TY - GEN

T1 - Faster algorithms for some optimization problems on collinear points

AU - Biniaz, Ahmad

AU - Bose, Prosenjit

AU - Carmi, Paz

AU - Maheshwari, Anil

AU - Munro, Ian

AU - Smid, Michiel

N1 - Publisher Copyright:
© Ahmad Biniaz, Prosenjit Bose, Paz Carmi, Anil Maheshwari, Ian Munro, and Michiel Smid; licensed under Creative Commons License CC-BY 34th Symposium on Computational Geometry (SoCG 2018).

PY - 2018/6/1

Y1 - 2018/6/1

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. 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 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. 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 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.

KW - Collinear points

KW - Range assignment

UR - http://www.scopus.com/inward/record.url?scp=85048965944&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SoCG.2018.8

DO - 10.4230/LIPIcs.SoCG.2018.8

M3 - Conference contribution

AN - SCOPUS:85048965944

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 81

EP - 814

BT - 34th International Symposium on Computational Geometry, SoCG 2018

A2 - Toth, Csaba D.

A2 - Speckmann, Bettina

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 34th International Symposium on Computational Geometry, SoCG 2018

Y2 - 11 June 2018 through 14 June 2018

ER -