Optimal Physical Diversity Algorithms and Survivable Networks zy Ramesh Bhandari AT&T Laboratories, Rm. 2B-41 OA Crawfords Corner Road Holmdel, NJ 07733, USA Tele. No. (908)949-0693 Fax. No. (908)949-4364 rbhandari@att.com Abstract zyxwvuts One way to improve the reliability zyxwvuts o f a network is through physical diversity, i.e., via routing of trafic between a given pair of nodes in the network over two or more physically-disjoint paths such that zyxwvut if a node or a physical link fails on one o f the disjoint paths, not all of the trafic is lost. Alternatively, enough spare capacity may be allocated on the individual paths such that the lost traflc due to a node or physical link failure can be routed immediately over the predetermined paths. In this paper, we present optimal algorithms for K-disjoint paths (K22) in a graph o f vertices (or nodes) and edges (or links). These algorithms are simpler than those given in the past. We discuss how such algorithms can be used in the design of survivable mesh networks based on the digital crossconnect systems {DCS). We also discuss the generation of optimal network topologies which permit K>2 disjoint paths and upon which survivable networks may be modeled. 1. Introduction As fiber is increasingly deployed in networks, reliability of a network is being called into question more than ever before. This is due to the fact that as more traffic is transported over the high bandwidth fiber network, any span (physical link) cut or node failure results in the loss of a large volume of traffic. One way to increase the reliability of a given network is through physical-diversity, i.e., via routing of traffic between a given pair of nodes in the network over two or more physically-disjoint paths such that if a node or a physical link (span) fails on one of the disjoint paths, not all of the traffic is lost. Alternatively, enough spare capacity may be allocated on the individual paths such that the zyxwvut 0-8186-7852-6/97 $10.00 0 1997 IEEE 433 lost traffic due to a node or physical link failure can be rerouted over the predetermined disjoint paths. Since most networks are bidirectional, we will represent networks (or graphs) by vertices (or nodes) and edges (or links). Unless otherwise stated, we will assume that multiple edges (two or more links between the same pair of nodes) are absent. Fig. 1 is an example of a bidirectional network of 8 nodes and 13 edges; each edge is the equivalent of two oppositely directed zy A z F 3 E Fig. 1 are the edge lengths. A bidirectional network; the numbers arcs, each of length equal to the edge length; length here has the general meaning in that it may represent the physical length of the edge (or link) in the network, or may be the cost of using the edge in transmission of data, and so on. A path or route between a pair of vertices is a sequence of arcs connecting them. When searching for diverse routes between a given pair of vertices, it is generally desirable to find the set of paths whose sum is a minimum. For example, if the length of an edge is the length of fiber over that physical link, then the optimal set of paths uses minimal amount of fiber between the pair of nodes. Similarly, if the length of a link in a graph represents cost of provisioning services along that link, then the optimal set of paths represents diverse routes