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Computing $k$-Crossing Visibility through $k$-levels (2312.02827v2)

Published 5 Dec 2023 in cs.CG

Abstract: Let $\mathcal{A}$ be a set of straight lines in the plane (or planes in $\mathbb{R}3$). The $k$-crossing visibility of a point $p$ on $\mathcal{A}$ is the set $Q$ of points in the elements of $\mathcal{A}$ such that the segment $pq$, where $q\in Q$, intersects at most $k$ elements of $\mathcal{A}$. In this paper, we present algorithms for computing the $k$-crossing visibility. Specifically, we provide $O(n\log n + kn)$ and $O(n\log n + k2n)$ time algorithms for sets of $n$ lines in the plane and arrangements of $n$ planes in $\mathbb{R}3$, which are optimal for $k=\Omega(\log n)$ and $k=\Omega(\sqrt{\log n})$, respectively. We also introduce an algorithm for computing $k$-crossing visibilities on polygons, which achieves the same asymptotic time complexity as the one presented by Bahoo et al. The techniques proposed in this paper can be easily adapted for computing $k$-crossing visibilities on other instances where the $(\leq k)$-level is known.

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References (24)
  1. Modem illumination of monotone polygons. Computational Geometry, 68:101–118, 2018.
  2. The number of small semispaces of a finite set of points in the plane. Journal of Combinatorial Theory, Series A, 41(1):154–157, 1986.
  3. A randomized algorithm for triangulating a simple polygon in linear time. Discrete & Computational Geometry, 26(2):245–265, 2001.
  4. Computing the k𝑘kitalic_k-crossing visibility region of a point in a polygon. In Charles J. Colbourn, Roberto Grossi, and Nadia Pisanti, editors, Combinatorial Algorithms, pages 10–21, Cham, 2019. Springer International Publishing.
  5. A hybrid metaheuristic strategy for covering with wireless devices. J. Univers. Comput. Sci., 18(14):1906–1932, 2012.
  6. Visibility-based planning of sensor control strategies. Algorithmica, 26(3):364–388, 2000.
  7. Optimal deterministic algorithms for 2-d and 3-d shallow cuttings. Discrete Comput. Geom., 56(4):866–881, 2016.
  8. Bernard Chazelle. Triangulating a simple polygon in linear time. Discrete & Computational Geometry, 6(3):485–524, 1991.
  9. Applications of random sampling in computational geometry, ii. Discrete & Computational Geometry, 4(5):387–421, 1989.
  10. Radiosity and realistic image synthesis. Morgan Kaufmann, 1993.
  11. Recognizing polygons, or how to spy. The Visual Computer, 3(6):344–355, 1988.
  12. Susan E Dorward. A survey of object-space hidden surface removal. International Journal of Computational Geometry & Applications, 4(03):325–362, 1994.
  13. An optimal algorithm for the (≤kabsent𝑘\leq k≤ italic_k)-levels, with applications to separation and transversal problems. International Journal of Computational Geometry Applications, 06(03):247–261, 1996.
  14. Three-dimensional computer vision: a geometric viewpoint. MIT press, 1993.
  15. Subir Kumar Ghosh. Visibility algorithms in the plane. Cambridge university press, 2007.
  16. Efficiently computing and representing aspect graphs of polyhedral objects. IEEE Trans. Pattern Anal. Mach. Intell., 13(6):542–551, jun 1991.
  17. Jean-Claude Latombe. Robot motion planning, volume 124. Springer Science & Business Media, 2012.
  18. An algorithm for planning collision-free paths among polyhedral obstacles. Communications of the ACM, 22(10):560–570, 1979.
  19. The superman problem. The Visual Computer, 10(8):459–473, 1994.
  20. Joseph O’Rourke et al. Art gallery theorems and algorithms, volume 57. Oxford University Press Oxford, 1987.
  21. M Peshkin and A Sanderson. Reachable grasps on a polygon: the convex rope algorithm. IEEE Journal on Robotics and Automation, 2(1):53–58, 1986.
  22. Thomas C Shermer. Recent results in art galleries (geometry). Proceedings of the IEEE, 80(9):1384–1399, 1992.
  23. Handbook of discrete and computational geometry. CRC press, 2017.
  24. Godfried T Toussaint. Shortest path solves edge-to-edge visibility in a polygon. Pattern Recognition Letters, 4(3):165–170, 1986.

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