On the total/sub k/-diameter of connection networks

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/sub k/-distance for a pair of nodes and total/sub k/-diameter of a connection network, and study the value TD/sub k/(d) which is the maximal such total/sub 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/sub k/(d) and a lower bound on the growth of TD/sub k/(d) as functions of k and d; those bounds are tight, /spl theta/(d/sup k/), when k is fired. Specifically, we prove that TD/sub k/(d)/spl les/2/sup k-1/d/sup k/, with the exceptions TD/sub 2/(1)=3, TD/sub 3/(1)=5, and that for every k, d/sub 0/>0, there exists (a) an integer d/spl ges/d/sub 0/ such that TD/sub k/(d)/spl ges/d/sup k/k/sup k/; and (b) a k-connected simple graph G with diameter d such that d/spl ges/d/sub 0/, and td/sub k/(G)/spl ges/(d-2)/sup k//k/sup k/.