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Issue No. 08 - August (2004 vol. 15)
ISSN: 1045-9219
pp: 687-398
Biing-Feng Wang , IEEE Computer Society
<p><b>Abstract</b>—Let <tmath>T=(V, E)</tmath> be an edge-weighted tree with <tmath>|V|=n</tmath> vertices embedded in the Euclidean plane. Let <tmath>{\hbox{\rlap{I}\kern 2.0pt{\hbox{E}}}}</tmath> denote the set of all points on the edges of <it>T</it>. Let <it>X</it> and <it>Y</it> be two subsets of <tmath>{\hbox{\rlap{I}\kern 2.0pt{\hbox{E}}}}</tmath> and let <it>r</it> be a positive real number. A subset <tmath>D\subseteq X</tmath> is an <tmath>X/Y/r{\hbox{-}}dominating set</tmath> if every point in <it>Y</it> is within distance <it>r</it> of a point in <it>D</it>. The <tmath>X/Y/r{\hbox{-}}dominating set problem</tmath> is to find an <tmath>X/Y/r{\hbox{-}}{\rm{dominating}}</tmath> set <tmath>D^*</tmath> with minimum cardinality. Let <tmath>p\ge 1</tmath> be an integer. The <tmath>X/Y/p{\hbox{-}}center</tmath><it>problem</it> is to find a subset <tmath>C^*\subseteq X</tmath> of <it>p</it> points such that the maximum distance of any point in <it>Y</it> from <tmath>C^*</tmath> is minimized. Let <it>X</it> and <it>Y</it> be either <it>V</it> or <tmath>{\hbox{\rlap{I}\kern 2.0pt{\hbox{E}}}}</tmath>. In this paper, efficient parallel algorithms on the EREW PRAM are first presented for the <tmath>X/Y/r{\hbox{-}}{\rm{dominating}}</tmath> set problem. The presented algorithms require <tmath>O(\log^2n)</tmath> time for all cases of <it>X</it> and <it>Y</it>. Parallel algorithms on the EREW PRAM are then developed for the <tmath>X/Y/p{\hbox{-}}{\rm{center}}</tmath> problem. The presented algorithms require <tmath>O(\log^3n)</tmath> time for all cases of <it>X</it> and <it>Y</it>. Previously, sequential algorithms for these two problems had been extensively studied in the literature. However, parallel solutions with polylogarithmic time existed only for their special cases. The algorithms presented in this paper are obtained by using an interesting approach which we call the dependency-tree approach. Our results are examples of parallelizing sequential dynamic-programming algorithms by using the approach.</p>
Trees, r{\hbox{-}}{\rm{dominating}} sets, p{\hbox{-}}{\rm{centers}}, network location theory, parallel algorithms, PRAM.

B. Wang, "Finding r-Dominating Sets and p-Centers of Trees in Parallel," in IEEE Transactions on Parallel & Distributed Systems, vol. 15, no. , pp. 687-398, 2004.
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