Cost vs. Quality: Tradeoffs in Communication Networks

This research studies design and optimization problems in communication networks. The goal is to design network structures and routing algorithms with good quality and low cost. A communication network is composed of a set of users connected by communication lines. Users communicate by sending information on the lines. Routing algorithms determine the routes in the network that the information will traverse. Design and optimization problems in this eld can be classi ed to two levels. In the rst level the target of the design is the topology of the network. In the second level, the topology is assumed to be given, and the target is the design of the routing which is performed on top of it. In this work we study problems in both levels of the design. In the design of network topologies, we are motivated by the recent developments in all-optical networks: A new and promising technology that supports high-bandwidth demands. All-optical networks provide a set of lightpaths, which can be viewed as high-bandwidth pipes on top of which routing is performed. Speci cally, this set of lightpaths de nes a virtual topology a graph in which two nodes are connected by an edge i there is a lightpath connecting them. The virtual topology can then be used obliviously for routing as if it was the actual topology of the network. Any desirable routing paradigm can be implemented on top of it. Since the capacity enabled by this new technology is enormous, it is crucial that the optical layer will provide a certain degree of protection against failures of hardware components. This means that the design of the virtual topology has to support such protection mechanisms. Speci cally, it must be guaranteed that it is possible to re-route tra c associated with impaired lightpaths (as a result of a failure of a component that they use) on other lightpaths in the network. We de ne a hierarchy of survivability conditions that enable di erent degrees of protection; the more restrictive a condition is, the faster and simpler the restoration it provides. We de ne several cost functions that can be used to measure the cost of a design. One possibility, which we term uniform cost, corresponds to the switching cost of the entire network, indicated by the number of lightpaths in the design. There is a tradeo between the quality and the cost of the design; the more restrictive the survivability condition is, the more expensive it is to implement. We concentrate mainly on the most restrictive ring partition condition, in which lightpaths are arranged in the form of rings. We study the problem of the construction of ring partition designs with low cost. The input is an initial set of lightpaths in a given optical topology, and the target is to augment this set by adding lightpaths, such that the result will satisfy the ring partition condition. We prove some negative results regarding the tractability and the approximability of this problem, and we 1 provide an approximation algorithm for it. We also study the characterization of networks on which survivable designs can be constructed. Once the topology of the network is given, virtual or physical, routing of users tra c is performed obliviously on top of it. In the second part of the work we study problems in the design of routing strategies on top of a given topology. We consider the classical model of routing, where communication is achieved by sending and receiving messages, in a connectionless style. Every message is viewed as an independent entity which is routed according to the information kept in its header. Once a node receives a message, it determines on which outgoing link to forward the message according to its destination, and by consulting a local data structure. The designer of the routing strategy have to determine the form and interpretation of the distributed data structure that is used for the routing. Once the routing information in every node is determined, it implies a route in the network between every ordered pair of nodes. The cost of a routing strategy is the amount of memory needed to encode the routing information. The quality is the time that it takes to route messages according to it, where the length of the routing paths is an indicator. We present a universal design which is applicable for every graph with good cost-quality tradeo . We prove lower bounds on the quality of the routing when some constraints are put on the form and size of the routing information. We also tackle the characterization problem of networks which support routing on shortest paths (i.e., optimal quality), with a xed amount of memory in each node, encoded in the form of intervals. The routing model that we consider is somewhat restricted by few parameters. For instance, headers of messages are written once by the originator of the transmission and cannot be re-written. In fact, this work is one of many that study this and other less restrictivemodels. We discuss the motivation for this model, compare it to other alternatives, and review the related work done so far. 2 List of Figures 1.1 A virtual topology. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 1.2 A ring-partition design. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 2.1 Line protection within the physical layer and the e ect on the routes of lightpaths. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 20 2.2 Protection mechanisms within the di erent layers. : : : : : : : : : : : : : : : 21 2.3 The connectivity condition. : : : : : : : : : : : : : : : : : : : : : : : : : : : 24 2.4 The protection path condition. : : : : : : : : : : : : : : : : : : : : : : : : : : 25 2.5 The cost of a design. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 28 2.6 The cost of the design. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 29 3.1 2-node connectivity is not enough for the protection path and ring partition conditions. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 31 3.2 An example of the transformation : : : : : : : : : : : : : : : : : : : : : : : : 34 3.3 (A) A graph and a set of connections. (B) The corresponding induced graph. 35 4.1 The MCRPD problem. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 38 4.2 A graph, a set of connections, a matching-set (where only matchings in nontrivial end-node graphs are shown), and the equivalent subgraph-partition. : 39 5.1 The Ring Partition Algorithm (RPA). : : : : : : : : : : : : : : : : : : : : : : 45 5.2 The maximum spanning tree Tmax does not necessarily include a maximum number of connections. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 49 5.3 (A) An instance of the vertex-cover problem. (B) The corresponding instance of the MCRPD problem (a graph and a set of connections). : : : : : : : : : 50 5.4 A cycle of length four in IGCnC? if both cv 2 C? and cu 2 C?. : : : : : : : : 51 5.5 A disconnected subgraph which includes the nodes in Sv, in the case that v 62 V ?. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 51 6.1 (A) An instance I of the circular arc coloring problem. (B) The instance full2(I) of the MCRPDR problem. : : : : : : : : : : : : : : : : : : : : : : : 54 3 6.2 (A) An instance of MCRPDR. (B) An instance of MCRPD with induced graph which is a set of chains. : : : : : : : : : : : : : : : : : : : : : : : : : : 56 6.3 (a) The input instance I, where m = 4, n = 2, and C = f(1; 2); (3; 1)g. (b) The new instance I 0, where = 1, i.e., a multiplication by 2. : : : : : : : : : 58 8.1 The four layers model in today's core network architecture. : : : : : : : : : : 64 9.1 An interval routing scheme (with compactness 1). : : : : : : : : : : : : : : : 68 10.1 The structure of the proofs. : : : : : : : : : : : : : : : : : : : : : : : : : : : 71 10.2 (A) The directed graph G and the equivalence classes of its arcs, constructed by the transformation from the formula '. (B) The legal orientation of G's arcs which corresponds to the truth-assignment a. : : : : : : : : : : : : : : : 72 10.3 A set of constraints S over U = fu1; :::; u6g and the corresponding graph GS 76 10.4 The order of the nodes in the case: Set1i < Set2i . : : : : : : : : : : : : : : : : 77 11.1 (A) The general form of a lithium graph. (B) The simplest lithium graph. : 82 11.2 For any 1-LIRS, the interval labels Iu(u; v) and Iu(u;w) cannot contain both 1 and n. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 83 11.3 The graph G3;4. There are three isomorphic wings. Only Wing 1 is detailed. 84 11.4 A piece of the graph G0j;k. : : : : : : : : : : : : : : : : : : : : : : : : : : : : 86 11.5 (A) The general structure of a petal graph. (B) Two simple petal graphs. : : 87 11.6 A representation of a non-lithium graph. Each of the Gis is a 2-edge-connected graph with leaves. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 90 11.7 The labeling of the edges outgoing a node x when (A) back(x) is an edge outgoing x (B) Otherwise (back(x) = fu; vg and u is in Ty1). : : : : : : : : : : : : : : : : 92 11.8 A graph G labeled by Algorithm Linear-Label. : : : : : : : : : : : : : : : : : : 97 11.9 An example of routing paths. : : : : : : : : : : : : : : : : : : : : : : : : : : 97 12.1 The route from u to a destination v which is not in u's ball. : : : : : : : : : 101 13.1 Complexity results under the various models : : : : : : : : : : : : : : : : : : 112 4 List of Terms S(P ) The set of connections in a virtual path P G = Gp [ Gc A subgraph partition NGv = (NVv; NEv) An end-node graph of a node v IGC = (IVC ; IEC) An induced graph DG The diameter of a graph G deg(v) The degree of a node v R = (L;[v2V Iv) An interval routing scheme c(I) The compactness of a set I lc(I) The linear compactness of a set I MemG(R;x) The memory requirements of a node x according to R MemG(R) The memory requirements of R TotMemG(R) The total memory requirements of R CompG(R) The compactness of R ComplG(R) The linear compactness of R DilG(R;x; y) The length of a routing path from x to y according to R DilG(R) The dilation of R StrG(R) The stretch factor of R AvStrG(R) The average stretch factor of R 5 Chapter