TY - JOUR

T1 - Faster algorithms for some optimization problems on collinear points

AU - Biniaz, Ahmad

AU - Bose, Prosenjit

AU - Carmi, Paz

AU - Maheshwari, Anil

AU - Munro, J. Ian

AU - Smid, Michiel

N1 - Funding Information: *A preliminary version of this paper has been presented at the 34th International Symposium on Computational Geometry (SoCG 2018). „University of Windsor, ahmad.biniaz@gmail.com, Supported by NSERC. …Carleton University, {jit, anil, michiel}@scs.carleton.ca, Supported by NSERC. §Ben-Gurion University of the Negev, carmip@gmail.com, Partially supported by Grant 2016116 from the United States – Israel Binational Science Foundation. ¶University of Waterloo, imunro@uwaterloo.ca, Supported by NSERC and Canada Research Chairs Program. Publisher Copyright: © 2020, Carleton University. All rights reserved.

PY - 2020/1/1

Y1 - 2020/1/1

N2 - We propose faster algorithms for the following three optimization problems on n collinear points, i.e., points in dimension one. The first two problems are known to be NP-hard in higher dimensions. 1. Maximizing total area of disjoint disks: In this problem the goal is to maximize the total area of nonoverlapping disks centered at the points. Acharyya, De, and Nandy (2017) presented an O(n2)-time algorithm for this problem. We present an optimal Θ(n)-time algorithm, provided that the points are given in sorted order. 2. Minimizing sum of the radii of client-server coverage: The n points are partitioned into two sets, namely clients and servers. The goal is to minimize the sum of the radii of disks centered at servers such that every client is in some disk, i.e., in the coverage range of some server. Lev-Tov and Peleg (2005) presented an O(n3)-time algorithm for this problem. We present an O(n2)-time algorithm, thereby improving the running time by a factor of Θ(n). 3. Minimizing total area of point-interval coverage: The n input points belong to an interval I. The goal is to find a set of n disks of minimum total area, covering I, such that every disk contains at least one input point. We present an algorithm that solves this problem in O(n2) time.

AB - We propose faster algorithms for the following three optimization problems on n collinear points, i.e., points in dimension one. The first two problems are known to be NP-hard in higher dimensions. 1. Maximizing total area of disjoint disks: In this problem the goal is to maximize the total area of nonoverlapping disks centered at the points. Acharyya, De, and Nandy (2017) presented an O(n2)-time algorithm for this problem. We present an optimal Θ(n)-time algorithm, provided that the points are given in sorted order. 2. Minimizing sum of the radii of client-server coverage: The n points are partitioned into two sets, namely clients and servers. The goal is to minimize the sum of the radii of disks centered at servers such that every client is in some disk, i.e., in the coverage range of some server. Lev-Tov and Peleg (2005) presented an O(n3)-time algorithm for this problem. We present an O(n2)-time algorithm, thereby improving the running time by a factor of Θ(n). 3. Minimizing total area of point-interval coverage: The n input points belong to an interval I. The goal is to find a set of n disks of minimum total area, covering I, such that every disk contains at least one input point. We present an algorithm that solves this problem in O(n2) time.

UR - http://www.scopus.com/inward/record.url?scp=85096023883&partnerID=8YFLogxK

M3 - Article

SN - 1920-180X

VL - 11

SP - 418

EP - 432

JO - Journal of Computational Geometry

JF - Journal of Computational Geometry

IS - 1

ER -