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Optimal algorithms for the single and multiple vertex updating problems of a minimum spanning tree. (English) Zbl 0860.68053
Summary: 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 $$O(\text{lg }n)$$ time using $$n/\text{lg }n$$ EREW PRAM processors, where $$n=|V|$$. 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 $$O(\text{lg }k\cdot\text{lg }n)$$ parallel time using $$(k\cdot n)/(\text{lg }k\cdot\text{lg }n)$$ EREW PRAM processors. This is optimal for graphs having $$\Omega(kn)$$ edges.

##### MSC:
 68W10 Parallel algorithms in computer science 68R10 Graph theory (including graph drawing) in computer science
##### Keywords:
vertex updating problem; minimum spanning tree
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##### References:
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