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

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 ≤ 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 + m³n) time, which is more efficient than the previous result of O(m²n^2.5 + mn³) time. Besides, their maximal length is also minimized in the worst case.