TY - GEN

T1 - Klee's measure problem made easy

AU - Chan, Timothy M.

N1 - Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

PY - 2013

Y1 - 2013

N2 - 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(nd/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(nd/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(nd/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.

AB - 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(nd/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(nd/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(nd/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.

KW - Boxes

KW - Computational geometry

KW - Volume

UR - http://www.scopus.com/inward/record.url?scp=84893469130&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84893469130&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2013.51

DO - 10.1109/FOCS.2013.51

M3 - Conference contribution

AN - SCOPUS:84893469130

SN - 9780769551357

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 410

EP - 419

BT - Proceedings - 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013

T2 - 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013

Y2 - 27 October 2013 through 29 October 2013

ER -