for Steiner Tree problem in graph

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1 Improved Minimum Spanning Tree Heuristics for Steiner Tree problem in graph Ali Nourollah,2, Elnaz Pashaei, and Mohammad Reza Meybodi 3 Department of Electrical, Computer and IT Engineering, Qazvin Islamic Azad University, Qazvin, Iran 2 Department of Electrical and Computer Engineering of Shahid Rajaee Teacher Training University, Tehran, Iran 3 Department of Computer & IT Engineering, Amirkabir University of Technology, Tehran, Iran Abstract - The minimum Steiner tree problem, a classical combinatorial optimization problem with a long history, is a NP-complete problem. Due to its wide application, study of heuristic algorithm about Steiner tree problem has important practical and theoretical significance. In this paper we first review one of the existing algorithms for solving the Steiner problem in graphs, Minimum Spanning Tree Heuristic algorithm, then presenting a new heuristic algorithm IMSTH to improve it. We describe our algorithm and its computational results. It is shown that our algorithm can effectively improve the performance on MSTH. Keywords: Steiner Tree problem, Heuristic Algorithm, Minimum Spanning Tree Heuristic Algorithm Introduction A great number of the recent applications often require the underlying network to provide multicasting capabilities. Multicast refers to the delivery of packets from a single source to multiple destinations. At the routing level, a multicast routing scheme is responsible for determining the packet delivery path from the source to all destinations, typically a multicast tree []. Generation and minimization of the cost of such tree have been traditionally formulated as the Steiner Tree Problem. The Steiner Tree Problem involves constructing the least cost tree that spans a given set of points. In addition to multicast routing in communication networks, the Steiner tree problem has numerous applications especially in the area of telecommunication, distribution and transportation systems. The computation of phylogenetic trees in biology and the routing phase in VLSI design are real life problems that have been modeled as the Steiner tree problem [2]. Another interesting application is in the billing strategies of large telecommunications network service providers. The bill isn t based on the actual number of circuits provided, which may change over time, but on a simple formula calculated for an ideal network which will provide all the facilities at minimum cost. Several other network design problems can be formulated as generalizations of the Steiner tree problem. Steiner tree problem or so called Steiner Problem in Graphs (SPG) is a classic combinatorial optimization problem. Karp showed that its decision version is NP - complete [3], although some well known special cases of the SPG can be solved in polynomial time. When N = 2 the problem reduces to the shortest path problem while when N= V the problem reduces to the minimum panning tree problem. Both these problems can be solved in polynomial time. On the other hand, the Steiner tree problem is NP-hard when the graph v is a chordal graph, a bipartite graph or a complete graph with edge weights either or 2. Thus in the general case the problem is an NP-hard problem. In this paper, a novel solution based on MST to the construction of Steiner tree is presented. The rest of this paper is organized as follows: Section 2 review the definition of the problem and section 3 present a survey of proposed algorithms for the Steiner problem in graphs. Section gives the description of MSTH algorithm. The proposed algorithm is presented in section 5. Section 6 describes experimental results and performance evaluation. Finally Section 7 draws a conclusion and makes suggestion for future works. 2 Basic definitions Let G = (V, E) be a connected undirected graph, where V is the set of nodes and E denote the set of edges. Given a non-negative weight function w: E R + associated with graph edges and a subset N V of terminal nodes, the Steiner Problem in Graphs, SPG (V, E, w, N), consists of finding a minimum weighted connected sub tree of G spanning all terminal nodes in N. The solution of SPG (V, E, w, N) is Steiner minimum tree. The non-terminal nodes that end up in the Steiner minimum tree are called Steiner nodes. Terminal Steiner Tree Problem is a variation in which all the terminal nodes must appear at leaves of the tree. This problem that is also proved to be NP-complete has been matter of concern because it has direct application in VLSI design [5]. In Complete Steiner Problem the input graph is assumed to be complete. Another variation is the Complete Steiner (, 2) in which the input is a complete graph with edge weights or 2. All of these variations are NP-complete [6]. SPG or its terminal version are sometimes said to be metric, i.e. the triangle inequality holds for edge weights in the input graph. This imposes no limitation on the Steiner problem itself, since we can replace any edge with the shortest path connecting its

2 ends [7, ]. The Steiner Network Problem generalizes the metric Steiner tree problem to higher connectivity requirements: Given a graph G = (V, E), a cost function on edges, and a function r mapping unordered pairs of vertices to Z +, find a minimum cost graph that has r(u, v) edge disjoint paths for each pair of vertices u and v []. The issue of multipoint routing for multimedia traffic has led to emergence of Constrained Steiner Tree Problem, in which the problem is to find a minimum cost tree such that the delay delay variation or both between the source and each of the destinations is bounded. The Dynamic Steiner Tree Problem is another generalization of the problem, in which the set of destination nodes changes over time by receiving join or delete requests from nodes, and problem asks for a sequence of optimal trees []. 3 Related Work As SPG is NP-complete, there is little hope to find a polynomial time solution for it. All the work done to find a solution so far falls into three categories: Exact Algorithms, Approximation Algorithms and Meta heuristics. Two popular exact algorithms, the Spanning Tree Enumeration Algorithm (STEA) which enumerates all possible combinations of Steiner nodes, and the Dynamic Programming Algorithm (DPA), present time complexities of O( p 2 ( ) + n ) and O(3 n + 2 n + n ) respectively, where n is the number of nodes in the network and p is the number of multicast members []. These algorithms require long computation time or huge computational power for solving bigger problems, like the branch and bound algorithm proposed in [], that makes use of computational grids. In [], the author offers an approximation algorithm with performance ratio 5/3 based on finding a minimum spanning tree in 3-uniform hyper graphs that finds the solution with probability at least /2 and claims that the algorithm runs in O(lg2 n) time, using O(n3 ) processors. The best approximation algorithm known so far is due to Robins and Zelikovsky whose performance ratio is about.55 and even better for special cases such as quasibipartite and complete(-2) graphs [7]. Among several heuristics proposed to find an Approximate solution, Traveling Salesman Problem Heuristic (TSPH), Average Distance Heuristic (ADH) have Performance ratio of 2. TSPH is a heuristic based on the traveling salesman problem (TSP) and involves finding a tour for the graph induced by the network followed by removing the most expensive link. Shortest path heuristic (SPH) computes the tree by connecting all the terminals to an arbitrary root through their shortest paths and then finding the minimum spanning tree of the graph induced by the union of these paths, repeatedly removing the non-terminal leaves. The algorithm presented in [] is a distributed algorithm based on an improved version of the ADH heuristic, known as ADH with Full connection (ADHF). [] Also provides an efficient approach that supports dynamic multicast membership, by means of periodic improvement of locally inefficient sub-trees. In [2] the author introduces a new algorithm using the Random Neural Networks to find potential Steiner vertices that are not already in the solution returned by the MSTH or ADH, starting with the solution of the MSTH or ADH. The first approximation algorithm for SPG having an approximation ratio constant lower than 2 was due to Zelikovsky [3] with performance ratio /6. Then he repeatedly improved this ratio to currently best known performance ratio:.55. The heuristics proposed to find the Steiner tree for routing applications are either centralized or distributed. In the centralized approach, a central node that is aware of the state of the whole network computes the tree. The computation is generally easy and fast. But impractical for large networks where the overhead of maintaining, in a single node, coherent information about the state of the entire network may be prohibitive. In a distributed approach, on the other hand, each node of the network actively contributes to the algorithm computation. Distributed routing algorithms can be slower and more complex than the centralized ones, but they become indispensable when the network nodes can't reach a complete knowledge of the topology and of the state of the network []. Some metaheuristics are proposed as the solution for the Steiner problem in graphs too. Among the most efficient ones, it is found implementations of metaheuristics such as genetic algorithms, tabu search, Grasp and simulated annealing [, 3]. Esbensen and Mazumder [5] proposed a genetic algorithm and discuss its application in global routing of VLSI layouts. The algorithm s encoding is based on the use of the Distance Network Heuristic (DNH) which is a deterministic heuristic for the SPG. The performance of algorithm is compared to that of two heuristics from the literature and it has been shown that the algorithm is superior. Di Fatta, Lo Presti, and Lo Re proposed a parallel genetic algorithm for the Steiner problem in networks. When solving Beasley s OR Library standard test problems, they obtain promising speedup values. Tabu Search was introduced by Glover in 6. TS is an extension of classical local search methods typically used to find approximate solutions to difficult combinatorial optimization problems [6]. Ribeiro and Souza [] proposed an improved tabu search for the Steiner problem in graphs. The important feature of the algorithm is that move estimations, elimination tests, and neighborhood reduction techniques are used to speed up the local search and lead to a much faster algorithm with similar performance in terms of solution quality. In the context of parallel tabu search for the Steiner problem in graphs, Bastos and Ribeiro [7] describe a two phase algorithm: in their approach, a parallel multi-thread reactive TS phase is followed by a distributed Path Relinking (PR) phase, i.e., all processes switch from TS to PR simultaneously. Martins, Ribeiro and Souza [], proposed a parallel grasp for the Steiner problem in graphs. A Greedy Randomized Adaptive Search Procedure (GRASP) is a meta-heuristic for combinatorial optimization. A GRASP is an iterative process, where each of iteration consists of two phases: construction and local search. The construction phase of the algorithm is based on a version of distance network heuristic which is improved by Mehlhorn. Some heuristics are used in order to speed up the local search. For parallelization of GRASP, each

3 slave processor performs a fixed number of GRASP iterations. Once all processors have finished their computations, the best solution is collected by the master processor. The results of computational experiments illustrate the effectiveness of the proposed parallel GRASP procedure for the Steiner problem in graphs. Verhoeven and Severens proposed sequential and parallel local search methods for the Steiner tree problem based on a novel neighborhood. They claimed their approach is better" than those known in the literature. Computational results indicated that good speedups could be obtained without loss in solution quality []. Minimum Spanning Tree Heuristic Algorithm In the minimum spanning tree heuristic (MSTH) suggested by Takahashi and Matsuyama [], the solution T is obtained by deleting from the minimum spanning tree for G non-terminals of degree (one at a time). The worst-case time complexity of the minimum spanning tree heuristic is O ( e + v log v). The worst-case error ratio T T (N) is tightly bounded by v n +. Hence, the minimum spanning tree heuristic can be considered as inferior to the shortest paths heuristic in the worst-case sense. It also performs poorly on average. The proposed heuristic algorithm in this paper consists of eight steps. In the first step, after assuming that T is equal with null, we obtained T by graph G. 5 Improve Minimum Spanning Tree Heuristic Algorithm 5. Description of the IMSTH Algorithm The proposed heuristic algorithm in this paper consists of eight steps. In the first step, after assuming that T is equal with null, we obtained T by graph G. In the second step of algorithm, we assume two divisions of edges and paths between two terminals in T : Existence direct edge in T Not existence direct edge in T and existence direct edge in graph. In the third step, for the first category, select the direct edges of two terminals which obtained by T and add into T. in the fourth step, for the second category, make compare between paths and direct edges of two terminals: If direct edge be shortest, add into T. If path be shortest, add into T. In the next step, we study T, to determine all disjoin terminals. In the sixth step obtained the shortest paths between disjoin terminal and other terminals in the graph with Dijkstra algorithm. Select shortest path between reached paths and add into T. In the next step, we study T, for connectivity. If we have forest, determine terminals with degree, then seek shortest path from this terminal to other terminals, and add it into T. At the end, study T for cycle. If we have cycle, delete cycle. Fig. shows our suggested algorithm. Suggested algorithm: (INPUT: G = (V, E) where w: E R +, N Vset of Terminals, OUTPUT: T approximate optimal tree) { Step : T Step : Compute T of the graph G. Step 2: Assume two divisions of edges and paths, between two terminals in T : a) Existence direct edge in T b) Not existence direct edge in T and existence direct edge in graph Step 3: For the first category, select the direct edges of two terminals which obtained by T and add into T. Step : For the second category, make compare between paths and direct edges of two terminals: a) If direct edge be shortest, add into T. b) If path be shortest, add into T. Step 5: Study T, to determine all disjoin terminals. Step 6: Determine the shortest paths between disjoin terminal and other terminals in the T with Dijkstra algorithm. Select shortest path between reached paths and add into T. Step7: Study T, for connectivity. If we have forest, determine terminals with degree, seek shortest path from this terminal to other terminals, and add it into the T. Step : Study T for cycle. If we have cycle, delete cycle.} Fig. : Suggested algorithm for SPG in graph 5.2 Illustrative Example In the example graph shown in Figure 2a, vertices v, v3,v, v6 and v are terminal vertices. We construct T by this graph (Fig. 2b). The cost of T is equal with. It can be observed that between two terminals v3, v6 and v, v exists a direct edge in T, so the direct edges< v3, v6> and < v, v> is selected and add into T IMSTH. at the other hand, between two terminals v3 and v a direct edge in graph exists that is shortest than the paths between this two terminals, so in the fourth step the direct edge < v3, v> is selected and add into the T (Fig. 2c). In the next step, we study T for disjoin terminals but don t find any disjoin terminal. Now Study T, for connectivity. It is observed that we have forest. so determine terminals with degree, seek shortest path from this terminal to other terminals, and add it into the T (Fig. 2d). At the end, the cost of T is equal with 73(Fig. 2e). Fig. 2 illustrates steps of our algorithm.

4 (a) 227 We have implemented our algorithms in C# programming language. The experiments are based on the STPs in graph from category B in the OR-library. In this paper, the experiments are made to test the performance of IMSTH with respect to Minimum Spanning Tree Heuristic (MSTH), Directed Convergence Heuristic (DCH), Optimal Shortest Paths Heuristic (OSPH) and Average Distance Heuristic (ADH). Table, show it. As illustrated in Table, our algorithms have achieved good results. Figure 3 show the performance of our suggested algorithm (b) (c) TABLE : Our Algorithm in Compare of B Problems NO V G E G V D OPT DCH ADH MSTH IMSTH cost of tree IMSTH OPT MSTH (d) number of group Fig. 3: Performance of our Algorithm (e) Fig. 2: Steiner tree that generated by our algorithm 6 Experimental Results 7 Conclusions In this paper, we concern with the problem of finding minimum steiner tree in graph, which is a NP-complete problem. This paper presents a heuristic Steiner tree algorithm IMSTH. Experiment results show that IMSTH can effectively reduce the cost of Steiner tree by MSTH. This is a substantial improvement over previous improved works. The performance of our suggested algorithm was compared with Minimum Spanning Tree Heuristic (MSTH) Directed Convergence Heuristic (DCH), Optimal Shortest Paths Heuristic (OSPH) and Average Distance Heuristic (ADH). Experiment results show that our suggested algorithm produces good results and is comparable with other existing solutions.

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