This Article 
 Bibliographic References 
 Add to: 
Cost-Optimal Parallel Algorithms for the Tree Bisector and Related Problems
September 2001 (vol. 12 no. 9)
pp. 888-898

Abstract—An edge is a bisector of a simple path if it contains the middle point of the path. Let T=(V, E) be a tree. Given a source vertex s \in V, the single-source tree bisector problem is to find, for every vertex v \in V, a bisector of the simple path from s to v. The all-pairs tree bisector problem is to find for, every pair of vertices u, v \in V, a bisector of the simple path from u to v. In this paper, it is first shown that solving the single-source tree bisector problem of a weighted tree has a time lower bound \Omega(n \log n) 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 O(\log n) time single-source algorithms are proposed. One uses O(n) work and is for unweighted trees. The other uses O(n \log n) work and is for weighted trees. Previous algorithms for the single-source problem could achieve the same time O(\log n) and the same optimal work, O(n) for unweighted trees and O(n \log n) 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 O( \log n) time using O(n^2) 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.

[1] K. Abrahamson, N. Dadoun, D.G. Kirkpatrick, and T. Przytycka, "A Simple Parallel Tree Contraction Algorithm," J. Algorithms, vol. 10, no. 2, pp. 287-302, 1989.
[2] E.A. Albacea, “Parallel Algorithm for Finding a Core of a Tree Network,” Information Processing Letters, vol. 51, pp. 223-226, 1994.
[3] M. Ajtai,J. Komlos,W.L. Steiger, and E. Szemeredi,"An O(n log n) sorting network," Proc. Ann. ACM Symp. Theory of Computing, pp. 1-9, 1983.
[4] R.I. Becker and Y. Perl, “Finding the Two-Core of a Tree,” Discrete Applied Math., vol. 11, no. 2, pp.103-113, 1985.
[5] O. Berkman, B. Schieber, and U. Vishkin, "Optimal Doubly Logarithmic Parallel Algorithms Based on Finding Nearest Smaller Values," J. Algorithms, vol. 14, no. 3, pp. 344-370, 1993.
[6] O. Berkman and U. Vishkin, “Finding Level-Ancestors in Trees,” J. Computer and Science Sciences, vol. 48, pp.231-254, 1994.
[7] R. Cole, "Parallel Merge Sort," SIAM J. Computing, vol. 17, pp. 770-785, 1988.
[8] R. Cole and U. Vishkin, "Approximate Parallel Scheduling. Part 1: The Basic Technique with Applications to Optimal Parallel List Ranking in Logarithmic Time," SIAM J. Computing, vol. 18, pp. 128-142, 1988.
[9] T.H. Cormen,C.E. Leiserson, and R.L. Rivest,Introduction to Algorithms.Cambridge, Mass.: MIT Press/McGraw-Hill, 1990.
[10] K. Diks and W. Rytter, “On Optimal Parallel Computations for Sequences of Brackets,” Theoretical Computer Science, vol. 87, pp. 251-262, 1991.
[11] B. Gavish and S. Sridhar, “Computing the 2-Median on Tree Networks in$O(n\lg n)$Time,” Networks, vol. 26, no. 4, pp. 305-317, 1995.
[12] A. Gibbons and W. Rytter, Efficient Parallel Algorithms. Cambridge Univ. Press, 1988.
[13] S.L. Hakimi, E.F. Schmeichel, and M. Labbe, “On Locating Path- or Tree-Shaped Facilities on Networks,” Networks, vol. 23, pp. 543-555, 1993.
[14] G.Y. Handler and P. Mirchandani, Location on Networks. Cambridge, Mass.: MIT Press, 1979.
[15] J. J'aJ'a, An Introduction to Parallel Algorithms.New York: Addison-Wesley, 1992.
[16] O. Kariv and S.L. Hakimi, “An Algorithmic Approach to Network Location Problems. II: The p-Medians,” SIAM J. Applied Math., vol. 37, no. 3, 1979.
[17] G.L. Miller and J. Reif, “Parallel Tree Contraction and Its Applications,” Proc. 26th Ann. IEEE Symp. Foundations of Computer Science, pp. 478-489, 1985.
[18] S. Peng and W.-T. Lo, “A Simple Optimal Algorithm for a Core of a Tree,” J. Parallel and Distributed Computing, vol. 20, pp. 388-392, 1994.
[19] S. Peng and W.-T. Lo, “The Optimal Location of a Structured Facility in a Tree Network,” Parallel Algorithms and Applications, vol. 2, pp. 43-60, 1994.
[20] S. Peng and W.-T. Lo, “Efficient Algorithms for Finding a Core of a Tree with a Specified Length,” J. Algorithms, vol. 20, pp. 445-458, 1996.
[21] F.P. Preparata and M.I. Shamos, Computational Geometry. Springer-Verlag, 1985.
[22] P.V. Rangan, H.M. Vin, and S. Ramanathan, “Designing an On-Demand Multimedia Service,” Comm. Magazine, vol. 30, no. 7, Jul. 1992.
[23] A. Tamir, “An$O(pn^2)$Algorithm for thep-median and Related Problems on Tree Graphs,” Operations Research Letters, vol. 19, no. 2, pp. 59-64, 1996.
[24] B.C. Tansel, R.L. Francis, and T.J. Lowe, “Location on Networks: A Survey,” Management Science, vol. 29, pp. 482-511, 1983.
[25] R.E. Tarjan and U. Vishkin, “Finding Biconnected Components and Computing Tree Functions in Logarithmic Parallel Time,” SIAM J. Computing, vol. 14, no. 4, pp. 81-874, 1985.
[26] B.-F. Wang, “Finding ak-Tree Core and ak-Tree Center of a Tree Network in Parallel,” IEEE Trans. Parallel and Distributed Systems, vol. 9, no. 2, pp. 186-191, Feb. 1998.
[27] B.-F. Wang, “Efficient Parallel Algorithms for Optimally Locating a Path and a Tree of a Specified Length in a Weighted Tree Network,” J. Algorithms, vol. 34, pp. 90-108, 2000.
[28] B.-F. Wang, S.-C. Ku, K.-H. Shi, T.-K. Hung, and P.-S. Liu, “Parallel Algorithms for the Tree Bisector Problem and Applications,” Proc. 1999 Int'l Conf. Parallel Processing, pp. 192-199, 1999.

Index Terms:
Parallel algorithms, trees, bisectors, location theory, EREW PRAM, tree contraction, the Euler-tour technique.
Biing-Feng Wang, Shan-Chyun Ku, Keng-Hua ShiI, "Cost-Optimal Parallel Algorithms for the Tree Bisector and Related Problems," IEEE Transactions on Parallel and Distributed Systems, vol. 12, no. 9, pp. 888-898, Sept. 2001, doi:10.1109/71.954619
Usage of this product signifies your acceptance of the Terms of Use.