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On approximating rectangle tiling and packing
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Khanna, Sanjeev, Muthukrishnan, S. and Paterson, Michael S. (1998) On approximating rectangle tiling and packing. University of Warwick. Department of Computer Science. (Department of Computer Science research report). (Unpublished)
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Abstract
Our study of tiling and packing with rectangles in two-dimensional regions is strongly motivated by applications in database mining, histogram-based estimation of query sizes, data partitioning, and motion estimation in video compression by block matching, among others. An example of the problems that we tackle is the following: given an n by n array A of positive numbers, find a tiling using at most p rectangles (that is, no two rectangles must overlap, and each array element must fall within some rectangle) that minimizes the maximum weight of any rectangle; here the "weight" of a rectangle is the sum of the array elements that fall within it. If the array A were one-dimensional, this problem could be easily solved by dynamic programming. We prove that in the two-dimensional case it is NP-hard to approximate this problem to within a factor of 1.25. On the other hand, we provide a near-linear time algorithm that returns a solution at most 2.5 times the optimal. Other rectangle tiling and packing problems that we study have similar properties: while it is easy to solve them optimally in one dimension, the two-dimensional versions become NP-hard. We design efficient approximation algorithms for these problems.
| Item Type: | Report | ||||
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| Subjects: | Q Science > QA Mathematics | ||||
| Divisions: | Faculty of Science > Computer Science | ||||
| Library of Congress Subject Headings (LCSH): | Tiling (Mathematics) | ||||
| Series Name: | Department of Computer Science research report | ||||
| Publisher: | University of Warwick. Department of Computer Science | ||||
| Official Date: | 1998 | ||||
| Dates: |
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| Number: | Number 339 | ||||
| Number of Pages: | 10 | ||||
| DOI: | CS-RR-339 | ||||
| Institution: | University of Warwick | ||||
| Theses Department: | Department of Computer Science | ||||
| Status: | Not Peer Reviewed | ||||
| Publication Status: | Unpublished | ||||
| Funder: | European Strategic Programme of Research and Development in Information Technology (ESPRIT) | ||||
| Grant number: | 20244 (ESPRIT) | ||||
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