Complexity results on untangling red-blue matchings

Das, Arun Kumar; Das, Sandip; Fonseca, Guilherme D. da; Gerard, Yan; Bastien, Rivier,

doi:10.1016/j.comgeo.2022.101974

Cited by 2 publications

(1 citation statement)

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“…Using only removal choice: If P is in convex position, then by using O(n) flips we can untangle a TSP tour [24,29], as well as a red-blue matching [7], while the best known bound for trees is O(n log n) [7]. If instead of convex position, we have colinear red points in a red-blue matching, then O(n 2 ) flips suffice [7,13].…”

Section: Introductionmentioning

confidence: 99%

On the Longest Flip Sequence to Untangle Segments in the Plane

Fonseca

Gerard

Bastien

2023

Lecture Notes in Computer Science

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A (multi)set of segments in the plane may form a TSP tour, a matching, a tree, or any multigraph. If two segments cross, then we can reduce the total length with the following flip operation. We remove a pair of crossing segments, and insert a pair of non-crossing segments, while keeping the same vertex degrees. The goal of this paper is to devise efficient strategies to flip the segments in order to obtain crossing-free segments after a small number of flips. Linear and near-linear bounds on the number of flips were only known for segments with endpoints in convex position. We generalize these results, proving linear and near-linear bounds for cases with endpoints that are not in convex position. Our results are proved in a general setting that applies to multiple problems, using multigraphs and the distinction between removal and insertion choices when performing a flip.

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