Abstract
A well-known problem in computational geometry is Klee’s measure problem, which asks for the volume of a union of axis-aligned boxes in d-space. In this paper, we consider Klee’s measure problem for the special case where a 2-dimensional orthogonal projection of all the boxes has a common corner. We call such a set of boxes 2-grounded and, more generally, a set of boxes is k-grounded if in a k-dimensional orthogonal projection they share a common corner.
Our main result is an O(n (d−1)/2log2 n) time algorithm for computing Klee’s measure for a set of n 2-grounded boxes. This is an improvement of roughly \(O(\sqrt{n})\) compared to the fastest solution of the general problem. The algorithm works for k-grounded boxes, for any k≥2, and in the special case of k=d, also called the hypervolume indicator problem, the time bound can be improved further by a logn factor. The key idea of our technique is to reduce the d-dimensional problem to a semi-dynamic weighted volume problem in dimension d−2. The weighted volume problem requires solving a combinatorial problem of maintaining the sum of ordered products, which may be of independent interest.
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Notes
x d denotes the dth coordinate of point x.
This name is inspired by color gradients, which are images with decreasing color intensity in one direction.
Note that \(\sum_{k=1}^{d}{{d-1\choose{k-1}} \times(k-1)!} = \sum_{k=0}^{d-1}{\frac{(d-1)!}{(d-1-k)!}} = \sum_{k=0}^{d-1}{\frac{(d-1)!}{k!}} \le\mathtt{e}(d-1)! \).
Since we assume that all halfspaces have distinct weights, all inner terms that contain equal indices are 0, and so we can safely ignore these terms and use a strict ordering on indices i 2,…,i k .
For simplicity of presentation only, we use a non-strict ordering of the array indices (i.e., i 1≤⋯≤i d ) in this lemma. The sum of products with strictly ordered indices (i.e., i 1<⋯<i d , as defined in Sect. 4.2) can be easily reduced to this form by shifting arrays. In particular, the strictly ordered sum of products for d arrays \(\mathcal{A} _{1},\ldots , \mathcal{A} _{d}\) is equal to the non-strictly ordered sum of products for arrays \(\mathcal {A} '_{1},\ldots, \mathcal{A} '_{d}\) where \(\mathcal{A} '_{s}[i]\) is defined to as \(\mathcal{A} _{s}[i+s-1]\).
In d dimensions, an axis-parallel strip has the form \(\{{\mathbf {x}}\in \mathbb{R} ^{d} \mid a \le{\mathbf{x}}^{k} \le b \}\) where a and b are reals and k is an integer between 1 and d.
References
Agarwal, P.: An improved algorithm for computing the volume of the union of cubes. In: Proceedings of the 26th Annual Symposium on Computational Geometry, pp. 230–239 (2010)
Agarwal, P., Kaplan, H., Sharir, M.: Computing the volume of the union of cubes. In: Proceedings of the 23rd Annual Symposium on Computational Geometry, pp. 294–301 (2007)
Bentley, J.L.: Solutions to Klee’s rectangle problems. Unpublished manuscript (1977)
de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications. Springer, Berlin (2008)
Bringmann, K.: Klee’s measure problem on fat boxes in time O(n (d+2)/3). In: Proceedings of the 26th Annual Symposium on Computational Geometry, pp. 222–229 (2010)
Bringmann, K., Friedrich, T.: Approximating the volume of unions and intersections of high-dimensional geometric objects. In: Proceedings of 19th International Symposium on Algorithms and Computation, pp. 436–447 (2008)
Chan, T.: Semi-online maintenance of geometric optima and measures. In: Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 474–483 (2002)
Chan, T.M.: A (slightly) faster algorithm for Klee’s measure problem. Comput. Geom. 43(3), 243–250 (2010)
Chew, L., Dor, D., Efrat, A., Kedem, K.: Geometric pattern matching in d-dimensional space. Discrete Comput. Geom. 21(2), 257–274 (1999)
Fleischer, M.: The measure of Pareto optima. Applications to multi-objective metaheuristics. In: Evolutionary Multi-criterion Optimization, Second International Conference, pp. 519–533 (2003)
Fonseca, C., Paquete, L., López-Ibáñez, M.: An improved dimension-sweep algorithm for the hypervolume indicator. In: IEEE Congress on Evolutionary Computation, pp. 1157–1163 (2006)
Huband, S., Hingston, P., While, L., Barone, L.: An evolution strategy with probabilistic mutation for multi-objective optimisation. In: The 2003 Congress on Evolutionary Computation, vol. 4, pp. 2284–2291 (2003)
Kaplan, H., Rubin, N., Sharir, M., Verbin, E.: Counting colors in boxes. In: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 785–794 (2007)
Klee, V.: Can the measure of \(\bigcup_{1}^{n}[a_{i},b_{i}]\) be computed in less than O(nlogn) steps? Am. Math. Mon. 84(4), 284–285 (1977)
van Leeuwen, J., Wood, D.: The measure problem for rectangular ranges in d-space. J. Algorithms 2(3), 282–300 (1981)
Overmars, M.H., Yap, C.K.: New upper bounds in Klee’s measure problem. SIAM J. Comput. 20(6), 1034–1045 (1991)
Zitzler, E., Künzli, S.: Indicator-based selection in multiobjective search. In: Proceedings of 8th International Conference on Parallel Problem Solving from Nature, pp. 832–842 (2004)
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Yıldız, H., Suri, S. Computing Klee’s Measure of Grounded Boxes. Algorithmica 71, 307–329 (2015). https://doi.org/10.1007/s00453-013-9797-9
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DOI: https://doi.org/10.1007/s00453-013-9797-9