On approximating rectangle tiling and packing

Sanjeev Khanna, Shanmugavelayutham Muthukrishnan, Mike Paterson

    Research output: Contribution to conferencePaper

    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×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 algorithm for these problems.

    Original languageEnglish (US)
    Pages384-394
    Number of pages11
    StatePublished - Dec 1 1998
    EventProceedings of the 1998 9th Annual ACM SIAM Symposium on Discrete Algorithms - San Francisco, CA, USA
    Duration: Jan 25 1998Jan 27 1998

    Other

    OtherProceedings of the 1998 9th Annual ACM SIAM Symposium on Discrete Algorithms
    CitySan Francisco, CA, USA
    Period1/25/981/27/98

    Fingerprint

    Tiling
    Rectangle
    Packing
    Approximation algorithms
    Motion estimation
    Image compression
    Dynamic programming
    NP-complete problem
    Data Partitioning
    Block Matching
    Video Compression
    Packing Problem
    Motion Estimation
    Linear-time Algorithm
    Histogram
    One Dimension
    Dynamic Programming
    Approximation Algorithms
    Overlap
    Mining

    ASJC Scopus subject areas

    • Software
    • Mathematics(all)

    Cite this

    Khanna, S., Muthukrishnan, S., & Paterson, M. (1998). On approximating rectangle tiling and packing. 384-394. Paper presented at Proceedings of the 1998 9th Annual ACM SIAM Symposium on Discrete Algorithms, San Francisco, CA, USA, .

    On approximating rectangle tiling and packing. / Khanna, Sanjeev; Muthukrishnan, Shanmugavelayutham; Paterson, Mike.

    1998. 384-394 Paper presented at Proceedings of the 1998 9th Annual ACM SIAM Symposium on Discrete Algorithms, San Francisco, CA, USA, .

    Research output: Contribution to conferencePaper

    Khanna, S, Muthukrishnan, S & Paterson, M 1998, 'On approximating rectangle tiling and packing' Paper presented at Proceedings of the 1998 9th Annual ACM SIAM Symposium on Discrete Algorithms, San Francisco, CA, USA, 1/25/98 - 1/27/98, pp. 384-394.
    Khanna S, Muthukrishnan S, Paterson M. On approximating rectangle tiling and packing. 1998. Paper presented at Proceedings of the 1998 9th Annual ACM SIAM Symposium on Discrete Algorithms, San Francisco, CA, USA, .
    Khanna, Sanjeev ; Muthukrishnan, Shanmugavelayutham ; Paterson, Mike. / On approximating rectangle tiling and packing. Paper presented at Proceedings of the 1998 9th Annual ACM SIAM Symposium on Discrete Algorithms, San Francisco, CA, USA, .11 p.
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