Abstract
In this paper, we study the problem of locating a median path of limited length on a tree under the condition that some existing facilities are already located. The existing facilities may be located at any subset of vertices. Upper and lower bounds are proposed for both the discrete and continuous models. In the discrete model, a median path is not allowed to contain partial edges. In the continuous model, a median path may contain partial edges. The proposed upper bounds for these two models are O(nlog n) and O(nlog nα(n)), respectively. They improve the previous ones from O(nlog2 n) and O(n 2), respectively. The proposed lower bounds are both Ω(nlog n). The lower bounds show that our upper bound for the discrete model is optimal and the margin for possible improvement on our upper bound for the continuous model is slim.
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References
S. Alstrup, J. Holm, and M. Thorup, Maintaining median and center in dynamic trees, in Proceedings of the SWAT 2000, Lecture Notes in Computer Science, vol. 1851, Springer-Verlag, pp. 46–56, 2000.
S. Alstrup, P.W. Lauridsen, P. Sommerlund, and M. Throup, Finding cores of limited length, Technical Report, The IT University of Copenhagen, a preliminary version of this paper appeared in Proceedings of the 5th International Workshop on Algorithms and Data Structures, Lecture Notes in Computer Science, vol. 1272, Springer-Verlag, pp. 45–54, 1997.
S. L. Hakimi, E. F. Schmeichel and M. Labbé, On locating path-or tree-shaped facilities on networks, Networks, vol. 23, pp. 543–555, 1993.
S. Hart and M. Sharir, Nonlinearity of Davenport-Schinzel sequences and of general path compression schemes, Combinatorica, vol. 6, pp. 151–177, 1986.
E. Minieka, Conditional centers and medians on a graph, Networks, vol. 10, pp.265–272, 1980.
E. Minieka, The optimal location of a path or tree in a tree network, Networks, vol. 15, pp. 309–321, 1985.
F. P. Preparata and M. I. Shamos, Computational Geometry: An Introduction, Springer-Verlag, 1985.
P. J. Slater, Locating central paths in a network, Transportation Science, vol. 16, No. 1, pp. 1–18, 1982.
A. Tamir, Fully polynomial approximation schemes for locating a tree-shaped facility: a generalization of the knapsack problem, Discrete Applied Mathematics, vol. 87, pp. 229–243, 1998.
A. Tamir, J. Puerto, J.A. Mesa, and A.M. Rodriguez-Chia, Conditional location of path and tree shaped facilities on trees, manuscript, 2001.
B.-F. Wang, Finding a two-core of a tree in linear time, SIAM Journal on Discrete Mathematics, accepted.
B.-F. Wang, Efficient parallel algorithms for optimally locating a path and a tree of a specified length in a weighted tree network, Journal of Algorithms, vol. 34, pp. 90–108, 2000.
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Wang, BF., Ku, SC., Hsieh, YH. (2002). The Conditional Location of a Median Path. In: Ibarra, O.H., Zhang, L. (eds) Computing and Combinatorics. COCOON 2002. Lecture Notes in Computer Science, vol 2387. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45655-4_53
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DOI: https://doi.org/10.1007/3-540-45655-4_53
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