Fault tolerant routing in toroidal networks

In this paper, we study the Following node-to-node and node-to-set routing problems in r-dimensional torus T-n(r) with r greater than or equal to 2 and n greater than or equal to 4 nodes in each dimension: given al most 2r-1 faulty nodes and non-faulty nodes s and t T-n(r), find a fault-free path s-->t; and given at most 2r-k faulty nodes and non-faulty nodes s and t(1),...,t(k) in T-n(r), find k fault-free node-disjoint paths s-->t(i), 1(l)ess than or equal to i less than or equal to k. We give an O(r(2)) time algorithm which finds a fault-free path s-->t of length at most d(T-n(r))+1 for the node-to-node routing, where d(T-n(r)) is the diameter of T-n(r). For node-to-set routing, we show an O(r(3)) time algorithm which finds k fault-free node-disjoint paths s --> t(i), 1 less than or equal to i less than or equal to k, of length at most d(T-n(r))+1. The upper bounds on the length of the found paths are optimal. From this, Rabin diameter of T-n(r) is d(T-n(r))+1.