A Heuristic Approach for Solving the Minimum Routing Cost Spanning Tree Problem

Transcription

1 A Heuristic Approach for Solving the Minimum Routing Cost Spanning ree Problem Quoc Phan an Department of Information echnology Saigon University, Vietnam Abstract-In this paper, we are proposing 3- algorithms in form of heuristic to solve the minimum routing cost spanning tree problem MRC: he idea of 1 st -algorithm is removing gradually bad edges until the remained graph is just a spanning tree. he idea of 2 nd -algorithm is finding a minimum spanning tree -MS, then replace gradually bad edges of this MS with other edges of graph so that the routing-cost of MS gets better. he idea of 3 rd -algorithm is separating graph into sub-graphs which connect together via bridges; this heuristic will show how to process pairs of sub-graphs corresponding to bridges. We have conducted experiments for above proposals as well as obtained good results if compare it with approximate algorithms at the moment to solve the MRC. Keywords: Routing Cost, Routing load, Spanning rees, Metaheuristic, NP-hard, Gradually Edge-Removal algorithm, Edge-Replacement algorithm, Bridge-processing Algorithm. I. INRODUCION he minimum routing cost spanning trees problem (MRC) belongs to NP-hard class. he problem can be approached in the way of finding exact solution (especially for Back-tracking, Branch-Bound,...) or in the way of finding approximate solutions (especially for heuristic algorithms, metaheuristic,... ) [1][2][3][10]. No matter how well we fine-tune exact algorithms, then the result can t improve anymore. o answer how to find exact solutions for the MRC is really a challenge. In this paper, we are only mentioning about finding approximate solutions for the MRC. At first, we would like presenting generally some present common approachs for the MRC. he first is WONG algorithm suggested by C.K. WONG[2][8][9]. he algorithm utilizes the minimum path trees (formed from minimum paths, which start at each vertex of graph and connect to the other vertices of graph). he idea of WONG algorithm is much clear as follows: Step 1: Find minimum paths which start at each vertex of graph using Dijkstra algorithm. Step 2: Compute cost for each SP. Step 3: Find SP which has the minimum cost. WONG s complexity: O(m.n+ n 2 logn). Secondly, group of authors BANG YE WU and KUN-MAO-CHAO, the group have approached the MRC with the idea of General Star ( possibly reaches to 4/3- approximation) [2][7]. he third is CAMPOS algorithm invented by group of authors RUI CAMPOS and MANUEL RICARDO [8][9], the authors have approached the MRC with 2- approximation, basing on the idea of mixing algorithms Add, Dijkstra and Prim CAMPOS algorithm is considered the best 2-approximation algorithm at present with time complexity O(m + nlog(n)) [8][9]. In section 2, we will remind some definitions concerning to computation of spanning tree s cost; in section 3, we will represent our proposals; in section 4, there is an experiment table of algorithms and the final is several conclusions as well as possible works in the future. II. HE MINIMUM ROUING-COS SPANNING REE PROBLEM Our heuristics base on computation of spanning tree s routing-cost, and to compute the routing-cost of a spanning tree with time complexity O(n), we should take advantage of the term of routing-load of an edge [2]. In the following, we are presenting some familiar definitions of the MRC as well as proposing several new definitions to serve for our algorithms. Definition 1: Routing-cost Given G= (V,E) is an undirected-connected graph which has non-negative weights (cost). V is the set of vertices, n= V is the number of vertices, m = E is the number of edges. Suppose is an arbitrary spanning tree in G and C() is the total routing-cost of all pairs of vertices of, let d (u,v) be the routing-cost of a pair of vertices (u,v), defined as the total edge-weights of the path between u and v, which obey the following formula: C ( ) = d ( u, v ) (1) u, v V Example 1: Given a spaninng tree as fig.1 Fig 1: his spanning tree having routing-cost 196 According to formula (1), we have: /11/$ IEEE V4-314

2 d (v 1,v 2 ) + d (v 1,v 3 ) + d (v 1,v 4 ) + d (v 1,v 5 ) + d (v 2,v 3 ) + d (v 2,v 4 ) + d (v 2,v 5 ) + d (v 3,v 4 ) + d (v 3,v 5 ) + d (v 4,v 5 ) = 98. Since d (v i, v j ) = d (v j, v i ), so C() = 98 x 2 = 196 Definition 2: Routing load Given is a spanning tree in graph G, suppose E() is the set of edges of, (u,v) E(). As soon as removing (u, v) from ; is separated into 2 sub-graphs having the number of vertices are V(X), V(Y) in turn. Routing-load of an arbitrary edge (u,v) defined as formula: l((u, v))=2. V(X). V(Y) and we have a formula being equivalent to formula (1) as follows: C ( ) = l(,( u, v)). d ( u, v) (2) ( u, v) E ( ) We will compute the routing cost of above tree () obeying formula (2) as follows: l(, (v 1, v 2 )) = = 8, l(, (v 2, v 3 )) = 12, l(, (v 3, v 4 )) = 8, l(, (v 3, v 5 )) = 8. So C() = = 196. Definition 3: Minimum routing-cost spanning tree problem Given G=(V,E) is an indirected- connected graph having non-negative weighs (costs). n= V is the number of vertices, E is the set of edges, m= E is the number of edges. By the previous definition of routing-cost, let s find a spanning tree which has the minimum routing-cost among all possible spanning tree in G. By Cayley theorem, in a complete graph, there are n n- 2 spanning trees [13][14]. Hence, it is impossible to take exhaust-traversing method (all possible cases) into this problem (even incomplete graphs) Our heuristics frequently take responsibility of adding/removing edge from given spanning tree. herefore, we are proposing the following definitions to solve the problems as soon as possible. o compute cost of once insert an edge(u,v) into or remove a certain edge(u,v) ( Of course, is still a spanning tree after adding/removing edges); we needn t start computing again like formulas (1) (2), but, we propose 2-definitions as follows. hese definitions can be easily inferred from definition 1. Definition 4: Given is a tree of graph. he tree {u,v} (where u is vertex in ) has the cost obeying as follows: C( { u, v}) = C(Τ) + 2 d ( v, u) u Definition 5:Given is a tree of graph, \{u,v} (where u is vertex in ) has the cost obeying as follows: C( { u, v}) = C(Τ) 2 d ( v, u) u We use above definitions to compute the cost of spanning tree under constructing process. III. APPLYING HEURISIC ALGORIHMS FOR APPROACHING HE MRC PROBLEM. In this section, we are proposing several heuristics[15]. 3.1 Gradually Edge-Removal algorithm (H 1 ) he idea of Gradually Edge-Removal algorithm is removing gradually bad edges of graph until the remained graph remains only one spanning tree. Step 1: Suppose e is alternately an edge in G. If G-{e} is a connected graph, let s compute the total cost of all pairs of vertices of graph G-{e}; Where the cost of pairs of vertices is their minimum distance. Step 2: Find edge e having minimum cost, and corresponding to graph G -{e} ; in the case if there many edges e that make G -{e} have the same minimum value, then we select the edge which has maximum cost. Step 3: Remove formally edge e from graph, Update G= G-{e} again, this is the input graph for the next iteration. Repeat all above steps until G remains only n-1 edges. Example 2: Given a graph as fig.2, compute its MRC. Fig 2: his fig was quoted from reference material [9]. Next, we are listing a table illustrating steps of removing gradually edge from graph in detail. Edges 1 st loop 2 st loop 3 rd loop 4 th loop 1(v 1,v 2 ) (v 1,v 4 ) (v 2,v 3 ) 184 4(v 2,v 6 ) (v 3,v 4 ) (v 3,v 5 ) (v 3,v 7 ) V4-315

3 8(v 4,v 5 ) (v 5,v 8 ) (v 6,v 7 ) (v 7,v 8 ) able 1: Illustrates steps of H 1 algorithm Attention: the sign - in the table indicates that it is unable to remove the edge being considered, since the remained edges does not form a connected graph. So, according to above table, once apply 1 st heuristic for example 2, we will remove edges (v 2,v 3 ), (v 3,v 5 ), (v 5,v 8 ), (v 6,v 7 ) respectively, and the remained spanning tree having the set of edges {(v 1,v 2 ), (v 1,v 4 ), (v 2,v 6 ), (v 3,v 4 ), (v 3,v 7 ), (v 4,v 5 ), (v 7,v 8 )} corresponding to the cost 232. When we put WONGs algorithm into example 2, WONGs returns 234, and CAMPOSs returns 232. heorem 1: H 1 algorithm has time complexity (m 2 n+ mn 2 logn). he numbers of loops to remove edges take O(n). At each loop, use Dijkstra s algorithm to find the minimum cost of pairs of vertices: Dijkstra s complexity is O(m + nlogn), thus finding cost of all pairs of vertices each loop takes O(mn+ n 2 logn). herefore, the time complexity of H 1 algorithm is O(m 2 n+ mn 2 logn). 3.2 Edge-Replacement Algorithm (H 2 ) he idea of Edge-Replacement bases on Prims and Kruskal algorithms to find the minimum spanning tree of graph, then replace gradually each edge of spanning tree with a better edge. Step 1: Look up a minimum spanning tree (SP) in G using Prim algorithm - O(m+nlogn), or Kruskal algorithm - (m.logm). Step 2: Insert respectively edge e of the sub-graph (G- ) into. Certainly, e will form a cycle. In this new cycle, we need find the best edge e so that - e e has better cost in ; if there exists edge e, then replace with e e. Step 3: Repeat step 2 until there is no replacement done. his Heuristic gives approximate optimal result (since, maybe is not the best but it is impossible to insert properly any edge into ). Example 3: Given a graph G as fig.3: Fig 3: he obtained spanning tree having routing-cost 250, obeying H 2 algorithm. If we apply Edge-Removal algorithm for fig.3, the first obtained minimum spanning tree(ms) having the set of edges {(v 1,v 2 ), (v 1,v 3 ), (v 3,v 5 ), (v 4,v 6 ), (v 5,v 6 )}. Apply each step of Edge-Removal Algorithm: Once replace the first 4-edges (v 1,v 2 ), (v 1,v 3 ), (v 2,v 3 ), (v 2,v 4 ) in, we also obtain a spanning tree having the routing-cost 266; once we replace 5 th -edge (v 3,v 4 ) with edge (v 5,v 6 ), and in that time, we obtained the routing-cost value 250. his is just the desired spanning tree. heorem 2: H 2 algorithm has time complexity O(mn 2 ). he stage of finding minimum spanning tree (SP) using Prim algorithm takes O(m+nlogn). here are m-times for inserting edge into spanning tree: O(n). At each loop, we need find generated cycle: O(n 2 ) and it takes O(n) to compute the routing-cost for each new spanning tree. herefore, H 2 algorithm has time complexity O(mn 2 ). 3.3 Bridge-processing Algorithm or Blossom-shrinking Sub-graphs Algorithm (H 3 ) By the application of the MRC in network designing, it is both natural and common to separate a graph into connected components linked together by bridges. By the following, we would like proposing a method having relative efficiency, in order to find the routing-cost for this form of graph, with approximate result. In this Bridge-processing algorithm, we separate graph into sub-graphs. he sub-graphs are linked together via bridges. Our idea is utilize algorithms H 1,H 2, CAMPOSs, WONGs or even exhaust-traversing to compute cost of subgraphs and continue to process other bridges until we get the cost of initial graph. Given a graph having bridge G = X Y e, where e=(u,v) is a bridge in G, u X, v Y, suppose n 1 = X is the numbers of vertices of sub-graph (X) and n 2 = Y is the numbers of vertices of sub-graph (Y). Let be an optimal spanning tree and x, y respectively are the desired SPs V4-316

4 of sub-graphs X,Y. a i (i =1..n 1 ) is the minimum distance between vertex u and other vertices in X, b i (i=1..n 2 ) is the minimum distance between vertex v and other vertices of Y, and w(e) is bridge weight. As yet, we have a formula: cost() = cost( x ) + cost( y ) + α (3) where: n 1 n2 α = 2. a + i w( e). n1. n2 + bi. n1 i= 1 i= 1 Following is the steps of H 3 algorithm. Step 1: Find connected components of graph using Depth First Search or Breadth First Search [13][14]. Step 2: Use either exact algorithm or approximate algorithm to compute routing-cost for these sub-graphs. Step 3: Process alternately each bridge in graph, For each bridge e(u,v) which connects 2 sub-graphs X and Y together, we proceed to compute a i and b i and α as described in formula (3) for computing cost of spanning tree corresponding to the sub-graph X Y. Next, we joint X and Y together, and consider X Y is a unique sub-graph (Blossom-shrinking) to continue to process in case of either X or Y still has bridges connecting to other sub-graphs. Example 4: Given a graph G as fig.4: Fig 4: he obtained spanning tree obeying H 3, having routing-cost Firstly, by bridge e 1, we compute feasibly a i ={14,11, 6, 2, 0} and b i ={0, 2, 3, 10, 7, 13}. We are able to compute cost(x)=148, cost(y)= 250, and α = 2.[( ) ] = 866. herefore, the graph comprising of the first 11 vertices, having the cost of the obtained spanning tree: cost() = = Afterward, we consider the whole G graph containing 3 above connected components. We next take bridge e 2 in process. At that time, the sub-graph comprising of vertices between v 1 and v 11, having list a i are 14,11,6,2,0, 2, 4, 5, 12, 9, 15 and the sub-graph comprising 8-remained vertices, having list b i are: 0,1,3,1,3,3,5,6. By formula (3), we have: cost() = [( ) )] = = his result is not exact, the exact result must be he valid spanning tree have to select edge (v 16,v 19 ) instead of edge (v 18,v 19 ). Even though, if edge (v 16,v 19 ) is selected, particularly, if the sub-graph contain the series of vertices between v 12 and v 19, this graph must return 234, not the exact value 232. heorem 3: H 3 algorithm finds spanning tree with time complexity O(mn 2 ). It takes O(n) time for finding out a bridge in graph. his is time complexity of the algorithm used for verifying connectivity of graph via Depth First Search or Breadth First Search. hus, to find out all possible bridges in graph, it takes time complexity about O(m.n). o find parameters a i, b i, α like Bridge-processing algorithm, we get a maximum of n-times of using Dijkstra algorithm. So, this stage takes time complexity O(m.n + n 2 logn). herefore, H 3 algorithm has time complexity O(m 2 n). 3.4 Approaching complete-traversing for the MRC For the basic of comparison, verifying approximate algorithms, solving the MRC using exact algorithms is an interesting and much challenge. At this point, we are only mentioning complete-traversing algorithm to solve the problem. Before going into exchanging about how to behave towards special edges in graph hese are edges that we can easily determine if neither it does or does not exist in the desired spanning tree. ake special edges into process [1]. Edges will be inserted into desired spanning trees if they are bridges. [2]. Consider each triple formed an triangle, if there exists an edge which is greater than or equal to the sum of that two edges, remove that edge from graph. Complete-raversing by using Back-racking algorithm. Put Back-racking algorithm into practice to find all possible spanning tree of graph. Compute the routing-cost for each obtained spanning tree and concurrently record the minimum routing cost spanning tree as yet. he spanning having minimum routing-cost is the desired result. Some basic optimizing ideas for applying branch-bound algorithm. If the on-going constructing set of edges is not enough n-1 edges, but formed a cycle, then we needn t find the next edge, we should have to form a new spanning tree. V4-317

5 If the on-going constructing set of edges is not enough n-1 edges, but have routing-cost being greater than the best at that time, then we needn t find the next edge, we should have to form a new spanning tree. IV. EXPERIMENS We have implemented algorithms, such as: Backtracking, H 1,H 2,H 3, WONGs and CAMPOSs and even GA. We have tested above programs with input packets corresponding to graphs in fig.2 (8 vertices), fig.3 (6 vertices), fig.4 (19 vertices). In addition, we have created random data for input files having respectively n = 50 vertices, 100 vertices, 200 vertices, 500 vertices with random edge weights in range of [1..20]. he result of experiments (routing-cost values and running-time) of above implementations are listed below (Be note that while implementing, we have not taken advantage of datastructure Fibonacci heaps). otal vertices otal edges Backtrac king H H H WONG CAMPOS Genetic able 2: Experiment table of algorithms (Routing-cost and running-time). All input data packets were uploaded in our site[16] According to this table, we have several comments: Our approximate algorithms H 1, H 2, H 3 return good result if compare it with algorithms WONG, CAMPOS, or Genetic. (Be note that our implementations of WONG, CAMPOS algorithms, all base on reference materials [8][9], and the implementations of Genetic algorithm base on some selection operators, hybrid operators, and cross-over operators. V. CONCLUSION AND FUURE WORKS hroughout this paper, we have proposed several heuristic algorithms for the MRC in sections 3.1, 3.2, 3.3. We have implemented ours as well as algorithms WONG, CAMPOS and Genetic. We are now searching the numbers of algorithms, such as: Genetic Algorithms, Menetic Algorithms, Harmony Search and Bees Algorithm for the MRC problem and initially have got a bit of interesting results. We hope that the performance of MRC problem will improve significantly if we apply hybrid algorithms or parallel algorithms. [4][5][6][11][12]. REFERENCES [1]. Stefan Ruzika, Horst W. Hamacher, A survey on Multiple Objective Minimum Spanning ree Problems,Springer,2009 [2]. Bang Ye Wu, Kun-Mao Chao, Spanning rees and Optimization Problems,Chapman&Hall/CRC,2004. [3]. J.Dréo, A.Pétrowski, P.Siarry, E. aillard, MetaHeuristics for hard optimization, Springer, [4]. Huynh hi hanh Binh, Nguyen Duc Nghia, Heuristic Algorithms for Solving Bounded Diameter Minimum Spanning ree Problem And Its Application to Genetic Algorithm Development, Advances in Greedy Algorithms, Austria, [5]. Ching-LuehChang, Yuh-Dauh Lyuu, Parallelized approximation algorithms for minimum routing cost spanning trees, [6]. eodor Gabriel Vrainic, Michel oulouse, Parallel Meta-Heuristics, CIRREL,2009. [7]. Bang Ye Wu, Giuseppe Lancia, Vineet Bafna, Kun-Mao Chao, R.Ravi, Chuan Yi ang. A Polynomial ime Approximation Scheme for Minimum Routing Cost Spanning rees. [8]. Rui Campos, Manuel Ricardo, A fast algorithm for computing minimum routing cost spanning trees,2008. [9]. Rui Campos, Spanning rees in Communication Networks. INESC Porto, Faculdade de Engenharia, Universidade do Porto [10]. Matteo Fischetti,Giuseppe Lancia,Paolo Serafini, Exact Algorithms for Minimum Routing Cost rees, [11]. S.N.Sivanandam, S.N. Deepa, Introduction to Genetic Algorithms,Springer,2008. [12]. Zong Woo Geem, Music-Inspired Harmony Search Algorithm heory and Applications, Springer, [13]. J.A Bondy, U.S.R Murty, Graph theory with applications, North- Holland, New York, Amsterdam,Oxford, [14]. Reinhard Diestel, Graph theory, Springer-Verlag Heidelberg, New York, [15]. Phan an Quoc, A heuristic approach for solving the Minimum Routing Cost Spanning rees Problem ", Report of science research department of information technology of SAI GON university, Viet Nam, 2010, unpublished. [16]. Our site: V4-318