Optimal Edge Congestion of Exchanged Hypercubes

Abstract

Topological properties have become a popular and important area of focus for studies that analyze interconnections between networks. The hypercube is one of the most widely discussed topological structures for interconnections between networks and is usually covered in introductions to the basic principles and methods for network design. The exchanged hypercube EH(s, t) is a new variant of the hypercube that has slightly more than half as many edges and retains several valuable and desirable properties of the hypercube. In this paper, we propose an approach for shortest path routing algorithms from the source vertex to the destination vertex in EH(s, t) with time complexity O(n), where n - s + t + 1 and 1 <= s <= t. We focus on edge congestion, which is an important indicator for cost analyses and performance measurements in interconnection networks. Based on our shortest path routing algorithm, we show that the edge congestion of EH(s, t) is 3.2(s+t+1) - 2(s+1) - 2(t+1). In addition, we prove that our shortest path routing algorithm is an optimal routing strategy with respect to the edge congestion of EH(s, t).