Abstract
Geometric spanner is a fundamental structure in computational geometry and plays an important role in many geometric networks design applications. In this paper, we consider the following generalized geometric spanner problem under L 1 distance: Given a set of disjoint objects S, find a spanning network G with minimum size so that for any pair of points in different objects of S, there exists a path in G with length no more than t times their L 1 distance, where t is the stretch factor. Specifically, we focus on three types of objects: rectilinear segments, axis aligned rectangles, and rectilinear monotone polygons. By combining ideas of t-weekly dominating set, walls, aligned pairs and interval cover, we develop a 4-approximation algorithm (measured by the number of Steiner points) for each type of objects. Our algorithms run in near quadratic time, and can be easily implemented for practical applications.
The research of the first three authors was supported in part by National Science Foundation through CAREER award CCF-0546509 and grant IIS-0713489. The research of the last two authors was supported by the project New Horizons in Computing, Grant-in-Aid for Scientific Research on Priority Areas,MEXT Japan.
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Aronov, B., de Berg, M., Cheong, O., Gudmundsson, J., Haverkort, H., Smid, M., Vigneron, A.: Sparse Geometric Graphs with Small Dilation. In: Proceedings of the 12th Computing: The Australasian Theroy Symposium, vol. 51 (2006)
Arya, S., Das, G., Mount, D.M., Salowe, J.S., Smid, M.: Euclidean Spanners: Short, Thin, and Lanky. In: Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing (STOC 1995), pp. 489–498 (1995)
Arya, S., Mount, D.M., Smid, M.: Dynamic Algorithms for Geometric Spanners of Small Diameter: Randomized Rolutions. Technical Report, Max-Planck-Institut für Informatik (1994)
Arya, S., Mount, D.M., Smid, M.: Randomized and Deterministic Algorithms for Geometric Spanners of Small Diameter. In: 35th IEEE Symposium on Foundtions of Computer Science, pp. 703–712 (1994)
Asano, T., de Berg, M., Cheong, O., Everett, H., Haverkort, H., Katoh, N., Wolff, A.: Optimal Spanners for Axis-Aligned Rectangles. Comput. Geom. Theory Appl. 30(1), 59–77 (2005)
Chandra, B., Das, G., Narasimhan, G., Soares, J.: New Spareness Results on Graph Spanners. In: Proceedings of the Eighth Annual Symposium on Computational Geometry, pp. 192–201 (1992)
Clarkson, K.L.: Approximation Algorithms for Shortest Path Motion Planning. In: Proceedings of the nineteenth annual ACM conference on Theory of computing, pp. 56–65 (1987)
Das, G., Heffernan, P., Narasimhan, G.: Optimally Sparse Spanners in 3-Dimensional Euclidean Space. In: Proceedings of the Ninth Annual Symposium on Computational Geometry, pp. 53–62 (1993)
Das, G., Narasimhan, G.: A Fast Algorithm for Constructing Sparse Euclidean Spanners. In: Proceedings of the Tenth Annual Symposium on Computational Geometry, pp. 132–139 (1994)
Gudmundsson, J., Levcopoulos, C., Narasimhan, G.: Fast Greedy Algorithms for Constructing Sparse Geometric Spanners. SIAM Journal on Computing 31(5), 1479–1500 (2002)
Keil, J.M.: Approximating the Complete Euclidean Graph. In: Proceedings of 1st Scandinavian Workshop on Algorithm Theory, pp. 208–213 (1988)
Keil, J.M., Gutwin, C.A.: Classes of Graphs which Approximate the Complete Euclidean Graph. Discrete and Computational Geometry 7, 13–28 (1992)
Rupper, J., Seidel, R.: Approximating the d-dimensional Complete Euclidean Graph. In: Proceedings of 3rd Canadian Conference on Computational Geometry, pp. 207–210 (1991)
Yang, Y., Zhu, Y., Xu, J., Katoh, N.: Geometric Spanner of Segments. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 75–87. Springer, Heidelberg (2007)
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Zhu, Y., Xu, J., Yang, Y., Katoh, N., Tanigawa, Si. (2008). Geometric Spanner of Objects under L 1 Distance. In: Hu, X., Wang, J. (eds) Computing and Combinatorics. COCOON 2008. Lecture Notes in Computer Science, vol 5092. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69733-6_39
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DOI: https://doi.org/10.1007/978-3-540-69733-6_39
Publisher Name: Springer, Berlin, Heidelberg
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