Fast Stabbing of Boxes in High Dimensions

We present in this paper a simple yet e cient algorithm for stabbing a set S of n axisparallel boxes in d-dimensional space with c(S) points in output-sensitive time O(dn log c(S)) and linear space. Let c∗(S) and b∗(S) be, respectively, the minimum number of points required to stab S and the maximum number of pairwise disjoint boxes of S. We prove that b(S)6c(S)6c(S)6b(S)(1+log2 b ∗(S))d−1. Since nding a minimal set of c∗(S) points is NP-complete as soon as d¿1, we obtain a fast precision-sensitive heuristic for stabbing S whose quality does not depend on the input size. In the case of congruent or constrained isothetic boxes, our algorithm reports, respectively, c(S)62d−1b∗(S) and c(S)=Od(b∗(S)) stabbing points. Moreover, we show that the bounds we get on c(S) are asymptotically tight and corroborate our results with some experiments. We also describe an optimal output-sensitive algorithm for nding a minimal-size optimal stabbing point-set of intervals. Finally, we conclude with insights for further research. c © 2000 Elsevier Science B.V. All rights reserved.