Characterization of Efficiently Parallel Solvable Problems on a Class of Decomposable Graphs
In this paper, we sketch characteristics of those problems which can be systematically solved on decomposable graphs. Trees, series-parallel graphs, outerplanar graphs, and bandwidth-k graphs all belong to decomposable graphs. Let T d (|V|,|E|) and P d (|V|,|E|) denote the time complexity and processor complexity required to construct a parse tree representation T G for a decomposable G=(V,E) on a PRAM model M d . We define a general problem-solving paradigm to solve a wide class of subgraph optimization problems on decomposable graphs in O(T d (|V|,|E|)+log |V(T G )|) time using O(P d (|V|,|E|)+|V(T G )|/log |V(T G )|) processors on M d . By using our paradigm, we show the following parallel complexities: (a) The maximum independent set problem on trees can be solved in O(log |V|) time using O(|V|/log |V|) processors on an EREW PRAM. (b) The maximum matching problem on series-parallel graphs can be solved in O(log |E|) time using O(|E|/log |E|) processors on an EREW PRAM. (c) The efficient domination problem on series-parallel graphs can be solved in O(log |E|) time using O(|E|/log |E|) processors on an EREW PRAM.
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