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.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
