A Variation of Minimum Latency Problem

Abstract

In this paper we study the variation of the minimum latency problem(MLP) [2]. The MLP is to nd a walk tour on the graph G(V;E) with a distance matrix di;j . Where di;j indicate the distance between vi and vj . Let `(vi) is the latency length of vi, dened to be the dis- tance traveled before rst visiting vi. The minimum la- tency tour is to minimize Pn i=1 `(vi). In some message broadcast and scheduling problem [8] the vertex also has latency time and weight. Those problem need to extend the objective function of the minimum latency tour as Pn i=1 `(vi)w(vi). The denition is equivalent to the MLP with no edge distance but vertex latency time and vertex weight. We give an linear algorithm for the unweighted full k-ary tree or k-path graphs, and O(n log n) time for general tree graphs. The time complexity in trees is the same as Adolphson's result; however, the algorithm given here is not only simpler, easier to understand, but also more exible and thus can be easily extended to other classes of graphs.