Output sensitive algorithms for covering many points

Discrete Algorithms In this paper we devise some output sensitive algorithms for a problem where a set of points and a positive integer, m, are given and the goal is to cover a maximal number of these points with m disks. We introduce a parameter, ρ, as the maximum number of points that one disk can...

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Vydáno v:Discrete Mathematics and Theoretical Computer Science Ročník 17 no. 1; číslo Discrete Algorithms; s. 309 - 316
Hlavní autoři: Ghasemalizadeh, Hossein, Razzazi, Mohammadreza
Médium: Journal Article
Jazyk:angličtina
Vydáno: DMTCS 01.01.2015
Discrete Mathematics & Theoretical Computer Science
Témata:
[info.info-dm] computer science [cs]/discrete mathematics [cs.dm]
[info.info-hc] computer science [cs]/human-computer interaction [cs.hc]
Algorithms
computational geometry
Computer Science
covering with disks
Discrete Mathematics
Fuzzy sets
Human-Computer Interaction
Mathematical research
output sensitive algorithm
Set theory
Storage media
computational geometry
output sensitive algorithm
covering with disks
ISSN:1365-8050, 1462-7264, 1365-8050
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Popis
Shrnutí:Discrete Algorithms In this paper we devise some output sensitive algorithms for a problem where a set of points and a positive integer, m, are given and the goal is to cover a maximal number of these points with m disks. We introduce a parameter, ρ, as the maximum number of points that one disk can cover and we analyse the algorithms based on this parameter. At first, we solve the problem for m=1 in O(nρ) time, which improves the previous O(n2) time algorithm for this problem. Then we solve the problem for m=2 in O(nρ + 3 log ρ) time, which improves the previous O(n3 log n) algorithm for this problem. Our algorithms outperform the previous algorithms because ρ is much smaller than n in many cases. Finally, we extend the algorithm for any value of m and solve the problem in O(mnρ + (mρ)2m - 1 log mρ) time. The previous algorithm for this problem runs in O(n2m - 1 log n) time and our algorithm usually runs faster than the previous algorithm because mρ is smaller than n in many cases. We obtain output sensitive algorithms by confining the areas that we should search for the result. The techniques used in this paper may be applicable in other covering problems to obtain faster algorithms.
ISSN:1365-8050
1462-7264
1365-8050
DOI:10.46298/dmtcs.2102