The decycling problem in hierarchical cubic networks

Abstract

Hierarchical cubic networks (HCN) have been introduced as interconnection networks for massively parallel systems. This topology is based on hypercubes, and thus benefits from their desirable properties such as a small diameter. However, an HCN supersedes a hypercube in many ways: for instance, the number of links per node in an HCN is significantly lower than that of a hypercube of the same size, while their diameters remain similar. In this paper we address the decycling problem in an HCN. It consists in finding a node set of minimum size such that excluding these nodes from the network ensures an acyclic (cycle-free) network. This is a critical problem with many applications, and notably in parallel computing as it allows for the prevention of resource allocation issues such as deadlocks, livelocks and starvations. We propose an efficient algorithm generating in an $n$-dimensional HCN a decycling set of at most $2^{2n-1} - (2^{2n-2}/n + \lfloor 2^{n-1}/n\rfloor )$ nodes.