Near-Linear Algorithms for Geometric Hitting Sets and Set Covers

• Published in: Discrete & computational geometry Vol. 63; no. 2; pp. 460 - 482
• Main Authors: Agarwal, Pankaj K., Pan, Jiangwei
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2020; Springer Nature B.V
• Subjects: 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
• DOI: 10.1007/s00454-019-00099-6

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 .