Guarding a Terrain by Two Watchtowers Pankaj K. Agarwal Sergey Bereg Ovidiu Daescu Email author Haim Kaplan Simeon Ntafos Micha Sharir Binhai Zhu Article

First Online: 07 January 2009 Received: 19 June 2006 Accepted: 17 December 2008 DOI :
10.1007/s00453-008-9270-3

Cite this article as: Agarwal, P.K., Bereg, S., Daescu, O. et al. Algorithmica (2010) 58: 352. doi:10.1007/s00453-008-9270-3 Abstract Given a polyhedral terrain T with n vertices, the two-watchtower 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 , semi-continuous , 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 two-watchtower problem in ℝ^{2} and ℝ^{3} : (1) We show that the discrete two-watchtowers 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 semi-continuous version, where one of the towers is allowed to be placed anywhere on T . (2) We show that the continuous two-watchtower 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 (mn log ^{4} n ) time. (4) We show that the discrete version of the two-watchtower 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 P.K. Agarwal was supported by NSF under grants CCR-00-86013, EIA-01-31905, CCR-02-04118, and DEB-04-25465, by ARO grants W911NF-04-1-0278 and DAAD19-03-1-0352, and by a grant from the US–Israel Binational Science Foundation.

O. Daescu was supported by NSF Grants CCF-0430366 and CCF-0635013.

M. Sharir was supported by NSF Grant CCR-00-98246, by a grant from the US–Israeli Binational Science Foundation (jointly with P.K. Agarwal), by a grant from the Israeli Academy of Sciences for a Center of Excellence in Geometric Computing at Tel Aviv University, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University.

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Authors and Affiliations Pankaj K. Agarwal Sergey Bereg Ovidiu Daescu Email author Haim Kaplan Simeon Ntafos Micha Sharir Binhai Zhu 1. Department of Computer Science Duke University Durham USA 2. Department of Computer Science University of Texas at Dallas Richardson USA 3. School of Computer Science Tel Aviv University Tel Aviv Israel 4. Courant Institute of Mathematical Sciences New York University New York USA 5. Department of Computer Science Montana State University Bozeman USA