Guarding a Terrain by Two Watchtowers
 Pankaj K. Agarwal,
 Sergey Bereg,
 Ovidiu Daescu,
 Haim Kaplan,
 Simeon Ntafos,
 Micha Sharir,
 Binhai Zhu
 … show all 7 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
Given a polyhedral terrain T with n vertices, the twowatchtower problem for T asks to find two vertical segments, called watchtowers, of smallest common height, whose bottom endpoints (bases) lie on T, and whose top endpoints guard T, in the sense that each point on T is visible from at least one of them. There are three versions of the problem, discrete, semicontinuous, and continuous, depending on whether two, one, or none of the two bases are restricted to be among the vertices of T, respectively.
In this paper we present the following results for the twowatchtower problem in ℝ^{2} and ℝ^{3}: (1) We show that the discrete twowatchtowers problem in ℝ^{2} can be solved in O(n ^{2}log ^{4} n) time, significantly improving previous solutions. The algorithm works, without increasing its asymptotic running time, for the semicontinuous version, where one of the towers is allowed to be placed anywhere on T. (2) We show that the continuous twowatchtower problem in ℝ^{2} can be solved in O(n ^{3} α(n)log ^{3} n) time, again significantly improving previous results. (3) Still in ℝ^{2}, we show that the continuous version of the problem of guarding a finite set P⊂T of m points by two watchtowers of smallest common height can be solved in O(mnlog ^{4} n) time. (4) We show that the discrete version of the twowatchtower problem in ℝ^{3} can be solved in O(n ^{11/3}polylog(n)) time; this is the first nontrivial result for this problem in ℝ^{3}.
 Aggarwal, A., Chazelle, B., Guibas, L., Ó’Dúnlaing, C., Yap, C. (1988) Parallel computational geometry. Algorithmica 3: pp. 293328 CrossRef
 BenMoshe, B., Carmi, P., Katz, M.J. (2004) Computing all large sumsofpairs in ℝ n and the discrete planar twowatchtower problem. Inf. Process. Lett. 89: pp. 137139 CrossRef
 BenMoshe, B., Katz, M.J., Mitchell, J.S.B. (2007) A constantfactor approximation algorithm for optimal terrain guarding. SIAM J. Comput. 36: pp. 16311647 CrossRef
 Bespamyatnikh, S., Chen, Z., Wang, K., Zhu, B.: On the planar twowatchtower problem. In: Proc. 7th Annu. Internat. Conf. Computing and Combinatorics, pp. 121–130 (2001)
 Chazelle, B., Guibas, L. (1989) Visibility and intersection problems in plane geometry. Discrete Comput. Geom. 4: pp. 551581 CrossRef
 Chen, D.Z., Daescu, O. (1998) Maintaining visibility of a polygon with a moving point of view. Inf. Process. Lett. 65: pp. 269275 CrossRef
 Clarkson, K., Varadarajan, K. (2007) Improved approximation algorithms for geometric set cover. Discrete Comput. Geom. 37: pp. 4358 CrossRef
 Cole, R. (1987) Slowing down sorting networks to obtain faster sorting algorithms. J. ACM 34: pp. 200208 CrossRef
 Cole, R., Sharir, M. (1989) Visibility problems for polyhedral terrains. J. Symb. Comput. 7: pp. 1130 CrossRef
 Eidenbenz, S.J., Stamm, C., Widmayer, P. (2001) Inapproximability results for guarding polygons and terrains. Algorithmica 31: pp. 79113 CrossRef
 Guibas, L., Hershberger, J., Leven, D., Sharir, M., Tarjan, R. (1987) Lineartime algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica 2: pp. 209233 CrossRef
 Gupta, P., Janardan, R., Smid, M. Computational geometry: Generalized intersection searching. In: Mehta, D.P., Sahni, S. eds. (2005) Handbook of Data Structures and Applications. CRC Press, Boca Raton, pp. 64.164.17
 Hershberger, J. (1989) Finding the upper envelope of n line segments in O(nlog n) time. Inf. Process. Lett. 33: pp. 169174 CrossRef
 Kaplan, H., Sharir, M., Verbin, E.: Colored intersection searching via sparse rectangular matrix multiplication. In: Proc. 22nd Annu. Sympos. Comput. Geom., pp. 52–60 (2006)
 Matoušek, J. (1993) Range searching with efficient hierarchical cuttings. Discrete Comput. Geom. 10: pp. 157182 CrossRef
 Megiddo, N. (1983) Applying parallel computation algorithms in the design of serial algorithms. J. ACM 30: pp. 852865 CrossRef
 Nilsson, B.: Guarding art galleries: Methods for mobile guards. Doctoral Thesis, Dept. Comput. Sci., Lund University (1994)
 Sharir, M. (1988) The shortest watchtower and related problems for polyhedral terrains. Inf. Process. Lett. 29: pp. 265270 CrossRef
 Sharir, M., Agarwal, P.K. (1995) Davenport Schinzel Sequences and Their Geometric Applications. Cambridge University Press, Cambridge
 Urrutia, J. Art gallery and illumination problems. In: Sack, J.R., Urrutia, J. eds. (2000) Handbook of Computational Geometry. Elsevier, Amsterdam, pp. 9731026 CrossRef
 Zhu, B. (1997) Computing the shortest watchtower of a polyhedral terrain in O(nlog n) time. Comput. Geom. Theory Appl. 8: pp. 181193
 Title
 Guarding a Terrain by Two Watchtowers
 Journal

Algorithmica
Volume 58, Issue 2 , pp 352390
 Cover Date
 20101001
 DOI
 10.1007/s0045300892703
 Print ISSN
 01784617
 Online ISSN
 14320541
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Computational geometry
 Visibility algorithms
 Terrain guarding
 Parametric search
 Industry Sectors
 Authors

 Pankaj K. Agarwal ^{(1)}
 Sergey Bereg ^{(2)}
 Ovidiu Daescu ^{(2)}
 Haim Kaplan ^{(3)}
 Simeon Ntafos ^{(2)}
 Micha Sharir ^{(3)} ^{(4)}
 Binhai Zhu ^{(5)}
 Author Affiliations

 1. Department of Computer Science, Duke University, Durham, NC, 27708, USA
 2. Department of Computer Science, University of Texas at Dallas, Box 830688, Richardson, TX, 75083, USA
 3. School of Computer Science, Tel Aviv University, Tel Aviv, 69978, Israel
 4. Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA
 5. Department of Computer Science, Montana State University, Bozeman, MT, 597173880, USA