Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6868458 | Computational Geometry | 2018 | 27 Pages |
Abstract
The maximum number of non-crossing straight-line perfect matchings that a set of n points in the plane can have is known to be O(10.0438n) and Ωâ(3n). The lower bound, due to GarcÃa, Noy, and Tejel (2000), is attained by the double chain, which has Î(3n/nÎ(1)) such matchings. We reprove this bound in a simplified way that uses the novel notion of down-free matchings. We then apply this approach to several other constructions. As a result, we improve the lower bound. First we show that the double zigzag chain with n points has Îâ(λn) non-crossing perfect matchings with λâ3.0532. Next we analyze further generalizations of double zigzag chains - double r-chains. The best choice of parameters leads to a construction that has Îâ(νn) matchings with νâ3.0930. The derivation of this bound requires an analysis of a coupled dynamic-programming recursion between two infinite vectors.
Related Topics
Physical Sciences and Engineering
Computer Science
Computational Theory and Mathematics
Authors
Andrei Asinowski, Günter Rote,