Optimal algorithms for the single and multiple vertex updating problems of a minimum spanning tree (original) (raw)
The vertex updating problem for a minimum spanning tree (MST) is de ned as follows: Given a graph G = (V; E G ) and an MST T for G, nd 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 = jV j. These algorithms 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(lg k lg n) parallel time using k n lg k lg n EREW PRAM processors. This is optimal for graphs having (kn) edges.
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