On the totalk-diameter of connection networks

Abstract We study connection networks in which certain pairs of nodes have to be connected by k edge-disjoint paths, and study bounds for the minimal sum of lengths of such k paths. We define the related notions of total k -distance for a pair of nodes and total k -diameter of a connection network, and study the value TD k (d) which is the maximal such total k -diameter of a network with diameter d . These notions have applications in fault-tolerant routing problems, in ATM networks, and in compact routing in networks. We prove an upper bound on TD k (d) and a lower bound on the growth of TD k (d) as functions of k and d ; those bounds are tight, θ(d k ) , when k is fixed. Specifically, we prove that TD k (d)⩽2 k−1 d k , with the exceptions TD 2 (1)=3 , TD 3 (1)=5 , and that for every k,d 0 >0 , there exists (a) an integer d⩾d 0 such that TD k (d)⩾d k /k k , and (b) a k -connected simple graph G with diameter d such that d⩾d 0 , and whose total k -diameter is at least (d−2) k /k k .