Fast Optimal Genome Tiling with Applications to Microarray Design and Homology Search
WABI '02 Proceedings of the Second International Workshop on Algorithms in Bioinformatics
Approximation algorithms for MAX-MIN tiling
Journal of Algorithms
Post-placement voltage island generation under performance requirement
ICCAD '05 Proceedings of the 2005 IEEE/ACM International conference on Computer-aided design
On communication protocols that compute almost privately
SAGT'11 Proceedings of the 4th international conference on Algorithmic game theory
Relations between two common types of rectangular tilings
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
On communication protocols that compute almost privately
Theoretical Computer Science
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We prove the following upper and lower bounds on the exact size of binary space partition (BSP) trees for a set of n isothetic rectangles in the plane: An upper bound of 3n-1 in general, and an upper bound of 2n-1 if the rectangles tile the underlying space. This improves the upper bounds of 4n in [V. Hai Nguyen and P. Widmayer, Binary Space Partitions for Sets of Hyperrectangles, Lecture Notes in Comput. Sci. 1023, Springer-Verlag, Berlin, 1995; F. d'Amore and P. G. Franciosa, Inform. Process. Lett., 44 (1992), pp. 255--259]. A BSP satisfying the upper bounds can be constructed in O(n log n) time. A worst-case lower bound of 2n-o(n) in general, and $\frac{3n}{2}-o(n)$ if the rectangles form a tiling. The BSP tree is one of the most popular data structures in computational geometry, and hence even "small" factor improvements of $\frac{4}{3}$ or 2 on the previously known upper bounds that we show improve the performances of applications relying on the BSP tree. As an illustration, we present improved approximation algorithms for certain dual rectangle tiling problems using our upper bounds on the size of the BSP trees.