Adaptive Computation of the Klee's Measure in High Dimensions

TL;DR

This paper introduces three techniques to efficiently compute Klee's measure in high dimensions by exploiting input degeneracies and structures, significantly improving computation times in specific cases.

Contribution

It presents novel algorithms that adaptively leverage input properties like small maxima, limited hyperplane intersections, and small treewidth of the intersection graph.

Findings

Algorithms run in sub-quadratic time for certain input structures.

Techniques improve computation time based on input degeneracy and structure.

Combined approach adapts to multiple input configurations.

Abstract

The KLEE'S MESURE of $n$ axis-parallel boxes in $\mathbb{R}^d$ is the volume of their union. It can be computed in time within $O(n^{d/2})$ in the worst case. We describe three techniques to boost its computation: one based on some type of "degeneracy'' of the input, and two ones on the inherent "easiness'' of the structure of the input. The first technique benefits from instances where the MAXIMA of the input is of small size $h$, and yields a solution running in time within $O(n\log^{2d-2}{h}+ h^{d/2}) \subseteq O(n^{d/2}$). The second technique takes advantage of instances where no $d$-dimensional axis-aligned hyperplane intersects more than $k$ boxes in some dimension, and yields a solution running in time within $O(n \log n + n k^{(d-2)/2}) \subseteq O(n^{d/2})$. The third technique takes advantage of instances where the \emph{intersection graph} of the input has small treewidth $\omega$. It yields an algorithm running in time within $O(n^4\omega \log \omega + n (\omega \log \omega)^{d/2})$ in general, and in time within $O(n \log n + n \omega ^{d/2})$ if an optimal tree decomposition of the intersection graph is given. We show how to combine these techniques in an algorithm which takes advantage of all three configurations.