Optimal algorithms for the single and multiple vertex updating problems of a minimum spanning tree

The vertex updating problem for a minimum spanning tree (MST) is defined as follows: Given a graph G = (V, E(G)) and an MST T for G, find a new MST for G to which a new vertex z has been added along with weighted edges that connect z with the vertices of G. We present a set of rules that produce simple optimal parallel algorithms that run in O(lg n) time using n/lg n EREW PRAM processors, where n = /V/. These alobrithms employ any valid tree-contraction schedule that can be produced within the stated resource bounds. These rules can also be used to derive simple linear-time sequential algorithms for the same problem. The previously best-known parallel result was a rather complicated algorithm that used n processors in the more powerful CREW PRAM model. Furthermore, we show how our solution can be used to solve the multiple vertex updating problem: Update a given MST when k new vertices are introduced simultaneously. This problem is solved in O (lgk . lg n) parallel time using (k . n)/(lgk . lg n) EREW PRAM processors. This is optimal for graphs having Omega(kn) edges.