Computing Minimum Spanning Trees with Uncertainty

Thomas Erlebach

Abstract

We consider the minimum spanning tree problem in a setting where information about the edge weights of the given graph is initially uncertain: For each edge e, a set A(e) called an uncertainty area is given, and it is only known that A(e) contains the actual weight w(e) of e. The algorithm can "update" e to obtain the edge weight w(e). The task is to output the edge set of a minimum spanning tree after a minimum number of updates. An algorithm is k-update competitive if it makes at most k times as many updates as the optimum. We present a 2-update competitive algorithm for the case that all areas A(e) are open or trivial, and show that this is best possible among deterministic algorithms. The condition on the areas A(e) is needed to exclude degenerate inputs for which no constant update competitive algorithm can exist. We also consider the Euclidean MST problem in a setting where the