Document

# Algorithms for Subpath Convex Hull Queries and Ray-Shooting Among Segments

## File

LIPIcs.SoCG.2020.69.pdf
• Filesize: 492 kB
• 14 pages

## Cite As

Haitao Wang. Algorithms for Subpath Convex Hull Queries and Ray-Shooting Among Segments. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 69:1-69:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.SoCG.2020.69

## Abstract

In this paper, we first consider the subpath convex hull query problem: Given a simple path π of n vertices, preprocess it so that the convex hull of any query subpath of π can be quickly obtained. Previously, Guibas, Hershberger, and Snoeyink [SODA 90'] proposed a data structure of O(n) space and O(log n log log n) query time; reducing the query time to O(log n) increases the space to O(nlog log n). We present an improved result that uses O(n) space while achieving O(log n) query time. Like the previous work, our query algorithm returns a compact interval tree representing the convex hull so that standard binary-search-based queries on the hull can be performed in O(log n) time each. Our new result leads to improvements for several other problems. In particular, with the help of the above result, we present new algorithms for the ray-shooting problem among segments. Given a set of n (possibly intersecting) line segments in the plane, preprocess it so that the first segment hit by a query ray can be quickly found. We give a data structure of O(n log n) space that can answer each query in (√n log n) time. If the segments are nonintersecting or if the segments are lines, then the space can be reduced to O(n). All these are classical problems that have been studied extensively. Previously data structures of Õ(√n) query time were known in early 1990s; nearly no progress has been made for over two decades. For all problems, our results provide improvements by reducing the space of the data structures by at least a logarithmic factor while the preprocessing and query times are the same as before or even better.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
• Theory of computation → Computational geometry
##### Keywords
• subpath hull queries
• convex hulls
• compact interval trees
• ray-shooting
• data structures

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. P.K. Agarwal. Ray shooting and other applications of spanning trees with low stabbing number. SIAM Journal on Computing, 21:540-570, 1992.
2. P.K. Agarwal and M. Sharir. Applications of a new space-partitioning technique. Discrete and Computational Geometry, 9(1):11-38, 1993.
3. R. Bar-Yehuda and S. Fogel. Variations on ray shootings. Algorithmica, 11:133-145, 1994.
4. B. Becker, P.G. Franciosa, S. Gschwind, S. Leonardi, T. Ohler, and P. Widmayer. Enclosing a set of objects by two minimum area rectangles. Journal of Algorithms, 21:520-541, 1996.
5. B. Becker, P.G. Franciosa, S. Gschwind, T. Ohler, T. Ohler, G. Thiemt, and P. Widmayer. An optimal algorithm for approximating a set of rectangles by two minimum area rectangles. In Workshop on Computational Geometry, pages 13-25, 1991.
6. T.M. Chan. Optimal partition trees. Discrete and Computational Geometry, 47:661-690, 2012.
7. B. Chazelle and L. Guibas. Fractional cascading: II. Applications. Algorithmica, 1(1):163-191, 1986.
8. B. Chazelle and E. Welzl. Quasi-optimal range searching in spaces of finite VC-dimension. Discrete and Computational Geometry, 4(5):467-489, 1989.
9. D.Z. Chen, J. Li, and H. Wang. Efficient algorithms for the one-dimensional k-center problem. Theoretical Computer Science, 592:135-142, 2015.
10. D.Z. Chen and H. Wang. Approximating points by a piecewise linear function. Algorithmica, 88:682-713, 2013.
11. S.W. Cheng and R. Janardan. Algorithms for ray-shooting and intersection searching. Journal of Algorithms, 13:670-692, 1992.
12. T. Christ, M. Hoffmann, Y. Okamoto, and T. Uno. Improved bounds for wireless localization. Algorithmica, 57:499-516, 2010.
13. B.-S. Dai, M.-J. Kao, and D.T. Lee. Optimal time-convex hull under the L_p metrics. In Proceedings of the 13rd Algorithms and Data Structures Symposium (WADS), pages 268-279, 2013.
14. J. Driscoll, N. Sarnak, D. Sleator, and R.E. Tarjan. Making data structures persistent. Journal of Computer and System Sciences, 38(1):86-124, 1989.
15. H. Edelsbrunner, L. Guibas, J. Hershberger, R. Seidel, M. Sharir, J. Snoeyink, and E. Welzl. Implicitly representing arrangements of lines or segments. Discrete and Computational Geometry, 4:433-466, 1989.
16. H. Edelsbrunner and E. Welzl. Halfplanar range search in linear space and O(n^0.695) query time. Information Processing Letters, 23:289-293, 1986.
17. R.L. Graham and F.F. Yao. Finding the convex hull of a simple polygon. Journal of Algorithms, 4:324-331, 1983.
18. L. Guibas, J. Hershberger, and J. Snoeyink. Compact interval trees: A data structure for convex hulls. International Journal of Computational Geometry and Applications, 1(1):1-22, 1991. First appeared in SODA 1990.
19. L. Guibas, M. Overmars, and M. Sharir. Intersecting line segments, ray shooting, and other applications of geometric partitioning techniques. In Proceedings of the 1st Scandinavian Workshop on Algorithm Theory (SWAT), pages 64-73, 1988.
20. J. Matoušek. Efficient partition trees. Discrete and Computational Geometry, 8(3):315-334, 1992.
21. J. Matoušek. Range searching with efficient hierarchical cuttings. Discrete and Computational Geometry, 10(1):157-182, 1993.
22. A. Melkman. On-line construction of the convex hull of a simple polygon. Information Processing Letters, 25:11-12, 1987.
23. M.H. Overmars, H. Schipper, and M. Sharir. Storing line segments in partition trees. BIT Numerical Mathematics, 30:385-403, 1990.
24. H. Wagener. Optimal parallel hull construction for simple polygons in O(log log n) time. In Proceedings of the 33rd Annual Symposium on Foundations of Computer Science (FOCS), pages 593-599, 1992.
25. H. Wang and W. Zhang. On top-k weighted sum aggregate nearest and farthest neighbors in the L₁ plane. International Journal of Computational Geometry and Applications, 29:189-218, 2019.
X

Feedback for Dagstuhl Publishing