## Timothy M. Chan's Publications: Klee's measure problem

### Faster algorithms for largest empty rectangles and boxes

We revisit a classical problem in computational geometry: finding the largest-volume axis-aligned empty box (inside a given bounding box) amidst n given points in d dimensions. Previously, the best algorithms known have running time O(n log^2 n) for d = 2 (by Aggarwal and Suri [SoCG'87]) and near n^d for d >= 3. We describe faster algorithms with running time

• O(n 2^{O(log^*n)} log n) for d = 2,
• O(n^{2.5+o(1)}) time for d = 3, and
• O~(n^{(5d+2)/6}) time for any constant d >= 4.
To obtain the higher-dimensional result, we adapt and extend previous techniques for Klee�s measure problem to optimize certain objective functions over the complement of a union of orthants.

### Klee's measure problem made easy

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.

### Maximum-weight planar boxes in O(n^2) time (and better)

(with
Jérémy Barbay, Gonzalo Navarro, and Pablo Pérez-Lantero)

Given a set P of n points in R^d, where each point p of P is associated with a weight w(p) (positive or negative), the Maximum-Weight Box problem consists in finding an axis-aligned box B maximizing the sum of w(p) over all points p in B. We describe algorithms for this problem in two dimensions that run in the worst case in O(n^2) time, and much less on more specific classes of instances. In particular, these results imply similar ones for the Maximum Bichromatic Discrepancy Box problem. These improve by a factor of log n on the best worst-case complexity previously known for these problems, O(n^2 lg n) [Cortes et al., J. Alg., 2009; Dobkin et al., J. Comput. Syst. Sci., 1996].

### A (slightly) faster algorithm for Klee's measure problem

Given n axis-parallel boxes in a fixed dimension d >= 3, how efficiently can we compute the volume of the union? This standard problem in computational geometry, commonly referred to as Klee's measure problem, can be solved in time O(n^{d/2} log n) by an algorithm of Overmars and Yap (FOCS 1988). We give the first (albeit small) improvement: our new algorithm runs in time n^{d/2} 2^{O(log* n)}, where log* denotes the iterated logarithm.

For the related problem of computing the depth in an arrangement of n boxes, we further improve the time bound to near O(n^{d/2} / log^{d/2-1} n), ignoring log log n factors. Other applications and lower-bound possibilities are discussed. The ideas behind the improved algorithms are simple.

### Space-efficient algorithms for Klee's measure problem

(with Eric Y. Chen)

We give space-efficient geometric algorithms for two related problems. Given a set of n axis-aligned rectangles in the plane, we calculate the area covered by the union of these rectangles (Klee's measure problem) in O(n^{3/2} log n) time with O(sqrt{n}) extra space. If the input can be destroyed and there are no degenerate cases and input coordinates are all integers, we can solve Klee's measure problem in O(n log^2 n) time with O(log^2 n) extra space. Given a set of n points in the plane, we find the axis-aligned unit square that covers the maximum number of points in O(n log^3 n) time with O(log^2 n) extra space.

### Semi-online maintenance of geometric optima and measures

We give the first nontrivial worst-case results for dynamic versions of various basic geometric optimization and measure problems under the semi-online model, where during the insertion of an object we are told when the object is to be deleted. Problems that we can solve with sublinear update time include the Hausdorff distance of two point sets, discrete 1-center, largest empty circle, convex hull volume in three dimensions, volume of the union of axis-parallel cubes, and minimum enclosing rectangle. The decision versions of the Hausdorff distance and discrete 1-center problems can be solved fully dynamically. Some applications are mentioned.