Counting and Enumerating Crossing-free Geometric Graphs

Wettstein, Manuel

doi:10.1145/2582112.2582145

Cited by 15 publications

(8 citation statements)

References 29 publications

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“…At a different level of complexity, the idea of working ahead occurs in an algorithm of Wettstein [9,Section 6]. This trick, credited to Emo Welzl, is used to achieve polynomial delay between successive solutions when enumerating non-crossing perfect matching of a planar point set, despite having to build up a network with exponential space in a preprocessing phase.…”

Section: Working Aheadmentioning

confidence: 99%

Loopless Gray code enumeration and the Tower of Bucharest

Herter

Rote

2018

Theoretical Computer Science

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We give new algorithms for generating all n-tuples over an alphabet of m letters, changing only one letter at a time (Gray codes). These algorithms are based on the connection with variations of the Towers of Hanoi game. Our algorithms are loopless, in the sense that the next change can be determined in a constant number of steps, and they can be implemented in hardware. We also give another family of loopless algorithms that is based on the idea of working ahead and saving the work in a buffer.

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Section: Working Aheadmentioning

confidence: 99%

Loopless Gray code enumeration and the Tower of Bucharest

Herter

Rote

2018

Theoretical Computer Science

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“…It is a challenging problem to determine the number of configurations faster than listing all such configurations (i.e., count faster than enumerate) [3]. Exponential-time algorithms have been developed for triangulations [4], planar graphs [22], and matchings [28] that count these structures exponentially faster than the number of structures. Recently, it has been shown that the number of triangulations on n points in the plane can be counted in subexponential time [19], and this result extends to counting noncrossing prefect matchings, spanning trees, spanning cycles, 3-regular graphs, and more.…”

Section: Counting Algorithmmentioning

confidence: 99%

Convex Polygons in Geometric Triangulations

Dumitrescu¹,

Tóth

2017

Combinator. Probab. Comp.

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We show that the maximum number of convex polygons in a triangulation of n points in the plane is O(1.5029 n ). This improves an earlier bound of O(1.6181 n ) established by van Kreveld, Löffler, and Pach (2012) and almost matches the current best lower bound of Ω(1.5028 n ) due to the same authors. Given a planar straight-line graph G with n vertices, we also show how to compute efficiently the number of convex polygons in G.

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“…Although there has been significant research on counting non-crossing configurations in the plane, including matchings, simple polygons, spanning trees, triangulations, and pseudotriangulations, the complexity of these problems has remained undetermined. Research on these problems has instead focused on determining the number of configurations for special classes of point sets [20,8,24], bounding the number of configurations as a function of the number of points [2,4,39,37,17,38,1,33,3,36,21,10], developing exponential-or subexponential-time counting algorithms [12,7,42,5,30,13], or finding faster approximations [6,27].…”

Section: Introductionmentioning

confidence: 99%