Abstract
We present a new algorithm for finding minimum-link rectilinear paths among h rectilinear obstacles with a total of n vertices in the plane. After the plane is triangulated, for any point s, our algorithm builds an O(n)-size data structure in \(O(n+h\log h)\) time, such that given any query point t, we can compute a minimum-link rectilinear path from s to t in \(O(\log n+k)\) time, where k is the number of edges of the path, and the query time is \(O(\log n)\) if we only want to know the value k. The previously best algorithm solves the problem in \(O(n\log n)\) time.
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References
Bar-Yehuda, R., Chazelle, B.: Triangulating disjoint Jordan chains. International Journal of Computational Geometry and Applications 4(4), 475–481 (1994)
de Berg, M.: On rectilinear link distance. Computational Geometry: Theory and Applications 1, 13–34 (1991)
Chazelle, B.: Triangulating a simple polygon in linear time. Discrete and Computational Geometry 6, 485–524 (1991)
Chen, D., Inkulu, R., Wang, H.: Two-point \(L_1\) shortest path queries in the plane. In: Proc. of the 30th Annual Symposium on Computational Geometry (SoCG), pp. 406–415 (2014)
Chen, Danny Z., Wang, Haitao: A Nearly Optimal Algorithm for Finding L \(_\text{1 }\) Shortest Paths among Polygonal Obstacles in the Plane. In: Demetrescu, Camil, Halldórsson, Magnús M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 481–492. Springer, Heidelberg (2011)
Das, G., Narasimhan, G.: Geometric searching and link distance. In: Proc. of the 2nd Workshop of Algorithms and Data Structures (WADS), pp. 261–272 (1991)
Edelsbrunner, H., Guibas, L., Stolfi, J.: Optimal point location in a monotone subdivision. SIAM Journal on Computing 15(2), 317–340 (1986)
Ghosh, S.: Computing the visibility polygon from a convex set and related problems. Journal of Algorithms 12, 75–95 (1991)
Hershberger, J., Snoeyink, J.: Computing minimum length paths of a given homotopy class. Computational Geometry: Theory and Applications 4, 63–97 (1994)
Imai, H., Asano, T.: Efficient algorithms for geometric graph search problems. SIAM Journal on Computing 15(2), 478–494 (1986)
Kapoor, S., Maheshwari, S., Mitchell, J.: An efficient algorithm for Euclidean shortest paths among polygonal obstacles in the plane. Discrete and Computational Geometry 18(4), 377–383 (1997)
Lingas, A., Maheshwari, A., Sack, J.R.: Parallel algorithms for rectilinear link distance problems. Algorithmica 14, 261–289 (1995)
Maheshwari, A., Sack, J.R., Djidjev, H.: Link distance problems. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 519–558. Elsevier, Amsterdam (2000)
Mitchell, J., Polishchuk, V., Sysikaski, M.: Minimum-link paths revisited. Computational Geometry: Theory and Applications 47, 651–667 (2014)
Mitchell, J., Polishchuk, V., Sysikaski, M., Wang, H.: An optimal algorithm for minimum-link rectilinear paths in triangulated rectilinear domains (2015). ar**v:1504.06842
Mitchell, J., Rote, G., Woeginger, G.: Minimum-link paths among obstacles in the plane. Algorithmica 8, 431–459 (1992)
Mitchell, J., Suri, S.: Separation and approximation of polyhedral objects. Computational Geometry: Theory and Applications 5, 95–114 (1995)
Sato, M., Sakanaka, J., Ohtsuki, T.: A fast line-search method based on a tile plane. In: Proc. of the IEEE International Symposium on Circuits and Systems, pp. 588–597 (1987)
Schuierer, S.: An optimal data structure for shortest rectilinear path queries in a simple rectilinear polygon. International Journal of Compututational Geometry and Applications 6, 205–226 (1996)
Suri, S.: A linear time algorithm with minimum link paths inside a simple polygon. Computer Vision, Graphics, and Image Processing 35(1), 99–110 (1986)
Suri, S.: Minimum link paths in polygons and related problems. Ph.D. thesis, Johns Hopkins University, Baltimore, MD (1987)
Suri, S.: On some link distance problems in a simple polygon. IEEE Transactions on Robotics and Automation 6, 108–113 (1990)
Acknowledgments
J. Mitchell is partially supported by grants from Sandia National Labs, the National Science Foundation (CCF-1018388), and the US-Israel Binational Science Foundation (award 2010074). V. Polishchuk is supported by grant 2014-03476 from the Sweden’s innovation agency VINNOVA. H. Wang is supported in part by NSF under Grant CCF-1317143.
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Mitchell, J.S.B., Polishchuk, V., Sysikaski, M., Wang, H. (2015). An Optimal Algorithm for Minimum-Link Rectilinear Paths in Triangulated Rectilinear Domains. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47672-7_77
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