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<p><b>Abstract</b>—An edge is a bisector of a simple path if it contains the middle point of the path. Let <tmath>T=(V, E)</tmath> be a tree. Given a source vertex <tmath>s \in V</tmath>, the single-source tree bisector problem is to find, for every vertex <tmath>v \in V</tmath>, a bisector of the simple path from <tmath>s</tmath> to <tmath>v</tmath>. The all-pairs tree bisector problem is to find for, every pair of vertices <tmath>u</tmath>, <tmath>v \in V</tmath>, a bisector of the simple path from <tmath>u</tmath> to <tmath>v</tmath>. In this paper, it is first shown that solving the single-source tree bisector problem of a weighted tree has a time lower bound <tmath>\Omega(n \log n)</tmath> in the sequential case. Then, efficient parallel algorithms are proposed on the EREW PRAM for the single-source and all-pairs tree bisector problems. Two <tmath>O(\log n)</tmath> time single-source algorithms are proposed. One uses <tmath>O(n)</tmath> work and is for unweighted trees. The other uses <tmath>O(n \log n)</tmath> work and is for weighted trees. Previous algorithms for the single-source problem could achieve the same time <tmath>O(\log n)</tmath> and the same optimal work, <tmath>O(n)</tmath> for unweighted trees and <tmath>O(n \log n)</tmath> for weighted trees, on the CRCW PRAM. The contribution of our single-source algorithms is the improvement from CRCW to EREW. One all-pairs parallel algorithm is proposed. It requires <tmath>O( \log n)</tmath> time using <tmath>O(n^2)</tmath> work. All the proposed algorithms are cost-optimal. Efficient tree bisector algorithms have practical applications to several location problems on trees. Using the proposed algorithms, efficient parallel solutions for those problems are also presented.</p>
Parallel algorithms, trees, bisectors, location theory, EREW PRAM, tree contraction, the Euler-tour technique.

K. ShiI, S. Ku and B. Wang, "Cost-Optimal Parallel Algorithms for the Tree Bisector and Related Problems," in IEEE Transactions on Parallel & Distributed Systems, vol. 12, no. , pp. 888-898, 2001.
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