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Conference Papers Year : 2023

On the Longest Flip Sequence to Untangle Segments in the Plane

Abstract

A set of segments in the plane may form a Euclidean TSP tour or a matching, among others. Optimal TSP tours as well as minimum weight perfect matchings have no crossing segments, but several heuristics and approximation algorithms may produce solutions with crossings. To improve such solutions, we can successively apply a flip operation that replaces a pair of crossing segments by non-crossing ones. This paper considers the maximum number D(n) of flips performed on n segments. First, we present reductions relating D(n) for different sets of segments (TSP tours, monochromatic matchings, red-blue matchings, and multigraphs). Second, we show that if all except t points are in convex position, then D(n) = O(tn^2), providing a smooth transition between the convex O(n^2) bound and the general O(n^3) bound. Last, we show that if instead of counting the total number of flips, we only count the number of distinct flips, then the cubic upper bound improves to O(n^8/3).
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Dates and versions

hal-04051496 , version 1 (30-03-2023)

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Guilherme da Fonseca, Yan Gerard, Bastien Rivier. On the Longest Flip Sequence to Untangle Segments in the Plane. WALCOM 17th International Conference and Workshops on Algorithms and Computation, Mar 2023, Hsinchu, Taiwan. pp.102-112, ⟨10.1007/978-3-031-27051-2_10⟩. ⟨hal-04051496⟩
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