, Volume 58, Issue 2, pp 352-390
Date: 07 Jan 2009

Guarding a Terrain by Two Watchtowers

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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 ℝ2 and ℝ3: (1) We show that the discrete two-watchtowers problem in ℝ2 can be solved in O(n 2log 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 ℝ2 can be solved in O(n 3 α(n)log 3 n) time, again significantly improving previous results. (3) Still in ℝ2, we show that the continuous version of the problem of guarding a finite set PT 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 two-watchtower problem in ℝ3 can be solved in O(n 11/3polylog(n)) time; this is the first nontrivial result for this problem in ℝ3.

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.