Fast stabbing of boxes in high dimensions

28Citations
Citations of this article
10Readers
Mendeley users who have this article in their library.

Abstract

We present in this paper a simple yet efficient algorithm for stabbing a set script capital L sign of n axis-parallel boxes in d-dimensional space with c(script capital L sign) points in output-sensitive time O(dn log c(script capital L sign)) and linear space. Let c*(script capital L sign) and b*(script capital L sign) be, respectively, the minimum number of points required to stab script capital L sign and the maximum number of pairwise disjoint boxes of script capital L sign. We prove that b*(script capital L sign)≤c*(script capital L sign)≤c(script capital L sign)≤b*(script capital L sign)(1+log2 b*(script capital L sign))d-1. Since finding a minimal set of c*(script capital L sign) points is NP-complete as soon as d>1, we obtain a fast precision-sensitive heuristic for stabbing script capital L sign whose quality does not depend on the input size. In the case of congruent or constrained isothetic boxes, our algorithm reports, respectively, c(script capital L sign)≤2d-1 b*(script capital L sign) and c(script capital L sign) = Od(b*(script capital L sign)) stabbing points. Moreover, we show that the bounds we get on c(script capital L sign) are asymptotically tight and corroborate our results with some experiments. We also describe an optimal output-sensitive algorithm for finding a minimal-size optimal stabbing point-set of intervals. Finally, we conclude with insights for further research. © 2000 Elsevier Science B.V. All rights reserved.

Cite

CITATION STYLE

APA

Nielsen, F. (2000). Fast stabbing of boxes in high dimensions. Theoretical Computer Science, 246(1–2), 53–72. https://doi.org/10.1016/S0304-3975(98)00336-3

Register to see more suggestions

Mendeley helps you to discover research relevant for your work.

Already have an account?

Save time finding and organizing research with Mendeley

Sign up for free