A Faster Algorithm for Finding Disjoint Ordering of Sets

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. We study the problem of finding a disjoint ordering of sets X₁, X₂, X_s where X_i ⊆{1, 2, ···, n} and s ≤ 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.