Optimal fault-tolerant routings with small routing tables for k-connected graphs

We study the problem of designing fault-tolerant routings with small routing tables for a kconnected network of n processors in the surviving route graph model. The surviving route graph R(G,ρ)/F for a graph G, a routing ρ and a set of faults F is a directed graph consisting of nonfaulty nodes of G with a directed edge from a node x to a node y iff there are no faults on the route from x to y. The diameter of the surviving route graph could be one of the fault-tolerance measures for the graph G and the routing ρ and it is denoted by D(R(G,ρ)/F). We want to reduce the total number of routes defined in the routing, and the maximum of the number of routes defined for a node (called route degree) as least as possible. In this paper, we show that we can construct a routing λ for every n-node k-connected graph such that n 2k2, in which the route degree is O(k√n), the total number of routes is O(k2n) and D(R(G,λ)/F) 3 for any fault set F (|F |< k). In particular, in the case that k = 2 we can construct a routing λ′ for every biconnected graph in which the route degree is O( √ n), the total number of routes is O(n) and D(R(G,λ′)/{f }) 3 for any fault f . We also show that we can construct a routing ρ1 for every n-node biconnected graph, in which the total number of routes is O(n) and D(R(G,ρ1)/{f }) 2 for any fault f , and a routing ρ2 (using ρ1) for every n-node biconnected graph, in which the route degree is O( √ n), the total number of routes is O(n √ n) and D(R(G,ρ2)/{f }) 2 for any fault f .  2004 Published by Elsevier B.V.