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 k-connected 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 non-faulty 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 routingρ1 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.