On the totalk-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 totalk-distance for a pair of nodes and totalk-diameter of a connection network, and study the value TDk(d) which is the maximal such totalk-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 TDk(d) and a lower bound on the growth of TDk(d) as functions of k and d; those bounds are tight, θ(d), when k is fixed. Specifically, we prove that TDk(d) ≤ 2k−1dk, with the exceptions TD2(1) = 3, TD3(1) = 5, and that for every k, d0 > 0, there exists (a) an integer d ≥ d0 such that TDk(d) ≥ d/k, and (b) a k-connected simple graph G with diameter d such that d ≥ d0, and whose totalk-diameter is at least (d− 2)/k. ∗Up to 1990, E. A. Dinic, Moscow. †Research supported by the Bernard Elkin Chair for Computer Science and by USIsrael BSF grant 95-00238 ‡Contact author; Tel: (+972)-4-8294368, Fax: (+972)-4-8221128, Email: [email protected]