Abstract
We study the following problem. Given an n × n array A of nonnegative numbers and a natural number p, partition it into at most p rectangular tiles, so that the maximal weight of a tile is minimized. A tile is any rectangular subarray of A. The weight of a tile is the sum of the elements that fall within it. In the partition the tiles must not overlap and are to cover the whole array. We give a \(2\frac{1}{8}-\)approximation algorithm, which is tight with regard to the only known and used lower bound. Although the proof of the ratio of the approximation is somewhat involved, the algorithm itself is quite simple and easy to implement. Its running time is linear in the size of the array, but can be also made to be near-linear in the number of non-zero elements of the array.
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Paluch, K. (2004). A \(2 \frac{1}{8}\)-Approximation Algorithm for Rectangle Tiling. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds) Automata, Languages and Programming. ICALP 2004. Lecture Notes in Computer Science, vol 3142. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27836-8_88
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DOI: https://doi.org/10.1007/978-3-540-27836-8_88
Publisher Name: Springer, Berlin, Heidelberg
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