# Rectangle Tiling

## Abstract

We consider two tiling problems for two-dimensional arrays: given an *n*x*n* array *A* of nonnegative numbers we are to construct an optimal partition of it into rectangular subarrays. The subarrays cannot overlap and they have to cover all array elements. The first problem (RTILE) consists in finding a partition using *p* subarrays that minimizes the maximum weight of subarrays (by weight we mean the sum of all elements covered by the subarray). The second, dual problem (DRTILE), is to construct a partition into minimal number of subarrays such that the weight of each subarray is bounded by a given value *W*. We show a linear-time 7/3-approximation algorithm for the RTILE problem. This improves the best previous result both in time and in approximation ratio. If the array A is binary (i.e. contains only zeroes and ones) we can reduce the approximation ratio up to 2. For the DRTILE problem we get an algorithm which achieves a ratio 4 and works in linear-time. The previously known algorithm with the same ratio worked in time O(*n* ^{5}). For binary arrays we present a linear-time 2-approximation algorithm.

## Preview

Unable to display preview. Download preview PDF.

## References

- 2.M. Grigni, F. Manne,
*On the complexity of generalized block distribution*, Proc. 3rd intern. workshop on parallel algorithms for irregularly structured problems (IRREGULAR’96), Springer, 1996, LNCS 1117, 319–326.CrossRefGoogle Scholar - 3.Y. Han, B. Narahari, H.-A. Choi,
*Mapping a chain task to chained processors*, Information Processing Letters 44, 141–148, 1992.MATHCrossRefMathSciNetGoogle Scholar - 4.S. Khanna, S. Muthukrishnan, M. Paterson,
*On approximating rectangle tiling and packing*, Proc. 19th SODA (1998), 384–393.Google Scholar - 5.S. Khanna, S. Muthukrishnan, S. Skiena,
*Efficient array partitioning*, Proc. 24th ICALP, 616–626, 1997.Google Scholar - 6.G. Martin, S. Muthukrishnan, R. Packwood, I. Rhee,
*Fast algorithms for variable size block matching motion estimation with minimal error*.Google Scholar - 7.S. Muthukrishnan, V. Poosala, T. Suel,
*On rectangular partitionings in two dimensions: algorithms, complexity, and applications*, Proc. ICDT’99, LNCS 1540, 236–256.Google Scholar