Optimal Fault-Tolerant Routings for k-Connected Graphs with Smaller Routing Tables

Abstract

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 nonfaulty nodes 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 ≥ 2k ², in which the route degree is O $(n\sqrt n )$, the total number of routes is O(k ² n) and D(R(Gλ)/F) ≤ 3 for any fault set F(|F| < k). We also show that we can construct a routing ρ1 for every n-node biconnected graphs, in which the total number of routes is O(n) and D(R(G, ρ1)/{f}) ≤ 2 for any fault f, and using ρ1 a routing ρ2 for every n-node biconnected graphs, in which the route degree is O $(k\sqrt n )$, the total number of routes is O $(k^2 n)$ and D(R(G, ρ ₂ )/{ f}) ≤ 2 for any fault f.