Long Alternating Paths Exist

Authors Wolfgang Mulzer , Pavel Valtr

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Wolfgang Mulzer
  • Institut für Informatik, Freie Universität Berlin, Germany
Pavel Valtr
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic


This work was initiated at the second DACH workshop on Arrangements and Drawings which took place 21. - 25. January 2019 at Schloss St. Martin, Graz, Austria. We would like to thank the organizers and all the participants of the workshop for creating a conducive research atmosphere and for stimulating discussions. Part of this work was done on the Seventh Annual Workshop on Geometry and Graphs, Bellairs Research Institute, Holetown, Barbados, 10. - 15. March 2019. We also thank Zoltán Király for pointing out the reference [Clemens Müllner and Andrew Ryzhikov, 2019] to us.

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Wolfgang Mulzer and Pavel Valtr. Long Alternating Paths Exist. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 57:1-57:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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).

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Non-crossing path
  • bichromatic point sets


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