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.








Similar content being viewed by others
References
Bansal, N., Williams, R.: Regularity lemmas and combinatorial algorithms. Theory Comput. 8(1), 69–94 (2012)
Björklund, A., Lingas, A.: Fast Boolean matrix multiplication for highly clustered data. In: Proceedings of WADS 2001, LNCS, vol. 2125, pp. 258–263
de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Santa Clara (2008)
Dielissen, V.J., Kaldewaij, A.: Rectangular partition is polynomial in two dimensions but NP-complete in three. Inf. Process. Lett. 38(1), 1–6 (1991)
Ga̧sieniec, L., Lingas, A.: An improved bound on boolean matrix multiplication for highly clustered data. In: Proceedings of WADS 2003, LNCS, vol. 2748, pp. 329–339
Indyk, P.: High-dimensional Computational Geometry. PhD dissertation, Stanford University, September (2000)
Keil, J.M.: Polygon Decomposition. Dept. Comput. Sc. Univ. Saskatchewan, Survey (1996)
Le Gall, F.: Powers of tensors and fast matrix multiplication. In: Proceedings of the 39th ISSAC 2014, pp. 296–303
Lingas, A.: A geometric approach to Boolean matrix multiplication. In: Proceedings of ISAAC 2002, LNCS, vol. 2518, pp. 501–510
Lingas, A., Sledneu, D.: A combinatorial algorithm for all-pairs shortest paths in directed vertex-weighted graphs with applications to disc graphs. In: Proceedings of SOFSEM 2012, LNCS, pp. 373–384
Lipski, W.: Finding a Manhattan path and related problems. Networks 13(3), 399–409 (1983)
Mehlhorn, K.: Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry. EATCS Monographs on Theoretical Computer Science. Springer, Berlin (1984)
Muthukrishnan, S., Poosala, V., Suel, T.: On rectangular partitionings in two dimensions: algorithms, complexity, and applications. In: Proceedings of ICDT’99, LNCS Vol. 1540, pp. 236–256
Sack, J.-R., Urrutia, J. (eds.): Handbook of Computational Geometry. Elsevier, Amsterdam (2000)
Acknowledgements
J.J. was funded by The Hakubi Project at Kyoto University. C.L. was supported in part by Swedish Research Council Grant 621-2011-6179.
Author information
Authors and Affiliations
Corresponding author
Additional information
An extended abstract of this article appeared in Proceedings of the 25th International Symposium on Algorithms and Computation (ISAAC 2014), volume 8889 of Lecture Notes in Computer Science, pp. 65–78, Springer International Publishing Switzerland, 2014.
Rights and permissions
About this article
Cite this article
Floderus, P., Jansson, J., Levcopoulos, C. et al. 3D Rectangulations and Geometric Matrix Multiplication. Algorithmica 80, 136–154 (2018). https://doi.org/10.1007/s00453-016-0247-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-016-0247-3