Slowing down sorting networks to obtain faster sorting algorithms
Journal of the ACM (JACM)
The shortest watchtower and related problems for polyhedral terrains
Information Processing Letters
Visibility problems for polyhedral terrains
Journal of Symbolic Computation
Visibility and intersection problems in plane geometry
Discrete & Computational Geometry
Finding the upper envelope of n line segments in O(n log n) time
Information Processing Letters
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Computing the shortest watchtower of a polyhedral terrain in O(nlogn) time
Computational Geometry: Theory and Applications
Maintaining visibility of a polygon with a moving point of view
Information Processing Letters
Applying Parallel Computation Algorithms in the Design of Serial Algorithms
Journal of the ACM (JACM)
Computing all large sums-of-pairs in Rn and the discrete planar two-watchtower problem
Information Processing Letters
Improved approximation algorithms for geometric set cover
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
A constant-factor approximation algorithm for optimal terrain guarding
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Colored intersection searching via sparse rectangular matrix multiplication
Proceedings of the twenty-second annual symposium on Computational geometry
Computing visibility on terrains in external memory
Journal of Experimental Algorithmics (JEA)
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Given a polyhedral terrain T with n vertices, the two-watchtower problem for T calls for finding 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. 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, where the bases are constrained to lie at vertices of T, can be solved in O(n2 log4n) time, significantly improving previous solutions. The algorithm works, without increasing its asymptotic running time, even if, one of the towers is allowed to be placed anywhere on T. (2) We show that the continuous two-watchtower problem in R2, where the bases can lie anywhere on T, can be solved in O(n3α(n)log3n) time, again significantly improving previous results. (3) Still in R2, we show that the continuous version of the problem of guarding a finite set P ⊂ T of m points by two watchtowers of smallest height can be solved in O(mn log4n) time. (4) The discrete version of the two-watchtower problem in R3 can be solved in O(n11/3 polylog(n)) time; this is the first nontrivial result for this problem in R3.