Lovasz's lemma for the three-dimensional k-level of concave surfaces and its applications (1999)
| Venue: | In Proc. 40th IEEE Sympos. Found. Comput. Sci |
| Citations: | 6 - 0 self |
BibTeX
@INPROCEEDINGS{Katoh99lovasz'slemma,
author = {Naoki Katoh and Takeshi Tokuyama},
title = {Lovasz's lemma for the three-dimensional k-level of concave surfaces and its applications},
booktitle = {In Proc. 40th IEEE Sympos. Found. Comput. Sci},
year = {1999},
pages = {389--398}
}
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Abstract
1 Introduction Investigation of the combinatorial properties of k-levels of arrangements of curves and surfaces is a central topic in combinatorial geometry. The k-level problem can be considered as the parametric version of the selection problem, which finds the k smallest elements among n weighted elements. If we replace the weight of each element by a function with d \Gamma 1 parameters, we have an arrangement of n hypersurfaces in d-dimensional space, and its k-level shows the trajectory of the k-th smallest element. Note that, in this paper, we count the levels in an arrangement from the bottom, unless explicitly specified. So far, only k-levels of arrangements of hyperplanes are investigated well in the literature. Lov'asz [12] gave an O(k1=2n) upper bound for the complexity of the k-level of an arrangement of n lines in the plane by using the following lemma3:
