Computing Minimum Spanning Trees with Uncertainty

We consider the minimum spanning tree problem in a setting where information about the edge weights of the given graph is uncertain. Initially, for each edge $e$ of the graph only a set $A_e$, called an uncertainty area, that contains the actual edge weight $w_e$ is known. The algorithm can `update' $e$ to obtain the edge weight $w_e in A_e$. 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 if all areas $A_e$ are open or trivial, which is the best possible among deterministic algorithms. The condition on the areas $A_e$ is to exclude degenerate inputs for which no constant update competitive algorithm can exist. Next, we consider a setting where the vertices of the graph correspond to points in Euclidean space and the weight of an edge is equal to the distance of its endpoints. The location of each point is initially given as an uncertainty area, and an update reveals the exact location of the point. We give a general relation between the edge uncertainty and the vertex uncertainty versions of a problem and use it to derive a $4$-update competitive algorithm for the minimum spanning tree problem in the vertex uncertainty model. Again, we show that this is best possible among deterministic algorithms.