An improvement in the size-depth tradeoff for strictly non-blocking generalized-concentration networks

Abstract

Generalized-concentrators are interconnection networks that provide vertex-disjoint directed trees to satisfy interconnection requests. An interconnection network is non-blocking in the strict sense if every compatible interconnection request can be satisfied regardless of any existing interconnections. We present a non-asymptotic lower bound size-depth tradeoff, $k(n - m + \tfrac{k}{{k + 1}}c)c^{\tfrac{1}{k}} - (k - 1)(n - c)$ if r<k − 1, and $\alpha _k (n - \tfrac{m}{r} + \beta _k \tfrac{k}{{k + 1}}\tfrac{c}{r})r^{\tfrac{{k - 1}}{k}} c^{\tfrac{1}{k}} - \tfrac{1}{2}(n - \tfrac{m}{r})r$ otherwise (where $\alpha _k = \tfrac{1}{2}\tfrac{k}{{(k - 1)^{\tfrac{{k - 1}}{k}} }}$ and $\beta _k = 1 - \tfrac{1}{{2^1 + \tfrac{1}{k}}}$), for synchronous strictly non-blocking (c, r)-limited (n, m)-generalized-concentrators with arbitrary depth k.