Abstract
Given a set S of n points in the plane, and an integer k such that 0 ≤ k < n, we show that a geometric graph with vertex set S, at most n – 1 + k edges, and dilation O(n / (k + 1)) can be computed in time O(n log n). We also construct n–point sets for which any geometric graph with n – 1 + k edges has dilation Ω(n / (k + 1)); a slightly weaker statement holds if the points of S are required to be in convex position.
This work was supported by LG Electronics and NUS research grant R-252-000-166-112. Research by B.A. has been supported in part by NSF ITR Grant CCR-00-81964 and by a grant from US-Israel Binational Science Foundation. Part of the work was carried out while B.A. was visiting TU/e in February 2004 and in the summer of 2005. MdB was supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 639.023.301.
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References
Alzoubi, K., Li, X.-Y., Wang, Y., Wan, P.-J., Frieder, O.: Geometric spanners for wireless ad hoc networks. IEEE Trans. Parallel Dist. Systems 14(4), 408–421 (2003)
Callahan, P.B., Kosaraju, S.R.: A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. J. ACM 42, 67–90 (1995)
Chandra, B., Das, G., Narasimhan, G., Soares, J.: New sparseness results on graph spanners. Internat. J. Comput. Geom. Appl. 5, 124–144 (1995)
Das, G., Heffernan, P.: Constructing Degree–3 Spanners with Other Sparseness Properties. Int. J. Found. Comput. Sci. 7(2), 121–136 (1996)
do Carmo, M.: Differential Geometry of Curves and Surfaces. Prentice Hall, Englewood Cliffs (1976)
Emek, Y., Peleg, D.: Approximating minimum max-stretch spanning trees on unweighted graphs (full version). In: Proc. ACM-SIAM Symp. Discrete Algo., pp. 261–270 (2004)
Eppstein, D.: Spanning trees and spanners. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 425–461. Elsevier Science Publishers, Amsterdam (2000)
Eppstein, D., Wortman, K.: Minimum dilation stars. In: Proc. ACM Symp. Comput. Geom., pp. 321–326 (2005)
Farley, A.M., Proskurowski, A., Zappala, D., Windisch, K.J.: Spanners and message distribution in networks. Discrete Appl. Math. 137(2), 159–171 (2004)
Fekete, S.P., Kremer, J.: Tree spanners in planar graphs. Discrete Appl. Math. 108, 85–103 (2001)
Keil, J.M., Gutwin, C.A.: Classes of graphs which approximate the complete Euclidean graph. Discrete Comput. Geom. 7, 13–28 (1992)
Levcopoulos, C., Lingas, A.: There are planar graphs almost as good as the complete graphs and almost as cheap as minimum spanning trees. Algorithmica 8, 251–256 (1992)
Li, X.-Y.: Applications of computational geomety in wireless ad hoc networks. In: Cheng, X.-Z., Huang, X., Du, D.-Z. (eds.) Ad Hoc Wireless Networking. Kluwer, Dordrecht (2003)
Navarro, G., Paredes, R.: Practical construction of metric t-spanners. In: Proc. 5th Workshop Algorithm Eng. Exp., pp. 69–81. SIAM Press, Philadelphia (2003)
Navarro, G., Paredes, R., Chávez, E.: t-spanners as a data structure for metric space searching. In: Laender, A.H.F., Oliveira, A.L. (eds.) SPIRE 2002. LNCS, vol. 2476, pp. 298–309. Springer, Heidelberg (2002)
Salowe, J.S.: Constructing multidimensional spanner graphs. Internat. J. Comput. Geom. 1, 99–107 (1991)
Smid, M.: Closest point problems in computational geometry. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 877–935. Elsevier Science Publishers, Amsterdam (2000)
Vaidya, P.M.: A sparse graph almost as good as the complete graph on points in K dimensions. Discrete Comput. Geom. 6, 369–381 (1991)
Yao, A.C.: On constructing minimum spanning trees in k-dimensional spaces and related problems. SIAM J. Comput. 11 (1992)
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Aronov, B., de Berg, M., Cheong, O., Gudmundsson, J., Haverkort, H., Vigneron, A. (2005). Sparse Geometric Graphs with Small Dilation. In: Deng, X., Du, DZ. (eds) Algorithms and Computation. ISAAC 2005. Lecture Notes in Computer Science, vol 3827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11602613_7
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DOI: https://doi.org/10.1007/11602613_7
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