Abstract
The problem of partitioning a rectilinear figure into rectangles with minimum length is NP-hard and has bounded heuristics. In this paper we study a related problem,Elimination Problem (EP), in which a rectilinear figure is partitioned into a set of rectilinear figures containing no concave vertices of a fixed direction with minimum length. We show that a heuristic for EP within a factor of 4 from optimal can be computed in timeO(n 2), wheren is the number of vertices of the input figure, and a variant of this heuristic, within a factor of 6 from optimal, can be computed in timeO(n logn). As an application, we give a bounded heuristic for the problem of partitioning a rectilinear figure into histograms of a fixed direction with minimum length.
An auxiliary result is that an optimal rectangular partition of a monotonic histogram can be computed in timeO(n 2), using a known speed-up technique in dynamic programming.
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References
D. Z. Du, On Fast Heuristics for Minimum Edge Length Rectangular Partition, Technique Report-03618-86, Mathematical Sciences Research Institute, Berkeley, CA, 1986.
D. Knuth, Optimum binary search trees,Acta Inform.,1 (1971), 14–25.
D. T. Lee and F. P. Preparata, Computational geometry: a survey,IEEE Trans. Comput.,33 (1984), 1072–1101.
C. Levcopoulos, Minimum length and thickest first rectangular partition of polygons,Proc. 23rd Allerton Conf. on Comm. Control and Computing, Illinois, 1985, pp. 664–673.
A. Lingas,Heuristics for Minimum Edge Length Rectangular Partition of Rectilinear Figures, Lecture Notes on Computer Science, vol. 195, Springer-Verlag, Berlin, 1983.
A. Lingas, R. Pinter, R. Rivest, and A. Shamir, Minimum edge length decomposition of rectilinear figures,20th Allerton Conf. on Comm. Control and Computing, Illinois, 1982, pp. 53–63.
E. Lodi, F. Luccio, C. Mugnai, L. Pagli, and W. Lipski, On two-dimensional data organization 2,Fund. Inform.,2 (1979), 245–260.
F. Yao, Speed-up in dynamic programming,SIAM J. Algebraic Discrete Methods,3 (1982), 532–540.
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Communicated by D. T. Lee.
Part of this work was done when the first author was a postdoctor fellow in MSRI, Berkeley, and supported in part by NSF Grant No. 8120790. The second author was supported in part by NSF Grant No. DCR-8411945.
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Du, D., Zhang, Y. On heuristics for minimum length rectilinear partitions. Algorithmica 5, 111–128 (1990). https://doi.org/10.1007/BF01840380
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DOI: https://doi.org/10.1007/BF01840380