Publication 2002 Issue No. 9 - September Abstract - An Efficient Parallel Algorithm for the Efficient Domination Problem on Distance-Hereditary Graphs
An Efficient Parallel Algorithm for the Efficient Domination Problem on Distance-Hereditary Graphs
September 2002 (vol. 13 no. 9)
pp. 985-993
 ASCII Text x Sun-yuan Hsieh, "An Efficient Parallel Algorithm for the Efficient Domination Problem on Distance-Hereditary Graphs," IEEE Transactions on Parallel and Distributed Systems, vol. 13, no. 9, pp. 985-993, September, 2002.
 BibTex x @article{ 10.1109/TPDS.2002.1036071,author = {Sun-yuan Hsieh},title = {An Efficient Parallel Algorithm for the Efficient Domination Problem on Distance-Hereditary Graphs},journal ={IEEE Transactions on Parallel and Distributed Systems},volume = {13},number = {9},issn = {1045-9219},year = {2002},pages = {985-993},doi = {http://doi.ieeecomputersociety.org/10.1109/TPDS.2002.1036071},publisher = {IEEE Computer Society},address = {Los Alamitos, CA, USA},}
 RefWorks Procite/RefMan/Endnote x TY - JOURJO - IEEE Transactions on Parallel and Distributed SystemsTI - An Efficient Parallel Algorithm for the Efficient Domination Problem on Distance-Hereditary GraphsIS - 9SN - 1045-9219SP985EP993EPD - 985-993A1 - Sun-yuan Hsieh, PY - 2002KW - Parallel algorithmKW - PRAMKW - distance-hereditary graphsKW - the efficient domination problemKW - binary tree contraction technique.VL - 13JA - IEEE Transactions on Parallel and Distributed SystemsER -

Abstract—In the literature, there are quite a few sequential and parallel algorithms for solving problems on distance-hereditary graphs. With an n-vertex and m-edge distance-hereditary graph G, we show that the efficient domination problem on G can be solved in \big. O(\log^{2} n)\bigr. time using \big. O(n+m)\bigr. processors on a CREW PRAM. Moreover, if a binary tree representation of G is given, the problem can be optimally solved in \big. O(\log n)\bigr. time using \big. O(n/\log n)\bigr. procssors on an EREW PRAM.

[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] H.J. Bandelt and H.M. Mulder, “Distance-Hereditary Graphs,” J. Combinatorial Theory Series B, vol. 41, no. 1, pp. 182-208, 1989.
[3] H.J. Bandelt, A. Henkmann, and F. Nicolai, “Powers of Distance-Hereditary Graphs,” Discrete Math., vol. 145, pp. 37-60, 1995.
[4] D. Bange, A. Barkauskas, and P. J. Slater, “Disjoint Dominating Sets in Trees,” Sandia Laboratories Report, SAND 78-1087J, 1978.
[5] D. Bange, A. Barkauskas, and P.J. Slater, “Efficient Dominating Sets in Graphs,” Application of Discrete Math., R.D. Ringeisen and F.S. Roberts, eds., pp. 189-199, 1988.
[6] N. Biggs, “Perfect Codes in Graphs,” J. Combinatorial Theory Series B, vol. 15, pp. 289-296, 1973.
[7] A. Brandstädt and F.F. Dragan, “A Linear Time Algorithm for Connected$\big. \gamma\hbox{-}{\rm{Domination}}\bigr.$and Steiner Tree on Distance-Hereditary Graphs,” Networks, vol. 31, pp. 177-182, 1998.
[8] M.S. Chang and Y.C. Liu, “Polynomial Algorithm for the Weighted Perfect Domination Problems on Chordal Graphs and Split Graphs,” Information Processing Letters, vol. 48, pp. 205-210, 1993.
[9] G.J. Chang, C.P. Rangan, and S.R. Coorg, “Weighted Independent Perfect Domination on Cocomparability Graphs,” Discrete Applied Math., vol. 63, pp. 215-222, 1995.
[10] M.S. Chang, S.Y. Hsieh, and G.H. Chen, “Dynamic Programming on Distance-Hereditary Graphs,” Proc. Seventh Int'l Symp. Algorithms and Computation (ISAAC'97), pp. 344-353, 1997.
[11] Y.D. Liang, C.L. Lu, and C.Y. Tang, “Efficient Domination on Permutation Graphs and Trapezoid Graphs,” Proc. Third Ann. Int'l Computing and Combinatorics Conf., (COCOON '97), pp. 232-241, 1997.
[12] A. D'atri and M. Moscarini, “Distance-Hereditary Graphs, Steiner Trees, and Connected Domination,” SIAM J. Computing, vol. 17, no. 3, pp. 521-538, 1988.
[13] F.F. Dragan, “Dominating Cliques in Distance-Hereditary Graphs,” Proc. Fourth Scandinavian Workshop Algorithm Theory (SWAT '94), pp. 370-381, 1994.
[14] A. Esfahanian and O.R. Oellermann, “Distance-Hereditary Graphs and Multidestination Message-Routing in Multicomputers,” J. Combinatorial Math. and Combinatorial Computing, vol. 13, pp. 213-222, 1993.
[15] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs.New York, Academic Press, 1980.
[16] P.L. Hammer and F. Maffray, “Complete Separable Graphs,” Discrete Applied Math., vol. 27, no. 1, pp. 85-99, 1990.
[17] X. He, “Efficient Parallel Algorithms for Solving Some Tree Problems,” Proc. 24th Allerton Conf. Communication, Control, and Computing, pp. 777-786, 1986.
[18] X. He, “Efficient Parallel Algorithms for Series-Parallel Graphs,” J. Algorithms, vol. 12, pp. 409-430, 1991.
[19] X. He and Y. Yesha, “Binary Tree Algebraic Computation and Parallel Algorithms for Simple Graphs,” J. Algorithms, vol. 9, pp. 92-113, 1988.
[20] E. Howorka, “A Characterization of Distance-Hereditary Graphs,” Quarterly J. Math. (Oxford), vol. 28, no. 2, pp. 417-420, 1977.
[21] S.-Y. Hsieh, C.W. Ho, T.-S. Hsu, M.T. Ko, and G.H. Chen, “Efficient Parallel Algorithms on Distance-Hereditary Graphs,” Parallel Processing Letters, vol. 9, no. 1, pp. 43-52, 1999.
[22] S.-Y. Hsieh, “Parallel Decomposition of Distance-Hereditary Graphs,” Proc. Fourth Int'l ACPC Conf. including Special Tracks on Parallel Numerics (ParNum'99) and Parallel Computing in Image Processing, Video Processing, and Multimedia, pp. 417-426, 1999.
[23] S.-Y. Hsieh, C.W. Ho, T.-S. Hsu, M.T. Ko, and G.H. Chen, “A Faster Implementation of a Parallel Tree Contraction Scheme and Its Application on Distance-Hereditary Graphs,” J. Algorithms, vol. 35 pp. 50-81, 2000.
[24] S.-Y. Hsieh, C.W. Ho, T.-S. Hsu, M.T. Ko, and G.H. Chen, “Characterization of Efficiently Solvable Problems on Distance-Hereditary Graphs,” Proc. 8th Int'l Symp. Algorithms and Computation (ISAAC '98), pp. 257-266, 1998.
[25] R.M. Karp and V. Ramachandran, "Parallel Algorithms for Shared-Memory Machines," Handbook of Theoretical Computer Science, J. van Leeuwen, ed., vol. A, pp. 869-941.Amsterdam: NorthHolland, 1990.
[26] M. Livingston and Q.F. Stout, “Perfect Dominating Sets,” Congressus Numerantium, vol. 79, pp. 187-203, 1990.
[27] C.L. Lu and C.Y. Tang, “Efficient Domination on Bipartite Graphs,” Manuscript, 1996.
[28] H. Müller and F. Nicolai, “Polynomial Time Algorithms for Hamiltonian Problems on Bipartite Distance-Hereditary Graphs,” Information Processing Letters, vol. 46, no. 5, pp. 225-230, 1993.
[29] F. Nicolai, “Hamiltonian Problems on Distance-Hereditary Graphs,” Technique Report SM-DU-264, Univ. Duisburg, Germany, 1994.
[30] O.R. Oellermann, “Computing the Average Distance of a Distance-Hereditary Graph in Linear Time,” Congressus Numerantium, vol. 103, pp. 219-223, 1994.
[31] H.G. Yeh and G.J. Chang, “Weighted Connected Domination and Steiner Trees in Distance-Hereditary Graphs,” Discrete Applied Math., vol. 87, pp. 245-253, 1998.
[32] C.C. Yen, “Algorithmic Aspects of Perfect Domination,” PhD thesis, Department of Computer Science, National Tsing Hua Univ., Taiwan, 1992.
[33] C.C. Yen, R.C.T. Lee, “The Weighted Perfect Domination Problem and Its Variants,” Discrete Applied Math., vol. 66, pp. 147-160, 1996.

Index Terms:
Parallel algorithm, PRAM, distance-hereditary graphs, the efficient domination problem, binary tree contraction technique.
Citation:
Sun-yuan Hsieh, "An Efficient Parallel Algorithm for the Efficient Domination Problem on Distance-Hereditary Graphs," IEEE Transactions on Parallel and Distributed Systems, vol. 13, no. 9, pp. 985-993, Sept. 2002, doi:10.1109/TPDS.2002.1036071