## 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 language | English (US) |
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Pages | 384-394 |

Number of pages | 11 |

State | Published - Dec 1 1998 |

Externally published | Yes |

Event | Proceedings of the 1998 9th Annual ACM SIAM Symposium on Discrete Algorithms - San Francisco, CA, USA Duration: Jan 25 1998 → Jan 27 1998 |

### Other

Other | Proceedings of the 1998 9th Annual ACM SIAM Symposium on Discrete Algorithms |
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City | San Francisco, CA, USA |

Period | 1/25/98 → 1/27/98 |

## All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)