Genetic algorithms and network ring design

Optimal network ring design is a difficult problem characterised by the requirement to compare a large number of potential solutions (network designs). The problem of network ring design can be described as consisting of three parts: routing, link capacity assignment and ring determination. It has traditionally been broken down into a number of subproblems, solved in sequence, and usually by heuristics thereby leading to locally-optimal design solutions. Genetic Algorithms (GAs) have shown themselves to be efficient at searching large problem spaces and have been successfully used in a number of engineering problem areas, including telecommunications network design. We present an approach of a GA to the network ring design problem in which the GA representation encapsulates all aspects of the problem and solves them simultaneously. A novel, hybrid bit and permutation representation is described along with the fitness function for the design problem. Results of applying this representation to a number of test networks are presented and compared with heuristic design methods. Introduction The design of an optimal, or cost effective, telecommunications network is a complex multifaceted task. Typical problem domains consider topology, connectivity and routing decisions. Restricting attention to the question of an optimal topology, for example, if we consider an n node network, and allow for p possible bandwidths for a link, the space of potential topologies is . The term (p+1) is used in order to allow for a link of zero capacity, i.e. not present, to be represented. For the values n = 10 and p = 3, this evaluates to 1.2x1027 possible designs. Even for this small problem, the search space is extremely large. Obviously, enumeration of all potential designs is impossible and effective heuristic search techniques need to be employed. One such technique is the Genetic Algorithm [1]. The above description of the problem does not include routing, or ring design, considerations which must be taken into account in any realistic solution. The actual problem that we wish to solve, with a search space that is correspondingly larger, is to determine (a) the routes used by all traffic in the network, (b) the link capacities, (c) and the multi-ring topology of the network such that the overall cost of the network is minimised. Cost here is taken to mean the weighted sum of parameters of interest such as the link length, the monetary cost of providing link bandwidth, and the deviation of each link’s utilisation from the average (in order to balance link use i.e. spread link usage as much as possible). In addition, cost will include a term dependent upon the multi-ring topology of the network, the main objective of p 1 + ( ) n n 1 – ( ) ( ) 2 ⁄ this aspect is to minimise inter-ring traffic as such traffic requires specialised crossconnect hardware in real networks. The corresponding weighting factors used in this multiple objective optimisation problem are determined by the priorities of the global design problem and, in turn, determine the trade-offs that the GA search process is willing to accept in the final design that is generated. Network survivability is considered crucial in the design of networks [2, 3, 4]. Thus, it is extremely important that a search algorithm be capable of generating cost effective, survivable network designs. Ring, or rather multi-ring, architectures are considered cost effective in that they offer high network survivability in the face of node failure and greater bandwidth sharing [5]. Ring topologies have received considerable attention from various sources as a consequence of the emergence of SONET standards. Bidirectional self healing ring (BSHR) networks are particularly difficult to design; in fact, the problem has been shown to be NP complete [6] by demonstrating a polynomial time transformation to the Undirected Hamiltonian Circuit Problem. While SONET standards identify logical and physical networks, this paper focuses on the design of the physical network and assumes that the logical networks defined as layers above it will have a greater connectivity than is present in the physical layer. Stated another way, all links that are present in the physical layer will be defined in the layer above it. The design of the logical layers above the physical layer is beyond the scope of this paper. This paper consists of the following sections. Firstly, a definition of the Network Ring Design Problem (NRDP) is provided and a mapping of this problem on to the Graph Ring Covering Problem (GRCP) demonstrated. The paper continues by describing heuristic approaches to the NRDP and the application of GAs to the NRDP, outlining the fitness function and hybrid representation used in its solution. Results of applying heuristic and GA approaches to a number of test networks are subsequently described, demonstrating the improved designs generated using GA search. Finally, a summary of the paper’s key messages is provided along with recommendations for future work. Formal statement of problem Let the graph G = (V,E) be a set of V nodes and a set of E edges. The number of nodes is n and the number of edges is m. The traffic requirements between the nodes is defined by an nxn matrix T, where Tij represents the traffic between nodes i and j. We wish to design a network such that the traffic requirements are satisfied by having link flows less than or equal to link capacity, every link belongs to some ring, and the cost of the network is minimised according to some objective function F. In order to solve this problem, the traffic must be routed, the individual links dimensioned and a ring covering computed for the graph G, i.e. the GRCP must be solved. A ring is simply a sequence of vertices v1, v2,..., vr, such that {vi, vi+1}, for all , is a link, and that v1 = vr and all other vertices are distinct. This is a simple cycle. A ring cover for a network G is a collection of rings C such that every link in G is included in at least one ring in C1. 1. It is easy to show that a ring covering with zero overlap of the edges is only possible if the degree of every node in the graph is even. 1 i r < ≤ Heuristic approaches to topology design Exact efficient solutions to the topology design problem are not known to exist and hence heuristics have been proposed. Essentially, topology design heuristics take two forms: either a fully (or partially) meshed network is reduced by reassigning traffic and removing redundant links or a network is built by adding links as needed when traffic demands are routed through the network. In the former case, a number of algorithms have been proposed [7, 8]. The Branch X-Change (BXC) method [7] starts from an arbitrary topological configuration and reaches a local minimum by means of a local transformation, consisting of the elimination of one or more old links and the insertion of one or more new links. The BXC has been applied to the design of minimum cost survivable networks [8] and centralised computer networks [9]. It has also been applied in the design of distributed computer networks [7]. The algorithm proceeds: Step 1: Perform a local transformation. A new link is added and an old link is deleted in such a way that two-connectivity is preserved. Step 2: Capacities and flows are assigned to the new topological configuration using a minimum link assignment [10], and cost and throughput are evaluated. If there is a cost-throughput improvement, then the transformation is accepted. Otherwise, it is rejected. Step 3: If all local transformations have been explored, stop. Otherwise, go to step 1. The Concave Branch Elimination (CBE) method can be applied whenever concave curves can be used to approximate discrete costs [11, 12]. This method begins with a fully meshed topology, using concave costs and applying the flow deviation algorithm [10] until a local minimum is reached. Generally, the flow deviation algorithm eliminates uneconomical links and compresses the topology. Hybrids of BXC and CBE have also been investigated. An excellent review of network design problems and algorithms can be found in [13]. Traditional network design algorithms and methodologies have not concerned themselves with survivability, an issue that has assumed increasing importance in modern optical, high capacity networks. In such networks, robust topological structures such as rings which exploit traffic pattern asymmetries are considered to be building blocks from which backbone networks can be constructed. Survivability can be viewed at many network levels. For example, providing alternate routes for traffic at a logical layer is one such mechanism. In fact, the requirement for a survivable network need not be stated in terms of rings but might be described such that any commodity might be served by two or more paths, with each path being node and link disjoint. However, in this paper we are concerned with the design of a high reliability network transport layer (e.g. SONET/SDH) such that routing at the ATM (or other similar layer) need not concern itself with the reliability of the underlying transport medium. Heuristic approaches to ring design The Ring Design Problem, and heuristics for its solution in the domain of Metropolitan Area Networks (MANs), can be found in [14]. The more general problem of designing unidirectional self healing rings (USHR) or bi-directional self healing rings (BSHR) has been extensively studied [2, 3, 4, 5, 26, 27, 28, 31 and others]. In [29] a methodology for SONET network design is presented which is of some practical value. In [14], the problem of ring design is formulated as one in which (1) a subset of vertices of the initial graph are selected and (2) a ring or Hamiltonian cycle is computed such that a predetermined budget is not exceeded and the total profit obtained from the network is maximised. In [30], the ring design problem is viewed as one in which ring coverings for a graph are computed and it is this philosophical approach that most closely resembles the ring design process described in this paper. This formulation is somewhat different to that presented in [6]. In [6], the formal description of the ring design problem is similar to that presented earlier in this paper and is equivalent to the computation of a minimal ring covering where no assumptions are made about the nature of the underlying link costs. The ring design process consists of two phases. In the first phase, the traffic is routed using a shortest path algorithm such as Dijkstra [16] and link loadings computed. These loadings are used to determine link bandwidths and associated link costs. In the second phase of the process, a modified depth first search (DFS) algorithm [21] is used to generate rings along with a decision step which concludes whether to include the ring in the ring cover or not. In this approach, a node is chosen as a start point for search based upon connectivity and edge cost information and the DFS is started, attempting to find a path to the remote end of the link of maximum cost. Once found, a decision is made as to whether to include the ring in the cover and, if included, node priorities are recalculated in order to remove the links included in the cover. A new node is chosen based upon recalculated priorities and DFS proceeds once again. The process terminates when all links are associated with at least one ring. In the case of BSHR networks the cost of the ring cover is the sum of the costs for the maximum cost link in each ring. The formulation of the problem in this paper is somewhat similar to that of the Selective Travelling Salesman Problem (STSP) studied previously by Laporte and Matello [15]. It is this algorithm that provides the motivation for the GA representation described in the rest of this paper. Solving the network design problem with GAs In this section, the solution of the network design problem using GAs is described. GAs have been applied to the design problem previously [17, 18, 19]. A hybrid bit and permutation representation is proposed consisting of three distinct elements; one to encode the routing problem, one to encode link capacities and the third used to solve the ring covering problem. In the following sections, the representation is described in detail, and the objective function is presented. A comprehensive introduction to GAs can be found in [1, 20 and others]. The representation Three distinct components are defined for the network ring design problem. The components represent solutions for the routing problem, link bandwidth assignment and an encoding used to determine ring covering. The representation is a hybrid one, with the first two components being encoded purely as bit strings while the final component consists of permutation entities. The representation is shown graphically in Figure 1. Routing Table Link Bandwidths Ring Covering Permutation Figure 1 Representation used for the Network Ring Design Problem The routing component of the representation is indirect. Symmetric routing is assumed; thus a route from i to j, Rij, is the same as a route from j to i. In other words, Rij = Rji. Hence, for an n node network, the routing table can be represented by n(n-1)/2 entries1. However, this limitation was imposed only to reduce the size of the potential solution space and is not a necessary limitation of the representation itself. The routing component of the encoding is thus kn(n-1)/2, where k is the number of bits used for each routing. The indirect representation arises from the fact that the k bits for each routing stand for an index into an array of possible routes. Hence, 2k routes can be associated with each source-destination pair. The 2k shortest paths and their costs for each source-destination pair are computed when the network topology is loaded. It is computed prior to the GA run. The value of k is an input to the application, and obviously determines the size of the routing table subproblem space that is explored. In this study, k = 3 was used, allowing 8 shortest paths to be explored for each source-destination pair. A brief investigation of the variation of design quality with the value of k has been undertaken [17]. However, more work remains in this area. The link capacity encoding is also indirect. For each of the m edges in the graph, an index into an array of possible bandwidths is represented. This bandwidth array, or bandwidth map, includes zero in order that the link can be assigned a zero capacity i.e. not actually be present in the network. In fact, the bandwidth map may contain multiple entries for the same bandwidth if search in this subspace of the overall design problem needs to be biased by the user. Similarly, it is possible to use two entries in the bandwidth map indicating the presence of a link rather than its bandwidth. In this case one indicates that a link is present with zero indicating no connection. When a binary bandwidth map is used the traffic loading on the various links indicated by the routing table is used to determine the bandwidth of the links which are indicated as being present. In essence, then, the graph provided as input to the GA defines the possible topologies that can be generated. Providing a fully-meshed graph as input to the GA design process implies that the space of all possible topologies can be explored. In some sense, by not specifying an edge in the input graph, the user is making the statement, “It is not possible to connect two particular nodes directly”. Thus, the representation avoids the regions of the space rendered infeasible by user constraint by not representing them in the bit string. This is a particularly attractive characteristic of a representation manipulated by a GA. If the bandwidth map contains p entries, the link capacity encoding requires mlog2p bits, where m is the number of edges in the graph. It is clear that the two segments of the representation affect each other. In the case where a non-binary bandwidth map is used, if the encoded link bandwidth is too low, and the routing of the various traffic elements are such that the load on the link exceeds the link’s bandwidth, an infeasible solution will be generated. While this is an undesirable characteristic for the representation, two possible mechanisms exist for resolving the problem. The first possibility is to fix the infeasible solution by increasing the bandwidth of the link. The second possibility is to penalise the solution heavily via the cost measured using the objective function. In this implementation, the second approach was chosen. Another possible approach to this problem with the representation is to encode route permutations for route selection and choose the first member of the permutation if possible but using later members of the permutation if link bandwidths make the route infeasible. This possibility may be explored in further work on this problem. However, as the results on the test networks clearly show, this interaction 1. We do not need an entry Rii. between elements of the representation chosen does not cause GA search to be ineffective in this problem domain. It seems evident that the penalty terms for infeasible solutions quickly cause search to be abandoned in inappropriate regions of the problem space. The third segment of the representa tion concerns the ring cover problem. This segment also provides an indirect representation in that the representation is used as an input to a modified depth first search algorithm [21] -a technique for traversing all m edges in a graph in O(m) time -in order to derive a ring covering. This segment of the representation consists of two smaller segments; the first concerning the nodes in the graph and the second the edges in the graph. While the first two segments in the representation are pure bit strings, these final segments are permutations. The nodal permutation segment represents a permutation on the node numbers for the graph. It is used to decide which node to start a search from for the determination of a ring in the graph ring covering. The edge permutation segment consists of n smaller permutations, one for each node in the graph. Each of these permutations is the size of the degree of the node and is used to decide the order in which edges are chosen when selecting an edge to traverse using the modified depth first search algorithm used in the ring cover computation. For example, the graph shown in Figure 2 might have the node permutation {n1, n5, n3, n2, n4, n7, n6} and edge permutation { { 1, 3, 2}, {2, 1}, {1, 2, 3}, {1, 2}, {2, 1}, {3, 2, 1}, {1, 2, 3} }, where the degree of the nodes is given by di = 3, 2, 3, 2, 2, 3, 3, for i = n1,..,n7. There is a subtle interaction between the edge permutation segment of the representation and the link capacity segment of the representation. A link bandwidth of zero represents no link which implies that the node degree can vary with choice of bandwidths. In this situation it does not make sense to include such a link in any ring covering for the graph. The modified depth first search algorithm thus ignores links of zero capacity when computing a ring covering. This node and edge permutation representation has two very desirable properties. Firstly, no infeasible solutions for the ring covering problem can be generated. It is possible for the link bandwidth representation to imply that a node is of degree one, i.e. has a single link. In this case a feasible ring covering will not be possible. In these cases a penalty is associated with such configurations. Secondly, all possible selections for depth first search node and edges choices are equally represented, implying no explicit representational bias. Assuming that a node number is encoded in a byte, i.e. in the range 0-255, the node permutation encoding requires n bytes. If the degree of a node is di, the edge permutation encoding requires bytes. The following algorithm is used to compute the ring covering. Figure 2 Example Graph