Abstract
The problem of partitioning an input rectilinear polyhedron \(P\) into a minimum number of 3D rectangles is known to be NP-hard. We first develop a \(4\)-approximation algorithm for the special case 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 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 \(\tilde{O}(n^2+ \min \{ r_Ar_B, n\min \{r_A,\ r_B\}\})\) 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.
Jesper Jansson: Funded by The Hakubi Project at Kyoto University.
Christos Levcopoulos: Research supported in part by Swedish Research Council grant 621-2011-6179.
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Floderus, P., Jansson, J., Levcopoulos, C., Lingas, A., Sledneu, D. (2014). 3D Rectangulations and Geometric Matrix Multiplication. In: Ahn, HK., Shin, CS. (eds) Algorithms and Computation. ISAAC 2014. Lecture Notes in Computer Science(), vol 8889. Springer, Cham. https://doi.org/10.1007/978-3-319-13075-0_6
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