On Rearrangeability of Multirate Clos Networks

Chung and Ross [SIAM J. Comput., 20 (1991), pp. 726–736] conjectured that the multirate three-stage Clos network C(n, 2n− 1, r) is rearrangeable in the general discrete bandwidth case; i.e., each connection has a weight chosen from a given finite set {p1, p2, . . . , pk} where 1 ≥ p1 > p2 > · · · > pk > 0 and pi is an integer multiple of pk, denoted by pk | pi, for 1 ≤ i ≤ k − 1. In this paper, we prove that multirate three-stage Clos network C(n, 2n− 1, r) is rearrangeable when each connection has a weight chosen from a given finite set {p1, p2, . . . , pk} where 1 ≥ p1 > p2 > · · · > ph > 1/2 ≥ ph+1 > · · · > pk > 0 and ph+2 | ph+1, ph+3 | ph+2, . . . , pk | pk−1. We also prove that C(n, 2n− 1, r) is two-rate rearrangeable and C(n, d 7n 3 e, r) is three-rate rearrangeable.