Solving cheap graph problems on Meshes
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Efficient mesh algorithms exist for ‘expensive’ graph problems like transitive closure and computing all shortest paths, taking O(n) time on an n×n mesh for a graph with n vertices. This is work-optimal. On the other hand, there are so far no efficient algorithms for ‘cheap’ graph problems like finding the connected components, constructing a minimum spanning tree, or finding a path between a pair of vertices. These problems are cheap in the sense that sequentially they can be solved in time (almost) linear in n and m, where m is the number of edges.
In this paper we show that the mentioned cheap problems can be solved on an √N×√N mesh in O((n·m/N)1/2+ log n· (√N+m/N)) time. We derive a lower bound of Ω((n·m/N)1/2+√N+m/N). Hence, the algorithms have optimal time order for most n,m and N. The algorithms are work-optimal to within a factor log n for N≤m/n.
KeywordsSpan Tree Minimum Span Tree List Ranking Sparse Graph Graph Problem
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