Abstract
In this paper, we present an experimental study comparing two algorithms, topological peeling and topological walk, for traversing arrangements of planar lines. Given a set H of n lines and a convex region R on a plane, both topological peeling and topological walk sweep the portion A R of the arrangement of H inside R in O(K + n log(n + r)) time and O(n + r) space, where K is the number of cells of A R and r is the number of boundary vertices of R. In our study, we robustly implemented these two algorithms using the LEDA library. Based on the implementation, we carried out experiments to conduct several comparisons, such as the arrangement traversal fashions, memory consumption, and execution time. In general, topological peeling exhibits a better control on the propagation of its sweeping curve (called the wavefront). For memory consumption, two types of measures, logical and physical memory, were examined. Our experiments showed that although both algorithms use nearly the same amount of logical memory, topological peeling could use twice as much physical memory as topological walk. For execution time, experiments revealed an interesting phenomenon that topological peeling has a 10% to 25% faster execution time than topological walk in most cases. Our analysis of this phenomenon indicates that the execution times of topological peeling and topological walk are both sensitive to the ratio of the lower input lines to all input lines. When the ratio of the lower lines to all input lines is around 85%, the two algorithms have roughly the same amount of execution time. Under this ratio, topological peeling considerably outperforms topological walk; above this ratio, topological walk slightly outperforms topological peeling.
The research was supported in part by the National Science Foundation under Grant CCR-9988468.
The research was supported in part by a faculty start-up fund from the CSE dept., SUNY at Buffalo, and an IBM faculty partnership award.
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References
E.G. Anagnostou, L.J. Guibas, and V.G. Polimenis, “Topological sweeping in three dimensions,” Lecture Notes in Computer Science, Vol. 450, Proc. Int’l Symp. on Algorithms, Springer-Verlag, 1990, pp. 310–317.
E.M. Arkin, Y.-J. Chiang, M. Held, J.S.B. Mitchell, V. Sacristan, S.S. Skiena, and T.-C. Yang, “On minimum-area hulls,” Algorithmica, Vol. 21, 1998, pp. 119–136.
T. Asano, L.J. Guibas, and T. Tokuyama, “Walking in an arrangement topologi-cally,” International Journal of Computational Geometry and Applications, Vol. 4, No. 2, 1994, pp. 123–151.
T. Asano and T. Tokuyama, “Topological walk revisited,” Proc. 6th Canadian Conf. on Computational Geometry, 1994, pp. 1–6.
P. Bose, W. Evans, D. Kirkpatrick, M. McAllister, and J. Snoeyink, “Approximating shortest paths in arrangements of lines,” Proc. 8th Canadian Conf. on Computational Geometry, 1996, pp. 143–148.
B. Chazelle, L.J. Guibas, and D.T. Lee, “The power of geometric duality,” BIT, Vol. 25, 1985, pp. 76–90.
D.Z. Chen, O. Daescu, X.S. Hu, X. Wu, and J. Xu, “Determining an optimal penetration among weighted regions in two and three dimensions,” Proc. 15th Annual ACM Symp. on Computational Geometry, 1999, pp. 322–331.
D.Z. Chen, O. Daescu, X.S. Hu, and J. Xu, “Finding an optimal path without growing the tree,” Proc. 6th Annual European Symp. on Algorithms, 1998, pp. 356–367.
D.Z. Chen, S. Luan, and J. Xu, “Topological peeling and implementation,” 12th Annual Int. Symp. on Algorithms and Computation, 2001, pp. 454–466.
D. Dobkin and A. Tal, “Efficient and Small Representation of Line Arrangements with Applications,” Proc. 17th Annual ACM symposium on Computational Geometry, 2001, pp. 293–301.
H. Edelsbrunner, Algorithms in Combinatorial Geometry, Springer-Verlag, New York, 1987.
H. Edelsbrunner and L.J. Guibas, “Topologically sweeping an arrangement,” Journal of Computer and System Sciences, Vol. 38, 1989, pp. 165–194.
H. Edelsbrunner, J. O’Rourke, and R. Seidel, “Constructing arrangements of lines and hyperplanes with applications,” SIAM J. Computing, Vol. 15, 1986, pp. 341–363.
H. Edelsbrunner and D. Souvaine, “Computing median-of-squares regression lines and guided topological sweep,” Journal of the American Statistical Association, Vol. 85, 1990, pp. 115–119.
H. Edelsbrunner and E. Welzl, “Constructing belts in two-dimensional arrangements with applications,” SIAM J. Computing, Vol. 15, 1986, pp. 271–284.
D. Eppstein and D. Hart, “An efficient algorithm for shortest paths in vertical and horizontal segments,” Proc. 5th Int. Workshop on Algorithms and Data Structures, 1997, pp. 234–247.
D. Eppstein and D. Hart, “Shortest paths in an arrangement with k line orientations,” Proc. 10th ACM-SIAM Symp. on Discrete Algorithms, 1999, pp. 310–316.
K. Hoffman, K. Mehlhorn, R. Rosenstiehl, and R. Tarjan, “Sorting Jordan sequences in linear time using level-linked search trees,” Information and Control, Vol. 68, 1986, pp. 170–184.
P.N. Klein, S. Rao, M.H. Rauch, and S. Subramanian, “Faster shortest-path algorithms for planar graphs,” Proc. 26th Annual ACM Symp. Theory of Computing, 1994, pp. 27–37.
J. Majhi, R. Janardan, M. Smid, and P. Gupta, “On some geometric optimization problems in layered manufacturing,” Proc. 5th Int. Workshop on Algorithms and Data Structures, 1997, pp. 136–149.
K. Miller, S. Ramaswami, P. Rousseeuw, T. Sellares, D. Souvaine, I. Streinu, and A. Struyf, “Fast implementation of depth contours using topological sweep,” Proc. 12th ACM-SIAM Symp. on Discrete Algorithms, 2001, pp. 690–699.
J. Nievergelt and F.P. Preparata, “Plane-sweep algorithms for intersecting geometric figures,” Comm. of the ACM, Vol. 25, No. 10, 1982, pp. 739–747.
F.P. Preparata and M.I. Shamos, Computational Geometry: An Introduction, Springer-Verlag, New York, 1985.
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Chen, D.Z., Luan, S., Xu, J. (2002). An Experimental Study and Comparison of Topological Peeling and Topological Walk. 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_49
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DOI: https://doi.org/10.1007/3-540-45655-4_49
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