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(mlogm) time, where m is the number of corners in P. We then apply it to compute the arithmetic matrix product of two n×n matrices A and B with nonnegative integer entries, yielding a method for computing A × B in Õ(n2 +min{rArB, n min{rA, rB}}) time, where Õ suppresses polylogarithmic (in n) factors and where rA and rB denote the minimum number of 3D rectangles into which the 3D histograms induced by A and B can be partitioned, respectively.