Connecting a Set of Circles with Minimum Sum of Radii

Chambers, Erin Wolf; Fekete, Sándor P.; Hoffmann, Hella-Franziska; Marinakis, Dimitri; Mitchell, Joseph S. B.; Srinivasan, Venkatesh; Stege, Ulrike; Whitesides, Sue

doi:10.1007/978-3-642-22300-6_16

Cited by 13 publications

(9 citation statements)

References 8 publications

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“…Does there exist a PTAS for the general case of CRA with bounded radii, a problem we have shown to be NP-hard? In the conference version of the paper [12], we sketched an approach for how to modify our PTAS for unbounded radii to address a special case of the CRA problem with bounded radii, if an additional assumption is made, that for any segment pq, with p and q within feasible disks, there exists a (connected) path of feasible disks whose sum of radii is O(|pq|). It would be interesting to obtain a PTAS for the CRA for the general case of bounded radii.…”

Section: Discussionmentioning

confidence: 99%

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Connecting a set of circles with minimum sum of radii

Chambers

Fekete

Hoffmann

et al. 2018

Computational Geometry

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We consider the problem of assigning radii to a given set of points in the plane, such that the resulting set of disks is connected, and the sum of radii is minimized. We prove that the problem is NP-hard in planar weighted graphs if there are upper bounds on the radii and sketch a similar proof for planar point sets. For the case when there are no upper bounds on the radii, the complexity is open; we give a polynomial-time approximation scheme. We also give constant-factor approximation guarantees for solutions with a bounded number of disks; these results are supported by lower bounds, which are shown to be tight in some of the cases. Finally, we show that the problem is polynomially solvable if a connectivity tree is given, and we conclude with some experimental results.

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Section: Discussionmentioning

confidence: 99%

“…We thank Ferran for being a great inspiration to all of us. A preliminary version of this work appears in the Algorithms and Data Structures Symposium (WADS), 2011 [12]. This work was started during the 2009 McGill/INRIA/University of Victoria Bellairs Workshop on Computational Geometry.…”

Section: Acknowledgmentsmentioning

confidence: 99%

Connecting a set of circles with minimum sum of radii

Chambers

Fekete

Hoffmann

et al. 2018

Computational Geometry

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“…We can consider other disk coverage problems with practical constraints such as the connectivity of the disks. Recently, Chambers et al [2] investigated a problem of assigning radii to a given set of points in the plane such that the resulting set of disks is connected and the sum of radii, i.e., α = 1 is minimized. When we bring such connectivity constraint to our problem for α ≥ 1, we need to find a "connected" set of disks which optimally covers the input points.…”

Section: Discussionmentioning

confidence: 99%

A note on minimum-sum coverage by aligned disks

Shin¹

2013

Information Processing Letters

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In this paper, we consider a facility location problem to find a minimum-sum coverage of n points by disks centered at a fixed line. The cost of a disk with radius r has a form of a non-decreasing function f (r) = r α for any α ≥ 1. The goal is to find a set of disks under L p metric such that the disks are centered on the x-axis, their union covers the points, and the sum of the cost of the disks is minimized. Alt et al. [1] presented an algorithm running in O(n 4 log n) time for any α > 1 under any L p metric. We present a faster algorithm for this problem running in O(n 2 log n) time for any α > 1 and any L p metric.

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“…In this problem the input is a set of n sensors and a finite set of m points on the line that are to be covered, and the goal is to find the Strip Cover was first considered by Bar-Noy and Baumer [3], who gave a 3 2 lower bound on the performance of RoundRobin, the algorithm in which the sensors take turns covering the entire region (see Observation 2), but were only able to show a corresponding upper bound of 1.82. The similar Connected Range Assignment (CRA) problem, in which radii are assigned to points in the plane in order to obtain a connected disk graph, was studied by Chambers et al [11]. They showed that the best one circle solution to CRA also yields a 3 2 -approximation guarantee, and in fact, the instance that produces their lower bound is simply a translation of the instance used in Observation 2.…”

Section: Introductionmentioning

confidence: 99%

Set It and Forget It: Approximating the Set Once Strip Cover Problem

2016

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We consider the Set Once Strip Cover problem, in which n wireless sensors are deployed over a one-dimensional region. Each sensor has a fixed battery that drains in inverse proportion to a radius that can be set just once, but activated at any time. The problem is to find an assignment of radii and activation times that maximizes the length of time during which the entire region is covered. We show that this problem is NP-hard. Second, we show that RoundRobin, the algorithm in which the sensors take turns covering the entire region, has a tight approximation guarantee of 3 2 in both Set Once Strip Cover and the more general Strip Cover problem, in which each radius may be set finitely-many times. Moreover, we show that the more general class of duty cycle algorithms, in which groups of sensors take turns covering the entire region, can do no better. Finally, we give an optimal O(n 2 log n)-time algorithm for the related Set Radius Strip Cover problem, in which sensors must be activated immediately.

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Connecting a Set of Circles with Minimum Sum of Radii

Cited by 13 publications

References 8 publications

Connecting a set of circles with minimum sum of radii

Connecting a set of circles with minimum sum of radii

A note on minimum-sum coverage by aligned disks

Set It and Forget It: Approximating the Set Once Strip Cover Problem

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