@MISC{Chan13klee’smeasure, author = {Timothy M. Chan}, title = {Klee’s measure problem made easy}, year = {2013} }

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Abstract

We present a new algorithm for a classic problem in computational geometry, Klee’s measure problem: given a set of n axis-parallel boxes in d-dimensional space, compute the volume of the union of the boxes. The algorithm runs in O(n d/2) time for any constant d ≥ 3. Although it improves the previous best algorithm by “just ” an iterated logarithmic factor, the real surprise lies in the simplicity of the new algorithm. We also show that it is theoretically possible to beat the O(n d/2) time bound by logarithmic factors for integer input in the word RAM model, and for other variants of the problem. With additional work, we obtain an O(n d/3 polylog n)-time algorithm for the important special case of orthants or unit hypercubes (which include the so-called “hypervolume indicator problem”), and an O(n (d+1)/3 polylog n)-time algorithm for the case of arbitrary hypercubes or fat boxes, improving a previous O(n (d+2)/3)-time algorithm by Bringmann. 1