eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-06-08
57:1
57:16
10.4230/LIPIcs.SoCG.2020.57
article
Long Alternating Paths Exist
Mulzer, Wolfgang
1
https://orcid.org/0000-0002-1948-5840
Valtr, Pavel
2
https://orcid.org/0000-0002-3102-4166
Institut für Informatik, Freie Universität Berlin, Germany
Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Let P be a set of 2n points in convex position, such that n points are colored red and n points are colored blue. A non-crossing alternating path on P of length 𝓁 is a sequence p₁, … , p_𝓁 of 𝓁 points from P so that (i) all points are pairwise distinct; (ii) any two consecutive points p_i, p_{i+1} have different colors; and (iii) any two segments p_i p_{i+1} and p_j p_{j+1} have disjoint relative interiors, for i ≠ j.
We show that there is an absolute constant ε > 0, independent of n and of the coloring, such that P always admits a non-crossing alternating path of length at least (1 + ε)n. The result is obtained through a slightly stronger statement: there always exists a non-crossing bichromatic separated matching on at least (1 + ε)n points of P. This is a properly colored matching whose segments are pairwise disjoint and intersected by common line. For both versions, this is the first improvement of the easily obtained lower bound of n by an additive term linear in n. The best known published upper bounds are asymptotically of order 4n/3+o(n).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol164-socg2020/LIPIcs.SoCG.2020.57/LIPIcs.SoCG.2020.57.pdf
Non-crossing path
bichromatic point sets