Fast algorithms for computing the largest empty rectangle
SCG '87 Proceedings of the third annual symposium on Computational geometry
Journal of Algorithms
Automatic subspace clustering of high dimensional data for data mining applications
SIGMOD '98 Proceedings of the 1998 ACM SIGMOD international conference on Management of data
A threshold of ln n for approximating set cover
Journal of the ACM (JACM)
Surface approximation and geometric partitions
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
Principles of data mining
Computers and Intractability; A Guide to the Theory of NP-Completeness
Computers and Intractability; A Guide to the Theory of NP-Completeness
The class cover problem and its applications in pattern recognition
The class cover problem and its applications in pattern recognition
Approximation Algorithms for the Class Cover Problem
Annals of Mathematics and Artificial Intelligence
Improved Approximation Algorithms for Geometric Set Cover
Discrete & Computational Geometry
The generalized MDL approach for summarization
VLDB '02 Proceedings of the 28th international conference on Very Large Data Bases
Hitting sets when the VC-dimension is small
Information Processing Letters
Small-Size $\eps$-Nets for Axis-Parallel Rectangles and Boxes
SIAM Journal on Computing
Tight lower bounds for the size of epsilon-nets
Proceedings of the twenty-seventh annual symposium on Computational geometry
The mono- and bichromatic empty rectangle and square problems in all dimensions
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
Classifying negative and positive points by optimal box clustering
Discrete Applied Mathematics
Hi-index | 0.00 |
In this paper we study the following problem: Given sets R and B of r red and b blue points respectively in the plane, find a minimum-cardinality set H of axis-aligned rectangles (boxes) so that every point in B is covered by at least one rectangle of H, and no rectangle of H contains a point of R. We prove the NP-hardness of the stated problem, and give either exact or approximate algorithms depending on the type of rectangles considered. If the covering boxes are vertical or horizontal strips we give an efficient algorithm that runs in O(rlogr+blogb+rb) time. For covering with oriented half-strips an optimal O((r+b)log(min{r,b}))-time algorithm is shown. We prove that the problem remains NP-hard if the covering boxes are half-strips oriented in any of the four orientations, and show that there exists an O(1)-approximation algorithm. We also give an NP-hardness proof if the covering boxes are squares. In this situation, we show that there exists an O(1)-approximation algorithm.