Compatible geometric matchings

Aichholzer, Oswin; Bereg, Sergey; Dumitrescu, Adrian; García, Alfredo; Huemer, Clemens; Hurtado, Ferrán; Kanō, Mikio; Márquez, Alberto; Rappaport, David; Smorodinsky, Shakhar; Souvaine, Diane L.; Urrutia, Jorge; Wood, David R.

doi:10.1016/j.comgeo.2008.12.005

Cited by 39 publications

(40 citation statements)

References 24 publications

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“…, k}. Aichholzer et al [5] proved that there is always a compatible sequence of O (log n) matchings that reconfigures any given matching into a canonical matching. Thus, the compatible matching graph, that has one node for each perfect planar matching and an edge between any two compatible matchings, is connected with diameter O (log n).…”

Section: Introductionmentioning

confidence: 98%

“…Compatible geometric matchings have been the object of study in both augmentation and reconfiguration problems. For example, the Disjoint Compatible Matching Conjecture [5] was recently solved in the affirmative [10]: every perfect planar matching M of 2n segments on 4n points can be augmented by 2n additional segments to form a PSLG that is the union of simple polygons.…”

Section: Introductionmentioning

confidence: 99%

“…, G k = G where each successive pair of PSLGs G i−1 , G i jointly satisfies some geometric constraints. In some situations, a bound on the value of k is desired as well [3][4][5][6][7][8][9]. One such solved problem is that of reconfiguring triangulations: given two triangulations T and T , one can compute a sequence of triangulations T = T 0 , .…”

Section: Introductionmentioning

confidence: 99%

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Bichromatic compatible matchings

Aloupis

Barba

Langerman

et al. 2015

Computational Geometry

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For a set R of n red points and a set B of n blue points, a BR-matching is a non-crossing geometric perfect matching where each segment has one endpoint in B and one in R. Two BR-matchings are compatible if their union is also non-crossing. We prove that, for any two distinct BR-matchings M and M , there exists a sequence of BR-matchings M = M 1 , . . . , M k = M such that M i−1 is compatible with M i . This implies the connectivity of the compatible bichromatic matching graph containing one node for each BR-matching and an edge joining each pair of compatible BR-matchings, thereby answering the open problem posed by Aichholzer et al. in [1].

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bichromatic compatible matchings

Aloupis

Barba

Langerman

et al. 2015

Computational Geometry

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“…Another related problem, that of transforming compatible perfect matchings, was very recently treated in [3]. The notion of compatible perfect matchings is defined analogously as for spanning trees.…”

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confidence: 99%

Transforming spanning trees: A lower bound

Buchin

Razen

Uno

et al. 2009

Computational Geometry

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For a planar point set we consider the graph whose vertices are the crossing-free straightline spanning trees of the point set, and two such spanning trees are adjacent if their union is crossing-free. An upper bound on the diameter of this graph implies an upper bound on the diameter of the flip graph of pseudo-triangulations of the underlying point set.We prove a lower bound of Ω(log n/ log log n) for the diameter of the transformation graph of spanning trees on a set of n points in the plane. This nearly matches the known upper bound of O (log n). If we measure the diameter in terms of the number of convex layers k of the point set, our lower bound construction is tight, i.e., the diameter is in Ω(log k) which matches the known upper bound of O (log k). So far only constant lower bounds were known.

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“…However, in many applications [5,6,22], the order of the reflex vertices and the rays is given online. If π is given (either in advance or online), then previously known best data structures require O(n 3/2−ε/2 ) time using O(n 1+ε ) space for any ε > 0.…”

Section: Applicationsmentioning

confidence: 99%

Shooting permanent rays among disjoint polygons in the plane

Ishaque

Speckmann

Tóth

2009

Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry

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Abstract. We present a data structure for ray shooting and insertion in the free space between disjoint polygonal obstacles with a total of n vertices in the plane, where each ray starts at the boundary of some obstacle. The portion of each query ray between the starting point and the first obstacle hit is inserted permanently as a new obstacle. Our data structure uses O(n log n) space and preprocessing time, and it supports m successive ray shooting and insertion queries in O((n + m) log 2 n + m log m) total time in the real RAM model of computation. We present two applications: (1) Our data structure supports efficient implementation of auto-partitions in the plane, that is, binary space partitions where each partition is done along the supporting line of an input segment. If n input line segments are fragmented into m pieces by an auto-partition, then it can now be implemented in O(n log 2 n + m log m) time. This improves the expected runtime of Patersen and Yao's classical randomized auto-partition algorithm for n disjoint line segments in the plane to O(n log 2 n). (2) If we are given disjoint polygonal obstacles with a total of n vertices in the plane, a permutation of the reflex vertices, and a half-line at each reflex vertex that partitions the reflex angle into two convex angles, then the convex partitioning algorithm draws a ray emanating from each reflex vertex in the prescribed order in the given direction until it hits another obstacle, a previous ray, or infinity. The previously best implementation (with a semidynamic ray shooting data structure) requires O(n 3/2−ε/2 ) time using O(n 1+ε ) space for any ε > 0. Our data structure improves the runtime to O(n log 2 n). Key words. ray shooting, tessellation, hierarchical decomposition AMS subject classifications. 68P05, 05C10DOI. 10.1137/1008043101. Introduction. Ray shooting data structures are a classical core component of computational geometry. They preprocess a set of objects in space such that one can efficiently find the first object hit by a query ray. A simple polygon with n vertices can be preprocessed in O(n) time to answer ray shooting queries in O(log n) time, using either a balanced geodesic triangulation [10] or a Steiner triangulation [21]. However, the free space between disjoint polygonal obstacles with a total of n vertices (e.g., n/2 disjoint line segments) cannot be handled as easily: the best ray shooting data structures can answer a query in O(n/ √ s) time using O(s) space and preprocessing, based on range searching data structures via parametric search. That is, an average query takes O( √ n) time using O(n) space (ignoring polylogarithmic factors). Geometric algorithms often rely on a ray shooting data structure, in which the result of

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Compatible geometric matchings

Cited by 39 publications

References 24 publications

Bichromatic compatible matchings

Bichromatic compatible matchings

Transforming spanning trees: A lower bound

Shooting permanent rays among disjoint polygons in the plane

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