A Faster Algorithm for Finding Disjoint Ordering of Sets

Abstract

Consider the problem of routing from a single source node to multiple target nodes with the additional condition that these disjoint paths be the shortest. This problem is harder than the standard one-to-many routing in that such paths do not always exist. Various sufficient and necessary conditions have been found to determine when such paths exist for some interconnection networks. And when these conditions do hold, the problem of finding such paths can be reduced to the problem of finding a disjoint ordering of sets. In addition to the applications in finding disjoint shortest paths in interconnection networks, the problem of finding disjoint ordering of sets is an interesting combinatorial problem in its own right. We study the problem of finding a disjoint ordering of sets A₁, A₂, ..., A_m where A_i ⊆ A = ｛a₁, a₂,…, a_n} and m ≤ n. We present an O(n³) algorithm for doing so, under certain conditions, thus improving the previously known O(n⁴) algorithm, and consequently, improving the corresponding one-to-many routing algorithms for finding disjoint and shortest paths.