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(mlog 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× n matrices A and B with nonnegative integer entries, yielding a method for computing A× B in O~(n2+min{rArB,nmin{rA,rB}}) time, where O~ suppresses polylogarithmic (in n) factors and where rAand rBdenote the minimum number of 3D rectangles into which the 3D histograms induced by A and B can be partitioned, respectively.
Original language | English |
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Pages (from-to) | 136-154 |
Number of pages | 19 |
Journal | Algorithmica |
Volume | 80 |
Issue number | 1 |
DOIs | |
Publication status | Published - 1 Jan 2018 |
Externally published | Yes |
Keywords
- Decomposition problem
- Matrix multiplication
- Minimum rectangulation
- Orthogonal polyhedron
- Time complexity
ASJC Scopus subject areas
- General Computer Science
- Computer Science Applications
- Applied Mathematics