Optimal Construction of All Shortest Node-Disjoint Paths in Hypercubes with Applications

Abstract

Routing functions had been shown effective in constructing node-disjoint paths in hypercube-like networks. In this paper, by the aid of routing functions, m node-disjoint shortest paths from one source node to other m (not necessarily distinct) destination nodes are constructed in an n-dimensional hypercube, provided the existence of such node-disjoint shortest paths which can be verified in O(mn^{1.5}) time, where {m} \leq {n}. The construction procedure has worst case time complexity O(mn), which is optimal and hence improves previous results. By taking advantages of the construction procedure, m node-disjoint paths from one source node to other m (not necessarily distinct) destination nodes in an n-dimensional hypercube such that their total length is minimized can be constructed in O(mn^{1.5} + {m}^3{n}) time, which is more efficient than the previous result of O(m^2 {n}^{2.5} + {mn}^3) time. Besides, their maximal length is also minimized in the worst case.