Parallel shortcutting of rooted trees - Thorup, Mikkel 1973- (Datalogisk Institut <København>: DIKU-Rapport, 1994)
Abstract: "First it is shown that for any rooted tree T with n vertices, and parameter m [> or =] n, there is a 'shortcutting' set of S of at most m arcs from the transitive closure T[superscript *] of T such for any (v, w)[element] T[superscript *], there is a dipath in T [union] S from v to w of length O([alpha](m, n)). An equivalent result has been achieved by Chazelle (1987), but our proof is algorithmically simpler, and, in particular, it lends itself well to parallelization. More precisely, suppose that weights from a semi-group are assigned to the arcs of T. Then we can preprocess T in time O(log n) with O(m/log n) processors on a CREW PRAM such that any (v, w) [element] T[superscript *], we can find the weight of the path from v to w in O([alpha](m, n)) sequential time. In an unpublished manuscript, Alon and Schieber (1987) have claimed that such a parallelization is possible for Chazelle's result. This claim is used in the optimal parallel sensitivity analysis for minimum spanning trees by Dixon (1993). However, Alon and Schieber did not give the details of the parallelization. Here we present a full proof, and our algorithms, both the sequential and the parallel versions, are rather simple, hence likely to be of practical relevance."