An Initial Study of the Multi-Constraint Routing Problem Using Genetic Algorithm

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1 An Initial Study of the Multi-Constraint Routing Problem Using Genetic Algorithm Zhongchao Yu Dept. of Computer Science University of Maryand College Park, MD ABSTRACT Multi-constrained quality-of-service routing (QoSR) is to find a feasible path that satisfies multiple constraints simultaneously, which has been proved to be an NPC problem. In this paper, we propose a GA-based algorithm to solve this problem. The main contribution of this paper is that it invented a new encoding scheme and associated genetic operations which could be used in a class of related problems. For the specific problem we are studying, the experiment result shows that with very small number of iterations and population, it could achieve very hight success ration (SR). Although it does not rival SA MCP in terms of average performance in the simulation we did, it is still valuable in that: (1) it is internally parallel; (2) it can be combined with SA MCP to yield a more efficient algorithm; (3) it has potential to tackle tougher problems. KEYWORDS genetic algorithm, simulation annealing, QoSR, parallel 1 INTRODUCTION The main function of quality-of-service routing (QoSR) is to find a feasible path that satisfies multiple constraints for QoS applications. Formally, this problem is defined as follows. Definition For a given graph G(V,E), a source node s, a destination node t and a constraint vector c =(c 1,c 2,,c k ) (k 2), we call that a path p from s to t is a multiconstrained path (MCP) if w l (p) c l for any 1 l k. We write w(p) c in short. Although QoSR was initially proposed for the IntServ model, it could also be used in the DiffServ model. QoS constraints can be divided into link constraints and path constraints. The link constraints of a path can be converted to the constraints of the bottleneck link in the path, such as bandwidth. It can be easily dealt with in a preprocessing step by pruning all links that do not satisfy these constraints and computing a path from the remaining sub-graph [5]. The path constraint is the overall restriction to all the links along the path, such as delay. We will focus on the path constraint problem in this paper. Many heuristics have been proposed for the multiconstrained QoSR problem because of its NP-completeness [5]. However, most of these algorithms have some or all of the following limitations: (1) High computation complexity, which prevents their practical applications; (2) Low success ration, which means that these algorithms sometimes cannot find a feasible path even when one does exist; (3) Some algorithms work only for a specific network. This paper provides a new heuristic using genetic algorithms. Although this algorithm is not the best in terms of the computation among the heuristic algorithms we have studied, it shows some intriguing characteristics. The main contributions of this paper are: (1) It is the first GA-based algorithm for the MCP problem; (2) It proposes a novel sequence based encoding method for a chromosome representing a path and provides related genetic operations (crossovers and mutations); (3) It can be combined with other heuristics such as SA MCP to increase performance as well as parallelism. 2 RELATED WORK It is known that this problem is NP-complete [4]. Many heuristic algorithms have been proposed to address this problem. Overviews on QoSR can be found in [2, 3]. If some scheduling schemes (e.g. Weighted Fair Queuing) are used, based on the dependencies between QoS parameters, the constraints of queuing delay, jitter, and loss can be formulated as a function of bandwidth. In this case the original NPC problem can be reduced to the standard shortest path problem [6, 7]. Based on this approach, Orda did an extensive study [8]. However, this is not the case for propagation delay, which must be taken into account for QoS routing in high-speed networks [9]. Furthermore, such algorithms can only be applied to networks with specific scheduling schemes. To address the multi-constrained optimization problem, Orda proposed a pre-computation algorithm by mapping the cost to a discrete space [9]. The time complexity is O( 1 Hmlog C), where H is the maximum ε number of hops and C is the upper bound of the cost. By proposing energy functions, multiple QoS weights are translated into a single metric. Jaffe proposed the linear function g(p) =a 1w 1(p)+a 2w 2(p) to solve the 2-constrained

2 problem [10]. He concluded that for given constraints of a QoS request, when a 2/a 1 = c 1/c2 the path found by Dijkstra s algorithm with minimized energy is a feasible solution with very high probability. Based on the nonlinear function g λ (p) = k (w l=1 l(p)/c l ) λ, Neve proposed the TAMCRA algorithm for the multi-constrained path problem for the PNNI protocol and its improved successor, SAMCRA, for IP networks [11, 12]. The algorithm tries to find the path with the minimum g λ (p) by heuristics. The algorithm uses a variation of Dijkstra s algorithm. It can find K shortest paths and store K undominated paths in each node. Its complexity is O(Knlog Kn + K 3 km). Based on reverse label marking, Korkmaz proposed H MCOP for the multi-constrained optimal-path problem [13]. This algorithm marks each node by running Dijkstra s algorithm reversely with g 1(p). Then it runs Dijkstra s algorithm forward with g λ (p), in which it considers both the labels marked reversely and the current forward path constructed partially. Our previous work in [1] used simulated annealing to achieve optimization with the complexity being IK(m + n log n), where I is the number of iterations. The proposed SA MCP algorithm turned out to be very efficient, with only a few number of iterations, SA MCP can achieve a very high success ratio (very close to 100% in the data we have tested). 3 IMPLEMENTATION OF GA MCP In this section, we illustrate the details of the implementation of the GA based algorithm called GA MCP. The high level logic is described as follows. Algorithm GA MCP 1. Generate initial population 2. if there is a feasible individual in current population 3. then return the corresponding path 4. for tm from 1 to T max 5. do generate the next population from current one using GA operations 6. if there is a feasible individual in current population 7. then return the corresponding path 8. return empty path This is only a conceptual logic and can be optimized further. In the real implementation, we check the feasibility of a new path after it is derived using genetic operations and return it immediately if it satisfies all the constraints. In the following, we describe each part concerning the GA implementation. 3.1 Encoding of the Chromosome A solution of the MCP problem is a path which satisfies all the constraints given a source and destination pair (s, t). Thus a chromosome represents such a path. A natural way to represent such a chromosome is to use a bit string with each bit indicating whether the corresponding edge is in the path. Although simple, this representation will result in many useless chromosomes after GA operations because they may not form a path from s to t at all. It implicitly increases the searching space, which results in poor performance. In GA MCP, we do not adopt bit string representing edges. Rather, we use a sequence to represent a path. For example, a path s v 1 v2... t is encoded as [s, v 1,v 2,...,t]. This encoding is very straightforward and more conforms to intuition. Also our implementation is written in Python, a high level scripting language and sequence is a built-in data type. Hence it is easy to prototype. Not all the sequences starting with s and ending with t are considered qualified chromosomes. A chromosome should really represent a path, that is, all the links in the path should exist in the graph. In our implementation, we guarantee that each chromosome we generate is valid in this sense. A path should have no cycles. Actually if any cycle exists, we can always reduce the path to a simpler one with no cycles. If the original one satisfy all the constraints, the reduced one must also satisfy the constraints because all the weights in question are positive real numbers. For example, suppose a feasible path is [s, v 1,...,v k,...,v k,v k+1,...,t]. The path can be reduced to [s, v 1,...,v k,v k+1,...,t]. This is very important for operations such as crossovers because cycles might occur in this process. We will elaborate on it later. 3.2 Fitness Function Fitness is defined as the reciprocal of the energy function. The energy function is defined as g(p) =max k l=1(w l (p)/c l ), where c =(c 1,c 2,...,c k ) is the constraint vector of a specific QoS application. The fitness is defined as fitness(p) = 1 because generally we find the chromosome with the highest g(p) fitness. 3.3 Generation of Initial Population The initial population is generated at raondom. As we said, all the paths we generate must be valid. It is too timeconsuming to generate arbitrary sequences and discard those invalid. We use a varied Dijkstra s algorithm to generate a random valid chromosome. It differs from Dijkstra s algorithm in that first it does not build a shortest-path-tree, rather it randomly selects one edge from the cutting edge set (represented by L in the following Random-Dijkstra algorithm) each time. Second, it does not do real relaxation except to set the parent. The random Dijkstra s algorithm is as follows. Algorithm Random-Dijkstra(G, s, t) 1. Initialize-Single-Source(G, s) 2. S {s} 3. Q V [G] {s} 4. L {(s, u) u Adj[s]} 5. while Q 6. do if L = 7. then return empty path 8. (u, v) random.choice(l) 9. π[v] u 10. Q Q {v} 11. S S + {v} 12. if v = t 13. then build the path and return it 14. for each vertex w Adj[v]

3 15. do if w S 16. then L = L {(w, v)} 17. else L = L + {(v, w)} 18. return empty path Note that any valid path could be possible built by the above algorithm. That is, any valid path is reachable from this algorithm. We do not give its formal proof here but its correctness should be obvious. This guarantees that GA MCP has the potential to search throughout the whole problem space. The time complexity of Random-Dijkstra as well as the following mutation algorithm is crucial to the overall time complexity of GA MCP. So we analyze Random-Dijkstra s time complexity now. For the standard Dijkstra s algorithm (as shown in Algorithm Standard-Dijkstra below for comparison), its running time depends on how the mini-priority queue is implemented [15]. By implementing the min-priority queue with a Fibonacci heap, the standard Dijkstra s algorithm could achieve a running time of O(V lg V + E). Since in Random-Dijkstra, Q does not need to be sorted, the running time is expected to be at least no worse than the standard Dijkstra s algorithm. Thus the time complexity of the standard Dijkstra s algorithm serves as a lower bound for Random-Dijkstra. In practise, it still depends on how Q, S and L are implemented. This in turn begs the question what is the most efficient way to implement a set which supports query, addition and deletion. In our implementation, all the vertices are actually numbered starting from 0 successively. This means that S and Q could be implemented as arrays whose elements can be indexed using the vertex. Since the algorithm does not require sorting, all of query, addition and deletion can be implemented in constant time. It is not the case for L however, whose elements are 2-tuples. If we assume that a perfect hash function exists, all the non-sorting operations can also be implemented in constant time. Thus we could view the time complexity of Random-Dijkstra as almost linear, i.e. O(V + E). Algorithm Standard-Dijkstra(G, w, s) 1. Initialize-Single-Source(G, s) 2. S 3. Q V [G] 4. while Q 5. do u Extract-Min(Q) 6. S S {u} 7. for each vertex v Adj[u] 8. do Relax(u, v, w) 3.4 Selection Method In order to create offsprings for the next generation, we need to make a decision on the selection method we use. GA MCP uses tournament selection. Each time we build up a pool of individuals by choosing np ool individuals at random. Here np ool, the size of pool, is a system parameter, which can be tuned in experiments. We then select the two individuals with the highest fitness values from the pool as the parents. Individuals in the pool are then returned to the original population and can be selected again. Each pair of parents selected are subject to three kinds of genetic operations: crossover, mutation and reproduction with probabilities P c, P m and P r respectively. Reproduction is quite straightforward. In the following, we focus on crossover and mutation. 3.5 Crossover The crossover operations are performed as follows. Given two parents P 1 and P 2, we first compute the set of vertices whichappearinbothp 1 and P 2 other than s or t. We call it the pivot set. We select a vertex v (a pivot) randomly from the pivot set if the set is not empty. Thus P 1 can be exclusively represented by P 1 =[S 1,v,T 1], where S 1 and T 1 are both sub paths. And similarly we have P 2 =[S 2,v,T 2]. The two resulting children will be C 1 =[S 1,v,T 2]andC 2 = [S 2,v,T 1] respectively after the crossover. Note that the pivot set might be empty. We could include s and t in the pivot set, but the resulting children will be the same as their parents after crossover. So in the case where the pivot set is empty, we forward the parents to the next generation without modification. Children may include cycles. For example, P 1 = [s, a, b, c, t] and P 2 = [s, b, d, a, t]. If we choose a as the pivot, the resulting children are C 1 = [s, a, t] and C 2 = [s, b, d, a, b, c, t] respectively. In this example, C 2 has a cycle [b, d, a, b]. For this reason, we need to check whether the children contain cycles after crossover. If they do, we reduce the children. In the above example, the reduced form for C 2 is C 2 =[s, b, c, t]. 3.6 Mutation The implementation of mutation operation is very similar to the above Random-Dijkstra algorithm. Given a path P, we randomly break one of the links in the path. Let the broken link be (u, v). This path is split into two sub-paths, which have the form of S = [s,...,u]andt = [v,...,t] respectively. Our goal in the mutation operation is to find anewpathwhichresembles the original one and does not contain the link (u, v). We use an algorithm similar to RandomDijstra to form the new sub-path. We have two options to build the new path: (1) Keep sub-path S and build a new path from u to t. Concatenate S and the new sub-path; (2) Keep sub-path T and build a new path from s to v. Concatenate the new sub-path and T. Note that the new sub-paths should not contain (u, v), which guarantees that the new path is different from the original one(if one exists).for a given path P subject to mutation, we do either (1) or (2) with equal probability but not both. If the new sub-path excluding (u, v) can not be found, which means that (u, v) is the only bridge that connects two disjointed sets in which s and t reside respectively, mutation can not be performed and we forward the parent to the next generation directly. 4 PERFORMANCE EVALUATION In this section, we will show the performance of GA MCP from two aspects: (1) the relationship between the distribution of QoS constraints and the performance of GA MCP with only two constraints; (2) the performance with multiple constraints. In either part, GA MCP is compared with SA MCP [1] and H MCOP [13, 14].

4 4.1 Theoretical Complexity This subsection analyzes the theoretical running time complexity of GA MCP. As we said in section 3.3, the running time of Random-Dijkstra could be viewed as O(V + E). Time spent in the generation of initial population is NO(V + E), where N is the number of population. In each iteration, the time spent in the selection of individuals is O(N). Both crossover and mutation can be bound by O(V +E). Thus the total time is NO(V + E)+T max(o(n)+o(n)o(v + E)) = NT maxo(v + E). 4.2 Methodology In the experiment, we simulate random network graphs with N nodes and generate k weights for each link, where w l (e) uniform[1, 1000] for l =1, 2,...,k and is independent of l and e. We simulate 10 graphs with N being 100. For each graph, we run the program on 100 different (s, t) pairsin the graph, in which we guarantee that the minimum hop is not less than two. For performance evaluation, we use the success ratio (SR), which is defined as the ratio of the number of requests satisfied to the total number of requests generated. SR is first computed for each graph. The average SR across the 10 graphs is computed thereafter. The evaluation depends heavily on the distribution of constraints. We generate requests with two constraints as follows. Based on the normalized weights in the whole graph, for a given request pair (s, t), we use the method of weighted constraint simulation to generate the constraints. First, we assume that each QoS application is concerned with the weight w l to a l degree. Then we use Dijkstra s algorithm to find the path p(s t) that minimizes the linear energy g(s, t) = k a l=1 lw l (s, t). Finally, we take the weightsofthepathp(s t) as the QoS constraints of the pair, i.e. c(s, t) = w(p(s, t)). In the case of two constraints, we let a 1 [0, 1] and a 2 =1 a 1. Because different QoS applications are concerned with weights to different degrees, we use the following three methods to generate a 1. (1) NORMAL: a 1 normal(0.5, 0.16); (2) UNIFORM: a 1 uniform(0, 1); (3)ABNORMAL: a 1 normal(0, 0.16) and a 1 [0, 0.5]. In order to guarantee that the difference between a 1 and the expectation is less than 0.5 with the probability of 99.7%, we set the standard deviation to be 0.16 in NORMAL and ABNORMAL distributions. For multiple constraints, we first take the random number b l uniform(0, 1) for 2 l k and calculate a l = b l / k b l=1 l. We then construct the least energy path from s to t according to the energy function g l (p) = k l=1 a lw l (p) and take the weights of the path as the constraints of the given (s, t), i.e. c(s, t) =w(p). All of the network graphs and QoS requests generated in simulations could be found at [16]. 4.3 Parameter Tuneup Before we start to do the performance evaluation, we need to figure out the best parameters for the GA algorithm. We are mainly concerned with the relationship between P c, P m and P r. To make life easier, we first fix some of the parameters in the kickoff test, which is shown in Table 1. Note that we also fix P r =0.0 firstbecausep r is relatively trivial compared with the other two. In addition, parents are implicitly reproduced to the next generation when either mutation or crossover fails. SR Parameter Value Size of population 10 Max running time 4 Size of tournament pool 2 P r 0.0 Numberofverticesineachgraph 50 Number of constraints 10 Table 1: Fixed parameters in tuneup test Pm Figure 1: Relationship between P m and SR The reason that we choose a small size of population and a small running time is that we hope it could match SA MCP, which has proved to be very efficient with a few number of iterations. Table 1 also shows that the group of data we chose contained 50 vertices in each graph. Each link has 10 different weights and thus it is a 10-constraints paradigm. The relationship between P m (P c =1 P m) and SR is plotted in Fig.1. From Fig.1, the best value for P m is 0.8 and hence P c = 0.2. We found that with only a few iterations and a small population, GA MCP can not match SA MCP. This is because with only 3 iterations, SA MCP performs very well. This does not mean GA MCP has not value however. We will address this problem later on. Table 2 shows the actual configuration (fixed parameters) in the performance evaluation. Note that we use more iterations and a slightly larger population in the actual simulation to see how well GA MCP can do. We also specify the size of the tournament pool to be 3 rather than 2 because we find 3 does slightly better when we increase the size of population. 4.4 Results and Analysis The final results are shown in Table 3. For each algorithm, we show the SR represented by the percentage of success in

5 Parameter Value Number of vertices in each graph 100 Number of graphs for each constraint type 10 Size of population 20 Max running time 10 Size of tournament pool 3 P m 0.8 P c 0.2 P r 0.0 Table 2: The configuration used in the evaluation meeting the request. For SA MCP, the results are the SR when using only 3 iterations. A very interesting phenomenon is that in multiple constraints when k = 2, which is actually a 2-constraint scenario, GA MCP outperforms H MCOP. However, when we compare their performance for real 2-constraint request (titled normal, uniform and abnormal respectively), GA MCP loses to H MCOP. The reason is not clear yet. It is probably caused by the fact that we use different methods to generate the data for different constraints. But the result wouldn t have been so dramatic. Although generally speaking, the introduction of GA to this domain of problems turn out to be successful, the results for this specific problem demonstrate that GA MCP does not rival SA MCP. But GA MCP is still valuable. First, the way in which the data generated in the simulation favors SA MCP since they are somewhat in the same spirit. Second, GA MCP is internally parallel but SA MCP does not. Third, GA MCP and other algorithms do not conflict with each other; rather, they can work cooperatively. For instance, we can combine GA MCP and SA MCP by always using an individual to perform SA MCP in each generation in GA MCP. At least one copy of the SA MCP individual is forwarded to the next generation. It can also perform mutation or crossover in conjunction with other individuals in the selection process. In this case, SA MCP could introduce some useful genes to accelerate the algorithm. We expect this scheme to be more efficient. Finally, the result only shows that SM MCP outperforms in real time average performance. In our simulation, GA MCP still uses fewer iterations and fewer number of population compared with other GA algorithms. If we increase NPOP and T max, the associated SR is increased greatly as demonstrated by Fig.1 and Table 3. Thus we expect that GA MCP shines in solving much harder problems. Using more iterations and more population, GA MCP is expected to solve more MCP problems than SA MCP although for most of them it is slower. We did not do any experiment in this regard now for not being able to find such a tough problem. All these optimizations will be implemented in our future work if possible. The most significant value of GA MCPisthatitisanew genetic algorithm paradigm for a class of of related problems. We have not seen any similar encoding scheme in the literature using sequence based representation of paths in graph related GA algorithms. It may serve as an effective alternative for some problems. 5 CONCLUSION In this paper, we invented a new encoding scheme and associated GA operations to successfully solve the MCP problem, which has been known to be NP complete. Although the simulation we did on the specific data shows that GA MCP does not rival SA MCP in terms of average performance in real time applications, it is still valuable for its internal parallelism and capability to search through very large problem space. In addition, GA MCP, combined with SA MCP could employee the advantages of both sides to achieve a better hybrid to solve the MCP problem. 6 ACKNOWLEDGEMENTS Thank Yong Cui for his implementation of SA MCP and H MCOP, the simulation data he generated and his constructive suggestions in this work. REFERENCES [1] Yong Cui, Ke Xu, Jianping Wu, and Zhongchao Yu. A Simulated Annealing Based Heuristic for Multiconstrained Routing. Technical Report. Tsinghua 2002 University, [2] Yong Cui, Jianping Wu, Ke Xu, et al. Research on Internetwork QoS Routing Algorithms: A Survey. To appear in Chinese Journal of Software. [3] S. Chen and K. Nahrstedt. An overview of qualityof-service routing for next-generation high-speed networks: Problems and solutions. IEEE Network, Volume: 12 Issue: 6, pp , November [4] Z. Wang and J. Crowcroft. Quality-of-service routing for supporting multimedia applications. IEEE Journal on Selected Areas in Communications, Volume: 14 Issue: 7, Sept. 1996, Page(s): [5] Z. Wang and J. Crowcroft. QoS Routing for Supporting Resource Reservation. IEEE Journal on Selected Areas in Communications. September [6] Q. Ma and P. Steenkiste. Quality-of-Service Routing with Performance Guarantees. 4th International IFIP Workshop on Quality of Service, May [7] C. Pornavalai, G. Chakraborty, and N. Shiratori. QoS based routing algorithm in integrated services packet networks, IEEE ICNP 97, Atlanta, GA. [8] A. Orda. Routing with end-to-end QoS guarantees in broadband networks. IEEE/ACM Transactions on Networking, vol. 7, no. 3, pp , [9] A. Orda and A. Sprintson. QoS routing: the precomputation perspective. IEEE INFOCOM 00, vol. 1, pp , [10] J. M. Jaffe. Algorithms for finding paths with multiple constraints. IEEE Networks, vol. 14, pp. 95-4, 1984.

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