Approximation algorithms for shortest path motion planning
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Approximating the complete Euclidean graph
No. 318 on SWAT 88: 1st Scandinavian workshop on algorithm theory
Steiner tree problem with minimum number of Steiner points and bounded edge-length
Information Processing Letters
An approximation algorithm for a bottleneck k-Steiner tree problem in the Euclidean plane
Information Processing Letters
Approximations for Steiner Trees with Minimum Number of Steiner Points
Journal of Global Optimization
A Divide-and-Conquer Algorithm for Min-Cost Perfect Matching in the Plane
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Worst and Best-Case Coverage in Sensor Networks
IEEE Transactions on Mobile Computing
Dynamic coverage in ad-hoc sensor networks
Mobile Networks and Applications
Approximation algorithm for bottleneck steiner tree problem in the euclidean plane
Journal of Computer Science and Technology
Geometric Spanner Networks
European Journal of Combinatorics
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Relay sensor placement in wireless sensor networks
Wireless Networks
Region-Fault Tolerant Geometric Spanners
Discrete & Computational Geometry
Introduction to Algorithms, Third Edition
Introduction to Algorithms, Third Edition
On exact solutions to the Euclidean bottleneck Steiner tree problem
Information Processing Letters
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We consider a geometric optimization problem that arises in network design. Given a set P of n points in the plane, source and destination points s,t ∈ P, and an integer k 0, one has to locate k Steiner points, such that the length of the longest edge of a bottleneck path between s and t is minimized. In this paper, we present an O(n log2 n)-time algorithm that computes an optimal solution, for any constant k. This problem was previously studied by Hou et al. [Hou10], who gave an O(n2log n)-time algorithm. We also study the dual version of the problem, where a value λ 0 is given (instead of k), and the goal is to locate as few Steiner points as possible, so that the length of the longest edge of a bottleneck path between s and t is at most λ. Our algorithms are based on two new geometric structures that we develop --- an (α,β)-pair decomposition of P and a floor (1+ε)-spanner of P. For real numbers β α 0, an (α,β)-pair decomposition of P is a collection W={(A1,B1),...,(Am,Bm)} of pairs of subsets of P, satisfying: (i) For each pair (Ai,Bi) ∈ W, the radius of the minimum enclosing circle of Ai (resp. Bi) is at most α, and (ii) For any p,q ∈ P, such that |pq| ≤ β, there exists a single pair (Ai,Bi) ∈ W, such that p ∈ Ai and q ∈ Bi, or vice versa. We construct (a compact representation of) an (α,β)-pair decomposition of P in time O((β/α)3 n log n). Finally, for the complete graph with vertex set P and weight function w(p,q) = ⌊|pq|⌋, we construct a (1+ε)-spanner of size O(n/ε4) in time O((1/ε4)n log2 n), even though w is not a metric.