cutting theorem

(definition)

Definition: For any set H of n hyperplanes in R^k, and any parameter r, 1 ≤ r≤ n, there always exists a (1/r)-cutting of size O(r^k). In two dimensions, a (1/r)-cutting of size s is a partition of the plane into s disjoint triangles, some of which are unbounded, such that no triangle in the partition intersects more than n/r lines in H. In R^k, triangles are replaced by simplices. Such a cutting can be computed in O(nr^k-1) time.

Note: From Algorithms and Theory of Computation Handbook, page 20-25, Copyright © 1999 by CRC Press LLC. Appearing in the Dictionary of Computer Science, Engineering and Technology, Copyright © 2000 CRC Press LLC.

Author: CRC-A

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Entry modified 17 December 2004.
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Cite this as:
Algorithms and Theory of Computation Handbook, CRC Press LLC, 1999, "cutting theorem", in Dictionary of Algorithms and Data Structures [online], Vreda Pieterse and Paul E. Black, eds. 17 December 2004. (accessed TODAY) Available from: http://www.nist.gov/dads/HTML/cuttingtherm.html