# Optimally computing the shortest weakly visible subedge of a simple polygon preliminary version

- First Online:

## Abstract

Given an *n*-vertex simple polygon *P*, the problem of computing the shortest weakly visible subedge of *P* is that of finding a shortest line segment *s* on the boundary of *P* such that *P* is weakly visible from *s* (if *s* exists). In this paper, we present new geometric observations that are useful for solving this problem. Based on these geometric observations, we obtain optimal sequential and parallel algorithms for solving this problem. Our sequential algorithm runs in *O(n)* time, and our parallel algorithm runs in *O*(log *n*) time using *O(n*/log *n*) processors in the CREW PRAM computational model. Using the previously best known sequential algorithms to solve this problem would take *O(n*^{2}) time. We also give geometric observations that lead to extremely simple and optimal algorithms for solving, both sequentially and in parallel, the case of this problem where the polygons are rectilinear.

## Preview

Unable to display preview. Download preview PDF.

### References

- [1]D. Avis and G. T. Toussaint. “An optimal algorithm for determining the visibility polygon from an edge,”
*IEEE Trans. Comput*, C-30 (12) (1981), pp. 910–914.Google Scholar - [2]B. K. Bhattacharya, D. G. Kirkpatrick, and G. T. Toussaint. “Determining sector visibility of a polygon,”
*Proc. 5-th Annual ACM Symp. Computational Geometry*, 1989, pp. 247–254.Google Scholar - [3]B. K. Bhattacharya, A. Mukhopadhyay, and G. T. Toussaint. “A linear time algorithm for computing the shortest line segment from which a polygon is weakly externally visible,”
*Proc. Workshop on Algorithms and Data Structures (WADS'91)*, 1991, Ottawa, Canada, pp. 412–424.Google Scholar - [4]B. Chazelle and L. J. Guibas. “Visibility and intersection problems in plane geometry,”
*Discrete and Computational Geometry*, 4 (1989), pp. 551–581.Google Scholar - [5]D. Z. Chen. “An optimal parallel algorithm for detecting weak visibility of a simple polygon,”
*Proc. of the Eighth Annual ACM Symp. on Computational Geometry*, 1992, pp. 63–72.Google Scholar - [6]D. Z. Chen. “Parallel techniques for paths, visibility, and related problems,” Ph.D. thesis, Technical Report No. 92-051, Dept. of Computer Sciences, Purdue University, July 1992.Google Scholar
- [7]Y. T. Ching, M. T. Ko, and H. Y. Tu. “On the cruising guard problems,” Technical Report, 1989, Institute of Information Science, Academia Sinica, Taipei, Taiwan.Google Scholar
- [8]J. I. Doh and K. Y. Chwa. “An algorithm for determining visibility of a simple polygon from an internal line segment,”
*Journal of Algorithms*, 14 (1993), pp. 139–168.CrossRefGoogle Scholar - [9]H. ElGindy. “Hierarchical decomposition of polygon with applications,” Ph.D. thesis, McGill University, 1985.Google Scholar
- [10]S. K. Ghosh, A. Maheshwari, S. P. Pal, S. Saluja, and C. E. V. Madhavan. “Characterizing weak visibility polygons and related problems,” Technical Report No. IISc-CSA-90-1, 1990, Dept. Computer Science and Automation, Indian Institute of Science.Google Scholar
- [11]M. T. Goodrich, S. B. Shauck, and S. Guha. “Parallel methods for visibility and shortest path problems in simple polygons (Preliminary version),”
*Proc. 6-th Annual ACM Symp. Computational Geometry*, 1990, pp. 73–82.Google Scholar - [12]L. J. Guibas, J. Hershberger, D. Leven, M. Sharir, and R. E. Tarjan. “Linear time algorithms for visibility and shortest paths problems inside triangulated simple polygons,”
*Algorithmica*, 2 (1987), pp. 209–233.CrossRefGoogle Scholar - [13]P. J. Heffernan and J. S. B. Mitchell. “Structured visibility profiles with applications to problems in simple polygons,”
*Proc. 6-th Annual ACM Symp. Computational Geometry*, 1990, pp. 53–62.Google Scholar - [14]J. Hershberger. “Optimal parallel algorithms for triangulated simple polygons,”
*Proc. 8-th Annual ACM Symp. Computational Geometry*, 1992, pp. 33–42.Google Scholar - [15]Y. Ke. “Detecting the weak visibility of a simple polygon and related problems,” manuscript, Dept. of Computer Science, The Johns Hopkins University, 1988.Google Scholar
- [16]C. P. Kruskal, L. Rudolph, and M. Snir. “The power of parallel prefix,”
*IEEE Trans. Comput.*, C-34 (1985), pp. 965–968.Google Scholar - [17]R. E. Ladner and M. J. Fischer. “Parallel prefix computation,”
*Journal of the ACM*, 27 (4) (1980), pp. 831–838.CrossRefGoogle Scholar - [18]D. T. Lee and A. K. Lin. “Computing the visibility polygon from an edge,”
*Computer Vision, Graphics, and Image Processing*, 34 (1986), pp. 1–19.Google Scholar - [19]S. H. Lee and K. Y. Chwa. “Some chain visibility problems in a simple polygon,”
*Algorithmica*, 5 (1990), pp. 485–507.CrossRefGoogle Scholar - [20]J.-R. Sack and S. Suri. “An optimal algorithm for detecting weak visibility of a polygon,”
*IEEE Trans. Comput*, C-39 (10) (1990), pp. 1213–1219.CrossRefGoogle Scholar - [21]S. Y. Shin. “Visibility in the plane and its related problems,” Ph.D. thesis, University of Michigan, 1986.Google Scholar
- [22]G. T. Toussaint. “A linear-time algorithm for solving the strong hidden-line problem in a simple polygon,”
*Pattern Recognition letters*, 4 (1986), pp. 449–451.CrossRefGoogle Scholar - [23]G. T. Toussaint and D. Avis. “On a convex hull algorithm for polygons and its applications to triangulation problems,”
*Pattern Recognition*, 15 (1) (1982), pp. 23–29.CrossRefGoogle Scholar