On anycast routing with bandwidth constraint

Transcription

1 Computer Communications 26 (2003) On anycast routing with bandwidth constraint Chor Ping Low*, Choon Leng Tan School of Electrical and Electronic Engineering, Nanyang Technological University, Blk S2, Nanyang Avenue, Singapore, Singapore Received 23 July 2002; revised 9 April 2003; accepted 9 April 2003 Abstract Anycast refers to the transmission of data from a source node to (any) one member in a group of designated recipients in a network. In this paper, we study the problem of finding a set of low cost paths for anycast routing with bandwidth constraint. The anycast routing problem (ARP) in this case involves the construction of a set of low-cost paths with bandwidth requirements, one for each source, for anycasting messages to one of the member in the anycast destination group. An optimal solution to ARP is a set of paths, one for each source node, that incurs the least overall cost. This problem is known to be NP-complete and hence a heuristic algorithm is proposed to find solutions for the problem. In addition, we derive a lower bound on the cost of an optimal solution for ARP by using Lagrangean Relaxation and Subgradient Optimization techniques. This lower bound is used to evaluate the performance of anycast routing algorithms in terms of their ability to find close-to-optimal solution. Simulation results show that the lower bound is tight and that our proposed algorithm is able to achieve good performance in terms of its ability of finding a set of low-cost paths for the problem. q 2003 Elsevier Science B.V. All rights reserved. Keywords: Anycast routing; Heuristic algorithm; NP-completeness; Lower bound 1. Introduction Anycast is a new communication service defined in Ipv6 [1]. An anycast message is one that should be delivered from a source node to one member in a group of designated recipients in a network. The traditional unicast message is a special case of an anycast message in the sense that for unicast message, the recipient group size is one. Using anycast communication service may considerably simplify some applications. For example, it is much easier for a client to send a message to an appropriate server when there are multiple servers for the same kind of service in a network. Another example is in the case of multiple mirrored Web sites. These Web sites can share a single anycast address and users could simply send a request with the anycast address in order to obtain information. An anycast packet is specified by the addresses of its source and destination. The destination for an anycast packet can be any member of the group of pre-defined hosts. Given an anycast packet with anycast (destination) address A; let GðAÞ denote the group of designated recipients. Let G s ðaþ denote the group of source hosts that may send * Corresponding author. Tel.: þ ; fax: þ address: icplow@ntu.edu.sg (C.P. Low). packets with anycast address A: The anycast routing problem (ARP) is that of finding a set of paths, one for each source node in G s ðaþ; to a member of GðAÞ such that the overall cost incurred is minimized. This problem can be shown to be NP-complete [2] and thus it is unlikely that optimal solutions for ARP can be found in polynomial time. Thus, heuristic algorithms, which run in polynomial time but are not able to guarantee optimal solutions, are likely to be the only viable approach to the problem. There are two types of approaches for routing anycast packets: Single-path routing and multi-path routing. Singlepath routing always uses the same path for transmitting anycast packets from a source to a member of GðAÞ; while multi-path routing splits anycast traffic into several different paths. The anycast routing protocols proposed in Refs. [3,4] are based on these two routing approaches. The shortest shortest path (SSP) method in Ref. [4] is based on singlepath routing. The minimum distance method (MIN_D), source based tree (SBT) method, and core based tree (CBT) method in Ref. [4] are multi-path routing approaches. In Ref. [5], an integrated approach that makes use of singlepath routing and multi-path routing is proposed. While single-path routing is simple and easy to implement, it suffers from the drawback of overloading some links in the selected path and cause traffic congestion /03/$ - see front matter q 2003 Elsevier Science B.V. All rights reserved. doi: /s (03)

2 1542 C.P. Low, C.L. Tan / Computer Communications 26 (2003) Thus, the throughput is limited. The multi-path routing approach addresses this problem by splitting traffic into several paths. However, the implementation of multiple path routing algorithms is more involved and additional processing has to be done at the receiving end to ensure that packets are received in the correct sequence. To take advantage from both approaches and overcome their shortcomings, we develop a routing algorithm that is based on the single-path routing approach but taking the link bandwidth into consideration. We define the residual bandwidth along a link to be the difference between the available bandwidth of the link and the sum of the source demands (for bandwidth) that are routed on that link. We define the bottleneck bandwidth of a path as the minimum residual bandwidth among all edges on the path. Our proposed routing algorithm is based on the approach of finding a shortest widest path (SWP) from a source to GðAÞ: In particular, we use breadth first search (BFS) to find paths with maximum bottleneck bandwidth from a source node to all the destination nodes and the path with the lowest cost among them is chosen as the candidate path for routing anycast packet. The chosen path is called the SWP from the source to GðAÞ: In order to evaluate the performance of our proposed algorithm in terms of their ability to generate close-tooptimal solutions, it is desirable to find a lower bound (which can be computed quickly) on the cost of an optimal solution. By comparing the cost of the solutions obtained from a heuristic algorithm with the lower bound, we would be able to evaluate how much the costs of these heuristic solutions differ from the optimal solutions. In this paper, we derive a lower bound for ARP by using the Lagrangean Relaxation technique [6] and problem decomposition. We are able to establish the fact that the lower bound is tight and can be efficiently computed. Empirical study is carried out to evaluate the performance of our proposed algorithm using this lower bound. The organization of this paper is as follows. Section 2 gives a formal definition of the ARP. Section 3 describes our proposed algorithms and the derivation of a lower bound for the ARP. Simulation results are presented in Section 4 and some concluding remarks are given in Section 5. of paths {P 1 ; P 2 ; ; P k } from each member of G s ðaþ to any (one) member of GðAÞ: Each path P i ¼ðV 0 ; E 0 Þ; where V 0, V and E 0, E; represents the path from the node s i [ G s ðaþ to one of the member node of GðAÞ: The path P i may also contain some nodes from the set V G s ðaþ GðAÞ; which we call relay nodes. To formulate the ARP, we introduce the following zero one variables: ( y t ij ¼ 1; if edge ði; jþ is included into the path from s t to d t 0; otherwise where d t denotes the destination node selected by the source s t : The ARP can then be formulated as follows: Minimize X X k c ij y t ij subject to: X y t ih 2 X y t ji h[v j[v 8 1; i ¼ s t ; s t [ G s ðaþ; t [ {1; 2; ; k} >< ¼ 21; i ¼ d t ; d t [ GðAÞ; u [ {1; 2; ; m} >: 0; i s or d t X k t¼1 ð1þ By t ij, b ij ; ði; jþ [ E ð2þ y t ij ¼ 0; 1; ði; jþ [ E; t [ {1; 2; ; k} ð3þ The variable to be determined is y t ij for every edge ði; jþ [ E and s t [ G s ðaþ: Constraint (1) ensures one unit of flow between the source node and the destination node. Constraint (2) ensures that the total bandwidth utilized on each link does not exceed its bandwidth capacity. y t ij is confined to zero one variables in constraints (3). A set of paths {P 1 ; P 2 ; ; P k } which satisfy constraints (1) (3) is called a feasible solution to the ARP. A feasible solution which incurs the least cost is known as an optimal solution. An edge is saturated if its residual bandwidth is less than the amount of bandwidth required by a source. 2. Network model and problem formulation Generally a network is modeled as a directed graph GðV; EÞ with node set V and edge set E; and lvl ¼ n; lel ¼ q: Each edge ði; jþ [ E has two parameters, namely bandwidth capacity b ij and cost c ij For each edge ði; jþ; b ij is known as the bandwidth of the link from node i to node j: Given a graph G ¼ðV; EÞ; let G s ðaþ ¼{s 1 ; s 2 ; ; s k }; GðAÞ ¼{d 1 ; d 2 ; ; d m };where G s ðaþ, V; GðAÞ, V and lg s ðaþl ¼ k; lgðaþl ¼ m: Each source node in G s ðaþ has a bandwidth requirement of B units on each edge that it uses to send data to destination. The ARP is that of finding a set 3. New proposed algorithm In this section, we outline a new algorithm for routing anycast packets. We first describe the approach for finding a path from source to destination. Following that we discuss how the problem of saturated edges can be handled by our proposed algorithm Finding a path from source to destination In the process of constructing a path for some source node s t [ G s ðaþ to a destination node d t [ GðAÞ; some

3 C.P. Low, C.L. Tan / Computer Communications 26 (2003) edges in the given network may become saturated. These saturated edges could not be used in the construction of paths for subsequent source nodes. The exclusion of these saturated edges may result in a disconnected network. This imply that we may not be able to find paths for subsequent source nodes to the set of recipient nodes and thus a feasible solution for ARP could not be found. To prevent the network from being disconnected, one possible strategy is to minimize to probability of having saturated edges in the network. One possible method is to remove all edges in E that have bandwidth smaller than the bandwidth required by all the sources [7]. However, this method is not efficient when the number of source nodes is large and it may disconnect the network. In addition, the cost of solution obtained using this method is typically much higher than the optimum. In this section, we introduce a procedure, called Algorithm Anycast Routing (AR), for finding a path with large bottleneck bandwidth to reduce the possibility of hitting saturated edges. This procedure is carried out by using breadth first search (BFS) to find paths with maximum bottleneck bandwidth from a source node to all the destination nodes. The path with the smallest cost among them is chosen as the candidate path for routing anycast packet from the source to the corresponding recipient. The selected path is called the SWP from the source to the corresponding destination. Fig. 1(b) (e) shows how the BFS tree is generated with node 1 as the source node for the network shown in Fig. 1(a). In Fig. 1(a), the number in the middle of the edge is cost and the numbers near the arrows at the two ends of an edge represent the bandwidth capacity of the edge in the direction of the arrows. In Fig. 1(b) (e), the two numbers in the square bracket represent the total cost and bottleneck bandwidth of the path from node 1 to the corresponding node respectively. Starting from node 1, its neighboring nodes, namely node 3 and node 5, that have not been visited are included into the BFS tree with bottleneck bandwidth 5 and 1 respectively, as shown in Fig. 1(b). Next, we examine node 3 s neighbors. Node 2 is included into the BFS tree and node 5 is shifted as a child of node 3 because the bottleneck bandwidth of the path 1! 3! 5 is 2, which is greater than the bottleneck bandwidth of the path 1! 5, as shown in Fig. 1(c). Similarly, node 6 and node 4 are included in the BFS tree when we examine node 2 and node 5 respectively, as shown in Fig. 1(d). When node 4 is examined, node 6 is shifted as a child of node 4 because the bottleneck bandwidth of the path 1! 3! 5! 4! 6 is 2, which is greater than the bottleneck bandwidth of the path 1! 3! 2! 6. The final BFS tree is shown in Fig. 1(e). If GðAÞ ¼{4; 6}; we will choose the path 1! 3! 5! 4 because this is the shortest-widest path. After a path from a source to destination is found, the link bandwidth is updated and the same process continues with the next source until either (i) all the source nodes have found a path to some destination node or (ii) edge is saturated and a path could not be found from some source Fig. 1. Generation of BFS tree rooted at node 1. node to any destination node. If situation (ii) occurs, we will use the algorithm described in the next section to handle it Resolving the problem of saturated edges In this section, we propose a heuristic algorithm to find a solution for the ARP when saturated edges disconnect the network as described in the previous section. Suppose we are unable to find any path from some source node s i to any of the destination nodes due to the existence of some saturated edges, where 1 # i # k: We first find the set of saturated edges that are included in the SWP from s i to some destination node in GðAÞ and we call it E st : This can be achieved by using the BFS strategy as described in the previous section. Next, we determine the set of paths S that pass through each of the saturated edge in E st : Note that the paths in S must have been used previously to route from some source node s r to some destination node in GðAÞ; where 1 # r, i: One possibility here is that there exist a path from some source s r ð1 # r, iþ that pass through all saturated edges in E st : The second possibility is that each path in S contains only some of the saturated edges in E st : For the first scenario, we will find an alternative path P 0 r for the source s r without including the saturated edges in E st :

4 1544 C.P. Low, C.L. Tan / Computer Communications 26 (2003) If there exists more that one path that pass through all edges in E st ; the path with minimum alternative overhead will have to give up the saturated edges and use the alternative path to get to destination. The alternative overhead is the difference in cost between the original path and the alternative path. For the second scenario, we will order the set of paths in S in decreasing order of the number of saturated edges that each path contains. Let P total denotes the resultant set of sorted paths. We begin by deleting the first path P l from the sorted list in P total (i.e. P l contains the most number of saturated edges among the paths in P total ). We will attempt to find an alternative path P 0 l for the corresponding source s l ; which does not contains any of the saturated edges in E st : If this path can be found, the set of saturated edges in P l will be deleted from the set E st : We next proceed to examine the second path in the sorted list. This process continues until there are no more saturated edges in E st ðe st ¼ fþ: This implies that a path can be constructed from the source s i to some destination node and a feasible solution is found. If P total ¼ f but E st f; then the heuristic algorithm fails to find a feasible solution for ARP. The outline of this algorithm is shown in Fig. 2. Lemma 1. The time complexity of Algorithm Anycast Routing is Oðk 2 n 2 Þ: Proof. The SWP in steps 2, 3, 7, 11 could be found by BFS in Oðn 2 Þ: The saturated edges in step 4 could be found in Fig. 2. Algorithm Anycast Routing (AR).

5 C.P. Low, C.L. Tan / Computer Communications 26 (2003) Fig. 3. Oðn 2 Þ: The paths through saturated edges in step 5 could be found in OðkeÞ: The for loop in step 6 could be done in Oðkn 2 Þ: The path with minimum overhead in step 8 could be computed in OðkÞ: In step 9, the paths could be arranged in OðkeÞ: Step 10 could be done in OðeÞ: Hence, the while loop in step 12 takes Oðkn 2 Þ: Therefore, each iteration of the for loop in step 1 takes Oðkn 2 Þ and the worst case time complexity of the algorithm is Oðk 2 n 2 Þ: A We will take the network model in Fig. 3 as an example. Let G s ðaþ ¼{1; 2; 3}; GðAÞ ¼{4; 6}: We assume that one unit of bandwidth will be consumed for every edge that is included into the path. By using the BFS strategy described above, we are able to find the shortest-widest path from node 1 and node 2 to any member in the destination group (P 1 ¼ 1! 3! 5! 4 and P 2 ¼ 2! 3! 5! 4; with cost equals to 30 and 110, respectively). Because the edge 3! 5 is saturated, we are unable to construct a path from node 3 to destination. Thus, edge 3! 5 is included in E st and the paths for nodes 1 and 2 pass through this saturated edge. Next, we will find an alternative path for node 1 and node 2 without including the saturated edge 3! 5, the alternative paths are P 0 1 ¼ 1! 5! 4 (cost is 90) and P 0 2 ¼ 2! 6 (cost is 260). The alternative overheads incurred for node 1 and node 2 are ¼ 60 and ¼ 150, respectively. Thus, node 1 which has smaller alternative overhead will have to give up the saturated edge 3! 5 and use the alternative path P 0 1 ¼ 1! 5! 4:With this, by applying the BFS strategy at node 3, the shortest-widest path found is P 3 ¼ 3! 5! 4: Thus, the resultant paths for nodes 1, 2, and 3 to the GðAÞ are P 0 1 ¼ 1! 5! 4; P 2 ¼ 2! 3! 5! 4; and P 3 ¼ 3! 5! 4; respectively. The total is 220, which is equal to the optimal cost. As mentioned above, if there are multiple saturated edges and none of the path passes through all the saturated edges, the path that passes through the saturated edges most frequently will have to give up the saturated edges and use the alternative path to the destination. We will take the network model in Fig. 4 as an example. Let G s ðaþ ¼{1; 2; 3; 4} and GðAÞ ¼{5; 9; 10}: By using the BFS method mentioned above, the shortest-widest path for node 1, 2, and 3 are P 1 ¼ 1! 6! 7! 8! 9; P 2 ¼ 2! 4! 6! 5; and P 3 ¼ 3! 7! 8! 10; respectively. We are unable to find a path from node 4 to destination because some edges are saturated. The saturated edges are (4! 6), (6! 5), (1! 6), (6! 7), (7! 8), and (8! 9). Next, we used the original graph (graph that without consider the path P 1 ; P 2 ; and P 3 ) and apply the BFS method to find the SWP for node 4, that is P 4 ¼ 4! 6! 7! 8! 9: From P 4 ; we can see that only saturated edge (4! 6), (6! 7), (7! 8), and (8! 9) should be included in E st : Thus, none of the path among P 1 to P 3 passes through all the saturated edges in E st : Next, we arrange the paths from P 1 to P 3 according to its appearance in the saturated edges in E st and store them in a list P total : Thus, P total ¼ {P 1 ; P 2 ; P 3 }: This is because P 1 passes through (6! 7), (7! 8), and (8! 9), appearances ¼ 3; P 2 passes through (4! 6), appearances ¼ 1; P 3 passes through (7! 8), appearances ¼ 1. Then, delete a path P form P total ; we get P ¼ P 1 and P total ¼ {P 2 ; P 3 }: We find an alternative path P 0 1 ¼ 1! 5 and delete saturated edges (6! 7), (7! 8), and (8! 9) form E st : Thus, E st ¼ {ð4! 6Þ}: Since only P 2 passes through (4! 6), we get P total ¼ {P 2 } and find an alternative path P 0 2 ¼ 2! 5: Saturated edge (4! 6) is then deleted from E st ; we get E st ¼ f and a feasible solution for ARP is found. Thus, the resultant paths for node 1, 2, 3, and 4 to the GðAÞ are P 0 1 ¼ 1! 5; P 0 2 ¼ 2! 5; P 3 ¼ 3! 7! 8! 10; and P 4 ¼ 4! 6! 7! 8! 9; respectively. The total cost is 840, which is 10.53% greater than the optimal cost (optimal solution is P 1 ¼ 1! 5; P 2 ¼ 2! 5; P 3 ¼ 3! 7! 8! 9; and P 4 ¼ 4! 6! 5; optimal cost ¼ 760) A lower bound for ARP Fig. 4. By applying Lagrangean decomposition technique [6] to ARP, we can obtain a lower bound on the cost of an optimal solution. We first introduce an artificial zero one variable x t ij that is equal to y t ij : x t ij ¼ y t ij ( ¼ 1; if edge ði;jþ is included into the path from s t to d t 0; otherwise

6 1546 C.P. Low, C.L. Tan / Computer Communications 26 (2003) The original problem is then transformed into: Minimize X X k c ij y t ij subject to: X x t ji x t ih 2 X h[v j[v 8 1; i ¼ s t ; s t [ S; t [ {1;2; ;k} >< ¼ 21; i ¼ d t ; d t [ D; u [ {1;2; ;m} >: 0; i s t ; i d t X k t¼1 ð4þ By t ij, b ij ; ði;jþ [ E ð5þ x t ij ¼ y t ij; ði;jþ [ E; t [ {1;2; ;k} ð6þ y t ij ¼ 0;1; ði;jþ [ E; t [ {1;2; ;k} ð7þ x t ij ¼ 0;1; ði;jþ [ E; t [ {1;2; ;k} ð8þ It is clear that the original problem and this transformed problem are equivalent. Using the Lagrangean decomposition method, we relax the equality constraint (6) and the resultant optimization problem, called the Lagrangean Program (LP) is as follows: Minimize: X X k c ij y t ij þ X X k l t ijðx t ij 2 y t ijþ subject to constrains (4), (5), (7) and (8), where l t ij are nonnegative Lagrangean multipliers. It can be shown [6] that the optimal solutions to problem LP must be lower bounds to ARP. It is clear that the LP is decomposable into the following two sub-problems: Sub-problem LP1: X X k ðc ij 2 l t ijþy t ij subject to constraints (5) and (7) Sub-problem LP2: X X k l t ijx t ij subject to constraints (4) and (8). The sum of the solutions to these two programs (for any value of l t ij) provides a lower bound on the optimal solution to the original problem. LP1 can be minimized as follows: For each edge ði; jþ; we first sort the list {c ij 2 l t ij; t ¼ 1; 2 ; k} in increasing order. The variable y t ij is set to one if ðc ij 2 l t ijþ, 0 and set to 0 otherwise. If y t ij is set to one and b units of bandwidth is requested by the source, the bandwidth capacity b ij on the edge ði; jþ will decremented by b units to reflect the fact that b units of bandwidth is consumed by the path from source t to destination node. This process continues with the next element in the sorted list and until (i) the available bandwidth b ij is less than b units or (ii) all elements in the sorted list have been considered. The outline of this algorithm is shown in Fig. 5. To minimize LP2, we can use l t ij as the weight of each link and find the SSP from each source node to any destination node based on these weight values. This can be done using Dijkstra s algorithm. Fig. 5. Procedure LP1.

7 C.P. Low, C.L. Tan / Computer Communications 26 (2003) Determination of Lagrange multiplier Choosing values for the Lagrange multipliers is of key importance in terms of the quality of the lower bound generated. We use the subgradient optimization method [6] to generate the best possible values of Lagrange multipliers iteratively. Subgradient optimization is an iterative procedure that generates further multipliers from an initial set of Lagrange multipliers in a systematic fashion. It can be viewed as a procedure that attempts to maximize the lower bound value obtained from LP by suitable choice of multipliers. The main idea is that we update the multiplier l t ij in each iteration by a specified step size along the gradient direction x t ij y t ij for our relaxed constraint to improve l t ij iteratively. If we cannot find a better l t ij in the N iterations, we reset l t ij to the multiplier value found in the previous Nth iteration and then improve l t ij again using a halved step size. When the step size becomes sufficiently small, we can stop the execution. To present the details of the subgradient optimization algorithm, we define the following notations: Let r an iteration counter p a user-defined parameter which specifies the step size and satisfying 0, p # 2 1 a close-to-zero threshold parameter z ub a upper bound cost for LP z lbmax the current best lower bound cost zðrþ the current lower bound in the rth iteration The procedure of subgradient optimization is as follows: 1. Initialize p; z ub ; l t ; and 1: ij 2. Solve the LP with the current set of multipliers to get a solution of value zðrþ and z lbmax : 3. Define subgradients g t ij for the relaxed constraint, evaluated at the current solution by: g t ij ¼ x t ij 2 y t ij 4. Define a step size T by T ¼ pðz ub 2 z lbmax Þ X X k g t2 ij This step size depends upon the gap between the current best lower bound cost z lbmax and the upper bound cost z ub and the user defined parameter p with P P kt¼1 ði;jþ[e g t2 ij being a scaling factor. 5. Define a variable f by f ¼ X X k g t ij If f ¼ 0; then a solution is found and the process is terminated. Otherwise, continue to next step. 6. Update l t ij using l t ij ¼ l t ij þ Tg t ij and go to step 2 to resolve LP with this new set of multipliers. 7. If we cannot find a better z lbmax value after N subgradient iterations with the current value of p; then reduce p by half and set l t ij to the value as the previous Nth iteration. The process continues until p is less than a specific small value 1: 4. Empirical studies 4.1. Random graph generation To ensure that simulations are fairly evaluated, random graphs are generated by using the method proposed by Waxman [8]. Graphs with low average degrees are constructed. The nodes are randomly placed on a rectangular grid and nodes are connected with the probability function: 2dðu; vþ Pðu; vþ ¼l exp rl where dðu; vþ is the distance between node u and v and L is the maximum possible distance between any pair of nodes. The parameters l and r are in the range (0,1]. They can be modified to create the desired network model. For example, a small value for l gives nodes with a low average degree, and a large value for r decrease the density of shorter links relative to longer ones. We define the cost of an edge ðu; vþ as the distance between node u and v: In our simulations, l is set to 0.25 and r is set to 0.2. The bandwidth capacity of each edge is allocated using the following function: Bðu; vþ ¼B 0 þ rð21þ k mod B 0 where B 0 is the mean bandwidth while r and k are random numbers. Using this function no bandwidth with negative value will be generated and the bandwidth capacity of all edges will range from 1 to 2B 0 2 1: Graphs are generated and tested until a single connected component is found. In the construction of each path from source to destination, we assume that one unit of bandwidth will be consumed for every edge that is included into the path. In our simulations, B 0 is set to Simulation results In order to evaluate the performance of our proposed heuristic algorithm (Algorithm Anycast Routing), we compare it with the SBT, CBT, and ISM. In addition, an exhaustive search algorithm (ES) is used to compute the optimal solutions for ARP. These optimal solutions will be used to evaluate the effectiveness of our proposed algorithm in finding a low-cost feasible solution for ARP. In addition, the performance of the proposed lower bound algorithm is evaluated by comparing its performance with that of the exhaustive search algorithm. To study the performance

8 1548 C.P. Low, C.L. Tan / Computer Communications 26 (2003) Fig. 6. Total cost vs. network size. of our proposed heuristic algorithm in large networks, we compare its performance with that of the lower bound algorithm. The empirical results will be discussed in the following sections. We will study the cost/delay performance of our proposed routing algorithm in comparison with the other existing algorithms. Our simulations are carried out in a 64 MB PIII 1 GHz PC Performance in small network Inthisempiricalstudy,wevarythenetworksize,numberof source nodes and number of destination nodes to evaluate the performance of our routing algorithm in response to changes in these parameters. The total cost shown at each data point is the average value calculated from 1000 feasible solutions. Fig. 6 shows the overall network cost vs. the network size. The number of source nodes and destination nodes are fixed at 3 and 2, respectively. The network size ranges from 5 to 15. In general, the solutions obtained from our proposed algorithm differ by no more than 32.36% from the corresponding optimal values. We also observe that our proposed algorithm performs better than the CBT, SBT, and ISM in that the cost obtained by our proposed algorithm is much lower than these three algorithms. In particular, the percentage of cost improvement achieved by our proposed algorithm as compared to CBT, SBT and ISM is at least 33.17, and 13.77% respectively. Fig. 7 shows the overall network cost vs. the number of source nodes. The number of destination nodes is fixed at 2 Fig. 8. Total cost vs. number of destinations. and the network size is fixed at 15 nodes. As expected, the total cost increases as the number of source nodes increases. This is because as the number of source nodes increases, more paths are needed and hence total cost will be larger. Again, we observe that our proposed algorithm performs better than CBT, SBT, and ISM. Fig. 8 shows the overall network cost vs. the number of destinations. The number of source nodes is fixed at 3 and the network size is fixed at 15 nodes. We observe that as the number of destinations increases, the total cost obtained decreases. This is because as the number of destination node increases there are more chances of finding lower cost paths to the set of destination nodes. We again observe that our proposed algorithm is able to achieve a better performance than the other three algorithms Performance of lower bound algorithm In this section, we study the performance of the proposed lower bound algorithm by comparing its performance with that of an exhaustive search algorithm. The results are tabulated in Tables 1 3. For each data point, 100 runs are taken and the average calculated. The set of parameters for the lower bound algorithm is initialized as follows: p ¼ 2; z ub ¼ 500; l t ij ¼ 1; and 1 ¼ : Table 1 shows the performance of the lower bound algorithm for different network sizes. The number of source nodes and destination nodes are fixed at 3 and 2, respectively. The network size ranges from 5 to 15. We see that the lower bound is very tight and it differs from the optimal solution by only 0.1% on average. Tables 2 and 3 shows the numerical results for different number of source nodes and destination nodes, respectively. The network size is fixed at 15 nodes in both cases. In Table 2, the number of Table 1 Numerical results for different network sizes Network size (nodes) Optimal cost Lower bound cost % of Cost difference % of Time needed, s Fig. 7. Total cost vs. number of sources

9 C.P. Low, C.L. Tan / Computer Communications 26 (2003) Table 2 Numerical results for different number of sources Number of sources Optimal cost Lower bound cost % of Cost difference % of Time needed, s destination nodes is fixed at 2 while in Table 3, the number of source nodes is fixed at 3. We observe that the lower bound cost follows the optimal cost very tightly in both tables. In Tables 1 3, the % of time needed, s is defined as below: running time of lower bound procedure s ¼ 100 running time of exhaustive search procedure Note that the running time of lower bound algorithm will be less than that of exhaustive search for values of s less than 100. From Tables 1 3, it is easy to see that the running time of the lower bound algorithm improves significantly over that of the exhaustive search algorithm for all cases. We observe that the value of s varies between and Thus, the experimental results shows that the lower bound algorithm is able to achieve between and % reduction in running time over that of the exhaustive search algorithm while the deviation in cost is only at most % Performance in large network Next, we compare the performance of the algorithms in large network. Note that as the network size increases, the computational time required by the exhaustive search technique can become exceedingly high due to the large solution space that the algorithm will have to search. Thus, lower bound cost will be used to evaluate the performance of the algorithms in large networks. Fig. 9 shows the performance of our proposed algorithm in large network. The network size is ranging from 200 to 1000 nodes. The number of source nodes and destination nodes are fixed at 60. Again, we observe that our proposed algorithm performs better that CBT, SBT, and ISM. In addition, we observe that the costs obtained from our proposed algorithm differ by no more than 27.30% of the lower bound value. From Fig. 9, we observe that the performance of the ISM is close to our proposed heuristic algorithm. To do a closer Table 3 Numerical results for different number of destinations Number of destinations Optimal cost Lower bound cost % of Cost difference % of Time needed, s comparison between these two algorithms, we next list down the mean values and standard deviations of cost obtained by each of these algorithms for different network sizes, as shown in Table 4. From Table 4, we observe that all the mean values and variances of ISM are greater than that of Algorithm Anycast Routing (AR). This implies that in general Algorithm AR is able to achieve reduction in cost in comparison with the ISM. Thus, we can conclude that Algorithm AR perform better than the ISM. 5. Conclusion Fig. 9. Total cost vs. network sizes. In this paper, we proposed a new routing algorithm, called Algorithm Anycast Routing, for the ARP. Our algorithm takes advantage of single-path routing, which is efficient and simple, and at the same time it takes bandwidth into consideration to reduce the possibility of congestion. The algorithm is based on BFS strategy to find the SWP. If saturated edges disconnect the network and a path from source to destination could not be found, the paths through the saturated edges will be examined and alternative paths will be used so that we are able to construct a set of paths from each of the source node to some destination node. We compare the performance of our proposed algorithm with the multi-path routing algorithms (SBT and CBT) and the integrated single and multi-path routing algorithm (ISM). Simulation results show that the proposed heuristic performs substantially better than these approaches. In addition, an exhaustive search algorithm and a lower bound algorithm are used to evaluate our proposed algorithm in terms of its ability to Table 4 Mean ðmþ and standard deviation ðsþ of AR and ISM Network size (nodes) AR ðm; sþ ISM ðm; sþ 200 ( , 28.35) ( , 35.32) 400 ( , 31.06) ( , 38.38) 600 ( ,33.10) ( , 39.74) 800 ( , 33.21) ( , 40.05) 1000 ( , 34.07) ( , 41.26)

10 1550 C.P. Low, C.L. Tan / Computer Communications 26 (2003) find close-to-optimal solutions. Our empirical studies shows that the lower bound obtained by our proposed algorithm is tight and that our proposed ARP algorithm is able to obtain close-to-optimal solutions. References [1] S. Deering, R. Hinden, Internet Protocol Version 6 (Ipv6) Specification, RFC 2460, December [2] C.P. Low, X. Song, On finding feasible solutions for the delay constrained group multicast routing problem, IEEE Transactions on Computers 51 (5) (2002) [3] D. Xuan, W. Jia, W. Zhao, Routing algorithms for anycast messages, Proceedings of IEEE Conference on Parallel Processing 122 (1998) 130. [4] D. Xuan, W. Jia, W. Zhao, H. Zhu, A routing protocol for anycast messages, IEEE Transactions on Parallel and Distributed Systems 11 (6) (2000) [5] D. Xuan, W. Zhao, Integrated routing algorithms for anycast messages, IEEE Communications Magazine 38 (2000) [6] C.R. Reeves, Modern Heuristic Techniques for Combinatorial Problems, Wiley, New York, 1993, pp [7] A. Bagchi, Route selection with multiple metrics, Information Processing Letter 64 (1997) [8] B.M. Wazman, Routing of multipoint connections, IEEE Journal on Selected Areas in Communications 6 (9) (1988)