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Tiling multi-dimensional arrays

  • Jonathan P. Sharp
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1684)

Abstract

We continue the study of the tiling problems introduced in [KMP98]. The first problem we consider is: given a d-dimensional array of non-negative numbers and a tile limit p, partition the array into at most p rectangular, non-overlapping subarrays, referred to as tiles, in such a way as to minimise the weight of the heaviest tile, where the weight of a tile is the sum of the elements that fall within it. For one-dimensional arrays the problem can be solved optimally in polynomial time, whereas for two-dimensional arrays it is shown in [KMP98] that the problem is NP-hard and an approximation algorithm is given. This paper offers a new (d 2 +2d−1)/(2d−1) approximation algorithm for the d-dimensional problem (d ≥ 2), which improves the (d+3)/2 approximation algorithm given in [SS99]. In particular, for two-dimensional arrays, our approximation ratio is 7/3 improving on the ratio of 5/2 in [KMP98] and [SS99]. We briefly consider the dual tiling problem where, rather than having a limit on the number of tiles allowed, we must ensure that all tiles produced have weight at most W and do so with a minimaln umber of tiles. The algorithm for the first problem can be modified to give a 2d approximation for this problem improving upon the 2d+1 approximation given in [SS99]. These problems arise naturally in many applications including databases and load balancing.

Keywords

Approximation Algorithm Load Balance Execution Plan Rectangular Tiling Tile Limit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [KMP98]
    Sanjeev Khanna, S. Muthukrishnan, and Mike Paterson. On approximating rectangular tiling and packing. In Proc. 9th Annual Symposium on Discrete Algorithms (SODA), pages 384–393, 1998.Google Scholar
  2. [KMS97]
    Sanjeev Khanna, S. Muthukrishnan, and S. Skiena. Efficient array partitioning. In Proc. 24th International Colloquium on Automata, Languages and Programming (ICALP), pages 616–626, 1997.Google Scholar
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    Fredrik Manne. Load Balancing in Parallel Sparse Matrix Computations. PhD thesis, Department of Informatics, University of Bergen, Norway, 1993.Google Scholar
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  5. [MPS99]
    S. Muthukrishnan, Viswanath Poosala, and Torsten Suel. On rectangular partitions in two dimensions: Algorithms, complexity, and applications. In 7th International Conference on Database Theory (ICDT), pages 236–256, January 1999.Google Scholar
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    Viswanath Poosala. Histogram-based estimation techniques in databases. PhD thesis, Department of Computer Science, University of Wisconsin-Madison, US, 1997.Google Scholar
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    Adam Smith and Subhash Suri. Rectangular tiling in multi-dimensional arrays. In Proc. 10th Annual Symposium on Discrete Algorithms (SODA), 1999.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Jonathan P. Sharp
    • 1
  1. 1.Department of Computer ScienceUniversity of WarwickCoventryUK

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