Publication 2004 Issue No. 8 - August Abstract - Finding r-Dominating Sets and p-Centers of Trees in Parallel
Finding r-Dominating Sets and p-Centers of Trees in Parallel
August 2004 (vol. 15 no. 8)
pp. 687-398
 ASCII Text x Biing-Feng Wang, "Finding r-Dominating Sets and p-Centers of Trees in Parallel," IEEE Transactions on Parallel and Distributed Systems, vol. 15, no. 8, pp. 687-398, August, 2004.
 BibTex x @article{ 10.1109/TPDS.2004.36,author = {Biing-Feng Wang},title = {Finding r-Dominating Sets and p-Centers of Trees in Parallel},journal ={IEEE Transactions on Parallel and Distributed Systems},volume = {15},number = {8},issn = {1045-9219},year = {2004},pages = {687-398},doi = {http://doi.ieeecomputersociety.org/10.1109/TPDS.2004.36},publisher = {IEEE Computer Society},address = {Los Alamitos, CA, USA},}
 RefWorks Procite/RefMan/Endnote x TY - JOURJO - IEEE Transactions on Parallel and Distributed SystemsTI - Finding r-Dominating Sets and p-Centers of Trees in ParallelIS - 8SN - 1045-9219SP687EP398EPD - 687-398A1 - Biing-Feng Wang, PY - 2004KW - TreesKW - r{\hbox{-}}{\rm{dominating}} setsKW - p{\hbox{-}}{\rm{centers}}KW - network location theoryKW - parallel algorithmsKW - PRAM.VL - 15JA - IEEE Transactions on Parallel and Distributed SystemsER -
Biing-Feng Wang, IEEE Computer Society

Abstract—Let T=(V, E) be an edge-weighted tree with |V|=n vertices embedded in the Euclidean plane. Let {\hbox{\rlap{I}\kern 2.0pt{\hbox{E}}}} denote the set of all points on the edges of T. Let X and Y be two subsets of {\hbox{\rlap{I}\kern 2.0pt{\hbox{E}}}} and let r be a positive real number. A subset D\subseteq X is an X/Y/r{\hbox{-}}dominating set if every point in Y is within distance r of a point in D. The X/Y/r{\hbox{-}}dominating set problem is to find an X/Y/r{\hbox{-}}{\rm{dominating}} set D^* with minimum cardinality. Let p\ge 1 be an integer. The X/Y/p{\hbox{-}}centerproblem is to find a subset C^*\subseteq X of p points such that the maximum distance of any point in Y from C^* is minimized. Let X and Y be either V or {\hbox{\rlap{I}\kern 2.0pt{\hbox{E}}}}. In this paper, efficient parallel algorithms on the EREW PRAM are first presented for the X/Y/r{\hbox{-}}{\rm{dominating}} set problem. The presented algorithms require O(\log^2n) time for all cases of X and Y. Parallel algorithms on the EREW PRAM are then developed for the X/Y/p{\hbox{-}}{\rm{center}} problem. The presented algorithms require O(\log^3n) time for all cases of X and Y. 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.

[1] K. Abrahamson, N. Dadoun, D.G. Kirkpatrick, and T. Przytycka, A Simple Parallel Tree Contraction Algorithm J. Algorithms, vol. 10, pp. 287-302, 1989.
[2] M. Atallah, R. Cole, and M. Goodrich, Cascading Divide-and-Conquer: A Technique for Designing Parallel Algorithms SIAM J. Computing, vol. 18, no. 3, pp. 499-532, 1998.
[3] B. Berger, J. Rompel, and P.W. Shor, Efficient NC Algorithms for Set Cover with Applications to Learning and Geometry J. Computer and System Sciences, vol. 49, no. 3, pp. 454-477, 1994.
[4] B. Bozkaya and B. Tansel, A Spanning Tree Approach to the Absolutep-Center Problem Location Science, vol. 6, pp. 83-107, 1998.
[5] R. Chandrasekaran and A. Tamir, Polynomially Bounded Algorithms for Locatingp-Centers on a Tree Math. Programming, vol. 22, pp. 304-315, 1982.
[6] R. Cole, Slowing Down Sorting Networks to Obtain Faster Sorting Algorithms J. ACM, vol. 34, pp. 200-208, 1987.
[7] R. Cole, An Optimally Efficient Selection Algorithm Information Processing Letters, vol. 26, pp. 295-299, 1988.
[8] R. Cole and U. Vishkin, Approximate Parallel Scheduling, Part I: The Basic Technique with Applications to Optimal Parallel List Ranking in Logarithmic Time SIAM J. Computing, vol. 17, no. 1, pp. 128-141, 1988.
[9] R. Cole and U. Vishkin, The Accelerated Centroid Decomposition Technique for Optimal Parallel Tree Evaluation in Logarithmic Time Algorithmica, vol. 3, pp. 329-346, 1988.
[10] G.N. Frederickson, Parametric Search and Locating Supply Centers in Trees Proc. Second Workshop Algorithms and Data Structures, pp. 299-319, 1991.
[11] G.N. Frederickson and D.B. Johnson, Findingkth Paths andp-Centers by Generating and Searching Good Data Structures J. Algorithms, vol. 4, pp. 61-80, 1983.
[12] A.J. Goldman, Optimal Center Location in Simple Networks Transportation Science, vol. 5, pp. 212-221, 1971.
[13] A.J. Goldman, Minmax Location of a Facility in an Undirected Tree Graph Transportation Science, vol. 6, pp. 407-418, 1972.
[14] 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.
[15] G.Y. Handler, Minimax Location of a Facility in an Undirected Tree Graph Transportation Science, vol. 7, pp. 287-293, 1973.
[16] G.Y. Handler, Finding Two-Centers of a Tree: The Continuous Case Transportation Science, vol. 12, pp. 93-106, 1978.
[17] X. He and Y. Yesha, Binary Tree Algebraic Computation and Parallel Algorithms for Simple Graphs J. Algorithms, vol. 9, pp. 92-113, 1988.
[18] X. He and Y. Yesha, Efficient Parallel Algorithms forr-Dominating Set andp-Center Problems on Trees Algorithmica, vol. 5, pp. 129-145, 1990.
[19] J. JáJá, An Introduction to Parallel Algorithms. Addison-Wesley, 1992.
[20] O. Kariv and S.L. Hakimi, An Algorithmic Approach to Network Location Problems SIAM J. Applied Math., vol. 37, pp. 513-538, 1979.
[21] Y.-B. Lin and I. Chlamtac, Wireless and Mobile Network Architecture. Wiley, 2001.
[22] J.Y.-B. Lee, On a Unified Architecture for Video-on-Demand Services IEEE Trans. Multimedia, vol. 4, no. 1, pp. 38-47, 2002.
[23] M. Luby and N. Nisan, A Parallel Approximation Algorithm for Positive Linear Programming Proc. 25th ACM Symp. Theory of Computing, pp. 448-457, 1993.
[24] N. Megiddo, Applying Parallel Computation Algorithms in the Design of Serial Algorithms J. ACM, vol. 30, pp. 852-865, 1983.
[25] N. Megiddo, A. Tamir, E. Zemel, and R. Chandrasekaran, An$O(n\log^2n)$Time Algorithm for thekth Longest Path in a Tree with Applications to Location Problems SIAM J. Computing, vol. 10, pp. 328-337, 1981.
[26] N. Megiddo and A. Tamir, New Results on the Complexity ofp-Center Problems SIAM J. Computing, vol. 12, no. 4, pp. 751-759, 1983.
[27] E. Minieka, The Optimal Location of a Path or Tree in a Tree Network Networks, vol. 15, pp. 309-321, 1985.
[28] C.A. Morgan and P.L. Slater, A Linear Time Algorithm for a Core of a Tree J. Algorithms, vol. 1, pp. 247-258, 1980.
[29] S. Peng and W.-T. Lo, A Simple Optimal Parallel Algorithm for a Core of a Tree J. Parallel and Distributed Computing, vol. 20, pp. 388-392, 1994.
[30] S. Peng and W. Lo, Efficient Algorithms for Finding a Core of a Tree with Specified Length J. Algorithms, vol. 15, pp. 143-159, 1996.
[31] S. Rajagopalan and V. Vazirani, Primal-Dual RNC Approximation Algorithms for Set Cover and Covering Integer Programs SIAM J. Computing, vol. 28, no. 2, pp. 525-540, 1999.
[32] 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. 862-874, 1985.
[33] A. Tamir, An$O(pn^2)$Time Algorithm for thep-Median and Related Problems on Tree Graphs Operations Research Letters, vol. 19, no. 2, pp. 59-64, 1996.
[34] A. Tamir, Thek-Centrum Multi-Facility Location Problem Discrete Applied Math., vol. 109, pp. 293-307, 2001.
[35] 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.
[36] B.-F. Wang, Finding a Two-Core of a Tree in Linear Time SIAM J. Discrete Math., vol. 15, no. 2, pp. 193-210, 2002.
[37] B.-F. Wang, S.-C. Ku, and K.-H. Shi, Cost-Optimal Parallel Algorithms for the Tree Bisector and Related Problems IEEE Trans. Parallel and Distributed Systems, vol. 12, no. 9, pp. 888-898, Sept. 2001.

Index Terms:
Trees, r{\hbox{-}}{\rm{dominating}} sets, p{\hbox{-}}{\rm{centers}}, network location theory, parallel algorithms, PRAM.
Citation:
Biing-Feng Wang, "Finding r-Dominating Sets and p-Centers of Trees in Parallel," IEEE Transactions on Parallel and Distributed Systems, vol. 15, no. 8, pp. 687-398, Aug. 2004, doi:10.1109/TPDS.2004.36