Edge congestion and topological properties of crossed cubes

An n-dimensional crossed cube, CQ(n), is a variation of hypercubes. In this paper, we give a new shortest path routing algorithm based on a new distance measure defined herein. In comparison with Efe's algorithm, which generates one shortest path in O(n(2)) time, our algorithm can generate more shortest paths in O(n) time. Based on a given shortest path routing algorithm, we consider a new performance measure of interconnection networks called edge congestion. Using our shortest path routing algorithm and assuming that message exchange between all pairs of vertices is equally probable, we show that the edge congestion of crossed cubes is the same as that of hypercubes. Using the result of edge congestion, we can show that the bisection width of crossed cubes is 2(n-1). We also prove that wide diameter and fault diameter are inverted right perpendicular n/2 inverted left perpendicular + 2. Furthermore, we study embedding of cycles in cross cubes and construct more types than previous work of cycles of length at least four.