3D Rectangulations and Geometric Matrix Multiplication

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.