Optimizing QoS resources in series parallel networks

The task of Quality of Service (QoS) routing is to find a path in the network that satisfies certain constraints on metrics such as bandwidth, while also achieving overall network resource efficiency. The focus of this paper is on a polynomial algorithm to find a feasible path that provides a QoS needed based on a bandwidth when a network is a series parallel graph. The approach is to divide a network topology graph into spanning tree sub-networks each with an optimized bandwidth while keeping the number of sub-networks to minimum. Minimizing the number of sub-networks will provide us sub-networks with maximal bandwidths. Introduction So far the Internet has offered only best effort service. All traffic is processed as quickly as possible and no preferences are given to any type of traffic. Today there are more and more applications that need service guarantees in order to function properly. These kinds of applications include for instance IP telephony, videoconference applications or video on-demand services. The timely delivery of digitized audio-visual information over local or wide area networks is now becoming realistic. In the current Internet, data packets of a session may follow different paths to the destination. The network resources, e.g. switch buffer and link bandwidth, are fairly shared by packets from different sessions. However, this architecture does not meet the requirements of the future integrated-service networks that carry heterogeneous data traffic. It does not support resource reservation which is vital for the provision of guaranteed end-to-end performance (bounded delay, delay jitter and loss ratio). The notion of Quality-of-Service (QoS) has been proposed to capture the qualitatively or quantitatively defined performance contract between the service provider and the user applications. A good introduction to QoS routing and routing in general can be found in (Lee et al., 1995 ; Steenstrup, 1995). The QoS requirement is given as a set of constraints, which can be either link constraints or path (tree) constraints (Lee et al., 1995). Research on QoS routing (Pornavalai et al., 1997 ; Salama, et al., 1997 ; San and Langandorfer, 1996 ; Jin and Nahrstedt, 2002) has been following two main directions: source routing and hop by hop or distributed routing. In source routing, a forwarding path is computed on-demand at the source and listed in the packet header. In hop-by-hop routing, packets are forwarded hop-by-hop at each node. Hop-by-hop routing can use fully distributed computation algorithms (Garcia-Luna-Aceves, 1989) which has lower memory requirements for the routers. Distributed applications such as teleconferencing, video-on-demand, and Internet phone have very diverse QoS constraints on delay, delay jitter, loss ratio, bandwidth, etc. Multiple metrics can certainly model a network more accurately. However, the problem is that finding a path subject to multiple constraints or to one constraint is NP-complete (Garey and Johnson, 1979), but the proof has never been published. Jaffe (Jaffe, 1984) investigated this particular problem further and proposed two approximation algorithms that solve the problem in pseudopolynomial time or polynomial time if the lengths and weights have a small range of values. The running time of such NP-complete problems for real-world network topologies is investigated by Partridge and Castineyra (Partridge and Castineyra, 1993). It is clear that any combination of two or more of delay, delay jitter, cost, loss probability as metrics are NP-complete. The only tractable combinations are bandwidth and one of the four other (delay, loss probability, cost and delay jitter) (Wang and Crowcroft, 1996). The proof of NP-completeness relies heavily on the correlation of the link weight metrics. It is been proved for the chain topology 18th National Computer Conference 2006 © Saudi Computer Society and assumed that the NP-complete nature will hold for all topologies. However QoS routing in realistic networks may not be NP-complete in nature as both chain topology and the correlated link metric vectors are unlikely to occur in realistic networks (Van Mieghem and Kuipers, 2003). For traffic with bandwidth guarantees, many studies have contributed to QoS path selection algorithms. Breslau et al. (Breslau et al., 1993) have suggested an adaptive load based source routing algorithm for traffic requiring bandwidth guarantees. Gawlick et al. (Gawlick et al., 1995) present an evaluation study of several routing algorithms, including minimal hop path, exponential path, and max-min path, for permanent connections with bandwidth guarantees. Matta and Shankar study delay and throughput based type of service routing (Matta and Shankar, 1995) and dynamic routing of real-time virtual circuits (Matta and Shankar, 1996). Wang and Crowcroft (Wang and Crowcroft, 1996) suggest shortest-widest path. Gawlick et al. (Gawlick et al., 1995) propose to use shortest path with exponential cost function for permanent connections. Guerin et al. (Guerin and Orda, 1999) suggest shortest-widest path. The dynamic-alternative path (Ma and Steenkiste, 1997a) is based on results of dynamic alternative path for telecommunications networks (Gibbens et al., 1988). For traffic with delay guarantees, several studies propose heuristics to tackle the NP-complete problem (Jaffe, 1984 ; Salama, 1997). Wang and Crowcroft (Wang and Crowcroft, 1996) study the complexity of QoS routing with multiple QoS constraints. They propose the shortest widest path algorithm as a way of minimizing the call blocking rate, but no performance evaluation is given. Rosen et al. (Rosen et al., 1991) propose an algorithm that iterates any single pair shortest-path algorithm over all possible residual link-bandwidth. Guerin et al. (Guerin et al., 1996) propose to use the widest shortest path to extend OSPF for QoS routing. In another paper (Guerin and Orda, 1999), Guerin and Orda study routing with inaccurate information and show that, using a shortest path algorithm; it can find the feasible path that is most likely to accommodate the requested bandwidth. Pornavilai et al. (Pornavalai et al., 1997) consider the problem of routing traffic with multiple QoS constraints, but they assume that the bandwidth to be reserved is known. Paths can either be selected on demand or they can be precomputed (Ma and Steenkiste, 1997b). With on demand routing, the path selection algorithm is executed for every request. With precomputed paths, the path selection algorithm is executed periodically as new routing information is received. The precomputed paths are stored in the routers, and routing requests result in a simple table look up. Since the requested bandwidths can be very diverse, using a path that meets all bandwidth requests results in using the widest path, which does not always perform well. An alternative is to use class based routing: several paths, each for a different bandwidth range, are precomputed. A new session selects the path with the lowest bandwidth satisfying the request. In practice, source routing often selects path on demand, while hop by hop routing uses precomputed paths. In our paper we will use a class based routing with one constraint: the bandwidth. For that we will give a polynomial algorithm to find a feasible path that provides a QoS needed in the case of series parallel networks topology. The approach is to divide a network topology graph into virtual spanning tree sub-networks each with an optimized bandwidth while keeping the number of sub-networks to minimum. Minimizing the number of sub-networks will provide us, sub-networks with maximal bandwidths. For the best of our knowledge this approach has not been done before. The paper is organized as follows: in section 2, we introduce definitions and notations needed to formulate the problem and to exhibit the solution. In section 3, we present intermediate results and our main result. Finally, the conclusion is given in section 4. Definitions and Notations Sets and their characteristic vectors will be not distinguished. We refer to Bondy and Murty book (Bondy and Murty, 1982) for any not defined term or not proved property known on graph theory. We present a computers network as an undirected graph G=(V, E) where V is the set of vertices (network devices) and E is the set of edges (links). If there is no confusion, we refer to n as the number of vertices of G and m as its number of edges. Bandwidths are presented by a vector x defined on E. For any subset F of E, x(F) is the sum of bandwidths on edges of F. For any edge e of E, we denote by G\e (i.e. deletion of e) the graph obtained from G by removing e and keeping all other edges and vertices, and by G/e (i.e. contraction of e) the graph obtained from G by removing e, identifying the two vertices adjacent to e and keeping all other edges and vertices. A spanning tree T of G is a minimal connected subset of edges (loop-free and connecting all nodes). Note that the number of edges in T is n–1. x is a packing of spanning trees of G if there exist positive numbers λ1, ..., λk and spanning trees T1, ..., Tk such that: 2 In this case, we say that x is integer if: all xj’s are integer and is integer. A graphical formulation of our problem can be as follows: Optimizing QoS Resources Problem Instance: Given a graph G (computers network) and an integer packing x of spanning trees of G (provided bandwidths with no residual). Question: Find positive integers λ1, ..., λk and spanning trees T1, ..., Tk such that: and k is minimum. (Finding virtual spanning tree sub-networks each one with the maximum possible bandwidth everywhere). The best known bound for k in the general case is m+n–2 (de Pina and Soares, 2003). And the best possible bound can be m. We are going to prove that this bound is achieved for series parallel graphs. A parallel extension to G is adding a parallel edge to one edge of G. A series extension of G is subdividing an edge and creating a new vertex and two new adjacent edges (see figure 1). A series parallel graph is obtained by applying series and parallel extensions repetitively as many as we want starting from one edge (see figure 2). Given a packing x of spanning trees of G and a spanning tree T: λ(x, T)=max{λ≥0 such that x–λT is again a packing of spanning trees of G} (2) i.e. the best service we can obtain for that virtual spanning tree sub-network given provided bandwidths x. The following result is a corollary of a theorem of Chaourar (Chaourar 2002): Theorem 1: If G is a series parallel graph then λ(x, T) is integer. Taking these maximums for successive spanning tree sub-networks gives an answer to the problem there above. Our main result is to compute these maximums using a polynomial time algorithm. A subset of edges F is an induced subgraph of G if all edges between vertices of F are in F (i.e. you take vertices and you get all edges connecting them in G). We denote by n(F) the number of vertices inducing F. In this case, F is connected if you can reach any vertex of F from any other one using edges of F. F is 2-connected if removing any vertex of F keep it connected. A nonempty induced subgraph F≠E is locked in G if F is 2-connected and E–F is connected. The class of all such subgraphs is denoted by Lock(G). Induced locked subgraphs were introduced by Chaourar (Chaourar 2002) in the more general context of matroids. In order to insure a zero residual bandwidth graph, we can construct x by summing as many as we want of spanning trees. Note that these spanning trees do not give the wanted optimum in general. Optimizing QoS resources 3 In this case, we say that x is integer if: all xj's are integer and 1 n x(E) � is integer. A graphical formulation of our problem can be as follows: Optimizing QoS Resources Problem: Instance: Given a graph G (computers network) and an integer packing x of spanning trees of G (provided bandwidths with no residual). Question: Find positive integers �1, ..., �k and spanning trees T1, ..., Tk such