Counting Plane Graphs with Exponential Speed-Up

Razen, Andreas; Welzl, Emo

doi:10.1007/978-3-642-19391-0_3

Cited by 13 publications

(11 citation statements)

References 9 publications

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“…it is easy to see that the number of plane graphs of S is not more than 2 3n times the number of plane triangulations of S. The fact proven by Razen and Welzl [20] that the number plane graphs of S is at least 2 3n/2 times the number of plane triangulations of S, however, is far from trivial. The same can be said for the result of Sharir, Sheffer, and Welzl that the number of plane perfect matchings of S is at most O(1.1067 n ) times the number of plane triangulations of S. Still, our understanding of these relationships is quite rudimentary.…”

Section: Introductionmentioning

confidence: 97%

“…There has been some interesting work [10,9,14,20,26] on relating the sizes of those classes: E.g. it is easy to see that the number of plane graphs of S is not more than 2 3n times the number of plane triangulations of S. The fact proven by Razen and Welzl [20] that the number plane graphs of S is at least 2 3n/2 times the number of plane triangulations of S, however, is far from trivial.…”

Section: Introductionmentioning

confidence: 99%

“…But the interesting problem is to determine the size N of such a class in time that is substantially less than N . So far this has been only accomplished for the class of all plane graphs in a recent paper of Razen and Welzl [20], who manage to determine |PG(S)| in time roughly O |PG(S)| /2 3n/2 . However, their method does involve enumerating all plane triangulations of S. This paper deals with the class of plane triangulations of S. We will refer to it by T (S) and we will use tr(S) = |T (S)|.…”

Section: Introductionmentioning

confidence: 99%

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A simple aggregative algorithm for counting triangulations of planar point sets and related problems

Álvarez

Seidel

2013

Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry

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We give an algorithm that determines the number tr(S) of straight line triangulations of a set S of n points in the plane in worst case time O(n 2 2 n ). This is the the first algorithm that is provably faster than enumeration, since tr(S) is known to be Ω(2.43 n ) for any set S of n points. Our algorithm requires exponential space.The algorithm generalizes to counting all triangulations of S that are constrained to contain a given set of edges. It can also be used to compute an optimal triangulation of S (unconstrained or constrained) for a reasonably wide class of optimality criteria (that includes e.g. minimum weight triangulations). Finally, the approach can also be used for the random generation of triangulations of S according to the perfect uniform distribution.The algorithm has been implement and is substantially faster than existing methods on a variety of inputs.

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Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A simple aggregative algorithm for counting triangulations of planar point sets and related problems

Álvarez

Seidel

2013

Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry

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“…These problems have also been studied from an algorithmic point of view, deriving algorithms for enumeration or counting of the plane graphs (or other graph types) that can be embedded over a given point set (such as in [13,19]). The combinatorial upper bounds are useful for analyzing the running times of such algorithms, and also to answer questions such as "how many bits are required to represent a triangulation (or any other kind of plane graphs)?…”

Section: Introductionmentioning

confidence: 99%

Counting plane graphs: Perfect matchings, spanning cycles, and Kasteleynʼs technique

Sharir

Sheffer

Welzl

2013

Journal of Combinatorial Theory, Series A

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We derive improved upper bounds on the number of crossing-free straight-edge spanning cycles (also known as Hamiltonian tours and simple polygonizations) that can be embedded over any specific set of N points in the plane. More specifically, we bound the ratio between the number of spanning cycles (or perfect matchings) that can be embedded over a point set and the number of triangulations that can be embedded over it. The respective bounds are O(1.8181 N ) for cycles and O(1.1067 N ) for matchings. These imply a new upper bound of O(54.543 N ) on the number of crossing-free straight-edge spanning cycles that can be embedded over any specific set of N points in the plane (improving upon the previous best upper bound O(68.664 N )). Our analysis is based on Kasteleyn's linear algebra technique.

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“…This technique involves giving charges to vertices (or edges, or faces, for graphs drawn in the plane) of a graph G, and then moving these charges between various vertices (or edges, or faces) of G. The novel approach of moving such charges between vertices and edges of different graphs over the same point set originated by Sharir and Welzl in 2006 [25], in studying the maximum number of triangulations that can be embedded over a specific set of N points in the plane. Since then, this technique has been extended in [18,19,23,24] to study various combinatorial and algorithmic properties of triangulations.…”

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confidence: 99%

Counting Plane Graphs: Cross-Graph Charging Schemes

Sharir

Sheffer

2013

Graph Drawing

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Abstract. We study cross-graph charging schemes for graphs drawn in the plane. These are charging schemes where charge is moved across vertices of different graphs. Such methods have been recently applied to obtain various properties of triangulations that are embedded over a fixed set of points in the plane. We show how this method can be generalized to obtain results for various other types of graphs that are embedded in the plane. Specifically, we obtain a new bound of 1 O * ( 187.53 N ) for the maximum number of crossing-free straight-edge graphs that can be embedded over any specific set of N points in the plane (improving upon the previous best upper bound 207.85 N in Hoffmann et al. [14]). We also derive upper bounds for numbers of several other types of plane graphs (such as connected and bi-connected plane graphs), and obtain various bounds on expected vertex-degrees in graphs that are uniformly chosen from the set of all crossing-free straight-edge graphs that can be embedded over a specific point set. We then show how to apply the cross-graph charging-scheme method for graphs that allow certain types of crossings. Specifically, we consider graphs with no set of k pairwise-crossing edges (more commonly known as k-quasi-planar graphs). For k = 3 and k = 4, we prove that, for any set S of N points in the plane, the number of graphs that have a straight-edge k-quasi-planar embedding over S is only exponential in N .

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Counting Plane Graphs with Exponential Speed-Up

Cited by 13 publications

References 9 publications

A simple aggregative algorithm for counting triangulations of planar point sets and related problems

A simple aggregative algorithm for counting triangulations of planar point sets and related problems

Counting plane graphs: Perfect matchings, spanning cycles, and Kasteleynʼs technique

Counting Plane Graphs: Cross-Graph Charging Schemes

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