Abstract
The problem of partitioning an orthogonal polyhedron P into a minimum number of 3D rectangles is known to be NP-hard. In this paper, we first develop a 4-approximation algorithm for the special case of the problem in which P is a 3D histogram. It runs in \(O(m \log m)\) time, where m is the number of corners in P. We then apply it to exactly compute the arithmetic matrix product of two \(n \times n\) matrices A and B with nonnegative integer entries, yielding a method for computing \(A \times B\) in \(\tilde{O}(n^2 + \min \{ r_A r_B,\, n \min \{r_A,\ r_B\}\})\) time, where \(\tilde{O}\) suppresses polylogarithmic (in n) factors and where \(r_A\) and \(r_B\) denote the minimum number of 3D rectangles into which the 3D histograms induced by A and B can be partitioned, respectively.
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Acknowledgements
J.J. was funded by The Hakubi Project at Kyoto University. C.L. was supported in part by Swedish Research Council Grant 621-2011-6179.
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An extended abstract of this article appeared in Proceedings of the 25th International Symposium on Algorithms and Computation (ISAAC 2014), volume 8889 of Lecture Notes in Computer Science, pp. 65–78, Springer International Publishing Switzerland, 2014.
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Floderus, P., Jansson, J., Levcopoulos, C. et al. 3D Rectangulations and Geometric Matrix Multiplication. Algorithmica 80, 136–154 (2018). https://doi.org/10.1007/s00453-016-0247-3
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DOI: https://doi.org/10.1007/s00453-016-0247-3