Parallel Construction of Independent Spanning Trees on Parity Cubes

Abstract

Zehavi and Itai (1989) proposed the following conjecture: every k-connected graph has k independent spanning trees (ISTs for short) rooted at an arbitrary node. An n-dimensional parity cube, denoted by PQn, is a variation of hyper cubes with connectivity n and has many features superior to those of hyper cubes. Recently, Wang et al. (2012) confirm the ISTs conjecture by providing an O(N log N) algorithm to construct n ISTs rooted at an arbitrary node on PQn, where N=2n is the number of nodes in PQn. However, this algorithm is executed in a recursive fashion and thus is hard to be parallelized. In this paper, we present a non-recursive and fully parallelized approach to construct n ISTs rooted at an arbitrary node of PQn in O(log N) time using N processors. In particular, the constructing rule of spanning trees is simple and the proof of independency is easier than ever before.