Multirate Rearrangeable Clos Networks and a Generalized Edge-Coloring Problem on Bipartite Graphs

Abstract

Chung and Ross [SIAM J. Comput., 20 (1991), pp. 726--736] conjectured that the minimum number m(n,r) of middle-stage switches for the symmetric 3-stage Clos network C(n,m(n,r),r) to be rearrangeable in the multirate environment is at most 2n-1. This problem is equivalent to a generalized version of the bipartite graph edge-coloring problem. The best bounds known so far on this function m(n,r) are $11n/9 \leq m(n,r) \leq 41n/16 + O(1)$, for $n, r \geq 2$, derived by Du et al. [SIAM J. Comput., 28 (1999), pp. 464--471]. In this paper, we make several contributions. First, we give evidence to show that even a stronger result might hold. In particular, we give a coloring algorithm to show that $m(n,r) \leq \lceil (r+1)n/2 \rceil$, which implies $m(n,2) \leq \lceil 3n/2 \rceil$---stronger than the conjectured value of 2n-1. Second, we derive that m(2,r) = 3 by an elegant argument. Last, we improve both the best upper and lower bounds given above: $\lceil 5n/4 \rceil \leq m(n,r) \leq 2n-1+\lceil (r-1)/2 \rceil$, where the upper bound is an improvement over $41n/16$ when r is relatively small compared to n. We also conjecture that $m(n,r) \leq \lfloor 2n(1-1/2^r) \rfloor$.