, Volume 58, Issue 2, pp 352390
First online:
Guarding a Terrain by Two Watchtowers
 Pankaj K. AgarwalAffiliated withDepartment of Computer Science, Duke University
 , Sergey BeregAffiliated withDepartment of Computer Science, University of Texas at Dallas
 , Ovidiu DaescuAffiliated withDepartment of Computer Science, University of Texas at Dallas Email author
 , Haim KaplanAffiliated withSchool of Computer Science, Tel Aviv University
 , Simeon NtafosAffiliated withDepartment of Computer Science, University of Texas at Dallas
 , Micha SharirAffiliated withSchool of Computer Science, Tel Aviv UniversityCourant Institute of Mathematical Sciences, New York University
 , Binhai ZhuAffiliated withDepartment of Computer Science, Montana State University
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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}.
Keywords
Computational geometry Visibility algorithms Terrain guarding Parametric search Title
 Guarding a Terrain by Two Watchtowers
 Journal

Algorithmica
Volume 58, Issue 2 , pp 352390
 Cover Date
 201010
 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