Computing Minimum Spanning Trees with Uncertainty

Abstract

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.