Near-Linear Algorithms for Geometric Hitting Sets and Set Covers

Given a finite range space Σ = ( X , R ) , with N = | X | + | R | , we present two simple algorithms, based on the multiplicative-weight method, for computing a small-size hitting set or set cover of Σ . The first algorithm is a simpler variant of the Brönnimann–Goodrich algorithm but more efficient...

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Vydané v:	Discrete & computational geometry Ročník 63; číslo 2; s. 460 - 482
Hlavní autori:	Agarwal, Pankaj K., Pan, Jiangwei
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.03.2020 Springer Nature B.V
Predmet:	Algorithms Combinatorics Computation Computational Mathematics and Numerical Analysis Data structures Disks Mathematics Mathematics and Statistics Zero sum games Rectangles Multiplicative weight method Disks Geometric set cover Near-linear algorithms 68U05 52C17
ISSN:	0179-5376, 1432-0444
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Abstract	Given a finite range space Σ = ( X , R ) , with N = \| X \| + \| R \| , we present two simple algorithms, based on the multiplicative-weight method, for computing a small-size hitting set or set cover of Σ . The first algorithm is a simpler variant of the Brönnimann–Goodrich algorithm but more efficient to implement, and the second algorithm can be viewed as solving a two-player zero-sum game. These algorithms, in conjunction with some standard geometric data structures, lead to near-linear algorithms for computing a small-size hitting set or set cover for a number of geometric range spaces. For example, they lead to O ( N polylog ( N ) ) expected-time randomized O (1)-approximation algorithms for both hitting set and set cover if X is a set of points and R a set of disks in R 2 .
AbstractList	Given a finite range space Σ=(X,R), with N=\|X\|+\|R\|, we present two simple algorithms, based on the multiplicative-weight method, for computing a small-size hitting set or set cover of Σ. The first algorithm is a simpler variant of the Brönnimann–Goodrich algorithm but more efficient to implement, and the second algorithm can be viewed as solving a two-player zero-sum game. These algorithms, in conjunction with some standard geometric data structures, lead to near-linear algorithms for computing a small-size hitting set or set cover for a number of geometric range spaces. For example, they lead to O(Npolylog(N)) expected-time randomized O(1)-approximation algorithms for both hitting set and set cover if X is a set of points and R a set of disks in R2. Given a finite range space Σ = ( X , R ) , with N = \| X \| + \| R \| , we present two simple algorithms, based on the multiplicative-weight method, for computing a small-size hitting set or set cover of Σ . The first algorithm is a simpler variant of the Brönnimann–Goodrich algorithm but more efficient to implement, and the second algorithm can be viewed as solving a two-player zero-sum game. These algorithms, in conjunction with some standard geometric data structures, lead to near-linear algorithms for computing a small-size hitting set or set cover for a number of geometric range spaces. For example, they lead to O ( N polylog ( N ) ) expected-time randomized O (1)-approximation algorithms for both hitting set and set cover if X is a set of points and R a set of disks in R 2 .
Author	Pan, Jiangwei Agarwal, Pankaj K.
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Cites_doi	10.1007/s00453-011-9517-2 10.1145/2462356.2462363 10.1016/j.comgeo.2015.12.002 10.1016/0196-6774(80)90015-2 10.1145/1542362.1542420 10.1137/0215051 10.1007/BF02187876 10.1007/s00454-006-1273-8 10.1007/s00454-010-9285-9 10.1007/BF02187743 10.1145/2390176.2390185 10.1137/1.9781611973068.21 10.1007/BF02293051 10.1017/CBO9780511814075 10.1145/1542362.1542366 10.1137/090762968 10.1016/0167-6377(95)00032-0 10.1090/S0894-0347-2012-00759-0 10.1145/201019.201036 10.1145/285055.285059 10.1145/1377676.1377708 10.1007/s00454-010-9323-7 10.1137/1.9781611973068.97 10.1007/s00453-013-9771-6 10.1007/978-3-540-77974-2 10.1006/game.1999.0738 10.1137/120891241 10.1007/BF02570718 10.1090/conm/223/03131 10.1016/0020-0190(81)90111-3 10.1137/060669474 10.1016/0925-7721(92)90006-E 10.1287/moor.20.2.257 10.1109/FOCS.2014.64 10.4086/toc.2012.v008a006
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Snippet	Given a finite range space Σ = ( X , R ) , with N = \| X \| + \| R \| , we present two simple algorithms, based on the multiplicative-weight method, for computing... Given a finite range space Σ=(X,R), with N=\|X\|+\|R\|, we present two simple algorithms, based on the multiplicative-weight method, for computing a small-size...
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SubjectTerms	Algorithms Combinatorics Computation Computational Mathematics and Numerical Analysis Data structures Disks Mathematics Mathematics and Statistics Zero sum games
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Title	Near-Linear Algorithms for Geometric Hitting Sets and Set Covers
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