# The Euclidean bottleneck Steiner path problem

A. Karim Abu-Affash, Paz Carmi, Matthew J. Katz, Michael Segal

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

4 Scopus citations

## Abstract

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. [11], who gave an O(n2 log 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 + ∈)-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. B i) 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((n/∈)3n log n). Finally, for the complete graph with vertex set P and weight function w(p, q) = b|pq|c, we construct a (1 + ")-spanner of size O(n/∈4) in time O((1/∈4) n log2 n), even though w is not a metric.

Original language English Proceedings of the 27th Annual Symposium on Computational Geometry, SCG'11 440-447 8 https://doi.org/10.1145/1998196.1998268 Published - 15 Jul 2011 27th Annual ACM Symposium on Computational Geometry, SCG'11 - Paris, FranceDuration: 13 Jun 2011 → 15 Jun 2011

### Conference

Conference 27th Annual ACM Symposium on Computational Geometry, SCG'11 France Paris 13/06/11 → 15/06/11

## Keywords

• Bottleneck path
• Geometric networks
• Geometric optimization
• Geometric spanners
• Pair decomposition
• Steiner points

## ASJC Scopus subject areas

• Theoretical Computer Science
• Geometry and Topology
• Computational Mathematics

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