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A better upper bound on the number of triangulations of a planar point set

Francisco Santos, Raimund Seidel
2003 Journal of combinatorial theory. Series A  
We show that a point set of cardinality n in the plane cannot be the vertex set of more than 59^n O(n^-6) straight-edge triangulations of its convex hull.  ...  This improves the previous upper bound of 276.75^n.  ...  We thank the organizers of this meeting, the University of Crete and the Anogia Academic Village for the invitation and financial support.  ... 
doi:10.1016/s0097-3165(03)00002-5 fatcat:7yti4f63nncpdcg6z6tncnhjcm

A lower bound on the number of triangulations of planar point sets

Oswin Aichholzer, Ferran Hurtado, Marc Noy
2004 Computational geometry  
We show that the number of straight-edge triangulations exhibited by any set of n points in general position in the plane is bounded from below by (2.33 n ).  ...  We highly appreciate the useful remarks and suggestions of two anonymous referees. Especially Eq. (7) , which lead to substantial improved bounds, is based on their valuable contribution.  ...  Introduction A triangulation of a finite planar point set S is a maximal non-crossing straight-edge graph with vertex set S.  ... 
doi:10.1016/j.comgeo.2004.02.003 fatcat:rvtqvkocf5ax5lmirhph2o5c7e

The Number of Triangulations on Planar Point Sets [chapter]

Emo Welzl
Graph Drawing  
We give a brief account of results concerning the number of triangulations on finite point sets in the plane, both for arbitrary sets and for specific sets such as the n × n integer lattice.  ...  yields the claimed bound on the number of triangulations.  ...  In [5] it is shown that the set of all triangulations of a point set can be enumerated in time O(t · poly(n)), where t is the number of triangulations.  ... 
doi:10.1007/978-3-540-70904-6_1 dblp:conf/gd/Welzl06 fatcat:p7dyyjh6jrbmncysof7dxahepm

On the average length of Delaunay triangulations

R. C. Chang, R. C. T. Lee
1984 BIT Numerical Mathematics  
We shall show that on the average, the total length of a Delaunay triangulation is of the same order as that of a minimum triangulation, under the assumption that our points are drawn from a homogeneous  ...  planar Poisson point distribution.  ...  Given a set P of N points, any triangulation of P has the same number of triangles, N t = 2(N-1)-Nh, and the same number of edges, N e = 3(N-1)-N h, where N h is the number of points on the convex hull  ... 
doi:10.1007/bf02136025 fatcat:cjbps7sqnjbsdfj2dh755bsvum

Incremental and batch planar simplification of dense point cloud maps

T. Whelan, L. Ma, E. Bondarev, P.H.N. de With, J. McDonald
2015 Robotics and Autonomous Systems  
Our results also show that the proposed planar simplification and triangulation algorithm removes more than 90% of the input planar points, leading to a triangulation with only 10% of the original quantity  ...  In this paper we present a method for incremental planar segmentation of a gradually expanding point cloud map and a method for efficient triangulation and texturing of planar surface segments.  ...  The polygon-based and point-based methods offer a trade-off depending on the desired number of triangles or the intended use of the final triangulation.  ... 
doi:10.1016/j.robot.2014.08.019 fatcat:rhfp5xdebvcozcapn6f5bmmg2a

Conceptual Framework for Finding Approximations to Minimum Weight Triangulation and Traveling Salesman Problem of Planar Point Sets

Marko Dodig, Milton Smith
2020 International Journal of Advanced Computer Science and Applications  
We provide motivation for our research and introduce the fields of triangulation and polygonization of planar point sets as theoretical bases of our approach, namely, we present the Isoperimetric Inequality  ...  We introduce a novel Conceptual Framework for finding approximations to both Minimum Weight Triangulation (MWT) and optimal Traveling Salesman Problem (TSP) of planar point sets.  ...  Triangulations, on the other hand, represent the most intuitive way one can partition a planar point set.  ... 
doi:10.14569/ijacsa.2020.0110403 fatcat:kswkleldebbmzmwzchflbwdosi

Diagonal Flips in Labelled Planar Triangulations

Zhicheng Gao, Jorge Urrutia, Jianyu Wang
2001 Graphs and Combinatorics  
Recently Komuro gives a linear bound on the maximum number of diagonal flips needed for such a transformation.  ...  A classical result of Wagner states that any two (unlabelled) planar triangulations with the same number of vertices can be transformed into each other by a finite sequence of diagonal flips.  ...  The graph of triangulations GT (P n ) of a point set P n is the graph whose vertex set is the set of triangulations of P n .  ... 
doi:10.1007/s003730170006 fatcat:lelsaokgirfw7mji5o7t7z77ku

A QPTAS for the Base of the Number of Triangulations of a Planar Point Set [article]

Marek Karpinski, Andrzej Lingas, Dzmitry Sledneu
2014 arXiv   pre-print
The number of triangulations of a planar n point set is known to be c^n, where the base c lies between 2.43 and 30.  ...  We present the first quasi-polynomial approximation scheme for the base of the number of triangulations of a planar point set.  ...  Acknowledgments We thank Victor Alvarez, Artur Czumaj, Peter Floderus, Miroslaw Kowaluk, Christos Levcopoulos and Mia Persson for preliminary discussions on counting the number of triangulations of a planar  ... 
arXiv:1411.0544v3 fatcat:hocdoh75cngdlh6thwjvtlrf6m

Experimental results on quadrangulations of sets of fixed points

Prosenjit Bose, Suneeta Ramaswami, Godfried Toussaint, Alain Turki
2002 Computer Aided Geometric Design  
We consider the problem of obtaining "nice" quadrangulations of planar sets of points.  ...  on the dual graph of the triangulations.  ...  A set of points admits a quadrangulation without Steiner points if and only if the number of points on the convex hull is even.  ... 
doi:10.1016/s0167-8396(02)00133-4 fatcat:cisaq4vpjzhg5dcocuv6psbs64

Novel parallel algorithm for constructing Delaunay triangulation based on a twofold-divide-and-conquer scheme

Wenzhou Wu, Yikang Rui, Fenzhen Su, Liang Cheng, Jiechen Wang
2014 GIScience & Remote Sensing  
To increase the efficiency when processing large data sets, a novel parallel algorithm is proposed for constructing the Delaunay triangulation of a planar point set based on a twofold-divide-and-conquer  ...  This algorithm automatically divides the planar point set into several non-overlapping subsets along the x-axis and y-axis directions alternately, according to the number of points and their spatial distribution  ...  Funding This work is supported by the National Natural Science Foundation of China [grant number 41371017] and the Program for New Century Excellent Talents in University (NCET-13-0280).  ... 
doi:10.1080/15481603.2014.946666 fatcat:jx6l4qc3xfbdfbe2xt4pnt6wcy

Planar simplification and texturing of dense point cloud maps

Lingni Ma, Thomas Whelan, Egor Bondarev, Peter H. N. de With, John McDonald
2013 2013 European Conference on Mobile Robots  
Experimental results show that our algorithm removes more than 90% of the input planar points, leading to a triangulation with only 10% of the original amount of triangles per planar segment, improving  ...  In this paper we present a method for efficient triangulation and texturing of planar surfaces in large point clouds.  ...  The polygonbased and point-based methods offer a trade-off depending on the desired number of triangles or the intended use of the final triangulation.  ... 
doi:10.1109/ecmr.2013.6698837 dblp:conf/ecmr/MaWBWM13 fatcat:auu2sw2atfcjfg4kphtx6wtkzy

Succinct Geometric Indexes Supporting Point Location Queries [chapter]

Prosenjit Bose, Eric Y. Chen, Meng He, Anil Maheshwari, Pat Morin
2009 Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms  
Our first and main result is a succinct geometric index that can answer point location queries, a fundamental problem in computational geometry, on planar triangulations in O(lg n) time 1 .  ...  in the data set permuted and stored elsewhere as a sequence.  ...  Denny and Sohler [15] showed how to encode the connectivity information of a planar triangulation by permuting its vertex set: Point Location in Planar Triangulations In this section, we show how  ... 
doi:10.1137/1.9781611973068.70 fatcat:ocva2u2vwvclxerdvnedmwxqi4

Succinct geometric indexes supporting point location queries

Prosenjit Bose, Eric Y. Chen, Meng He, Anil Maheshwari, Pat Morin
2012 ACM Transactions on Algorithms  
Our first and main result is a succinct geometric index that can answer point location queries, a fundamental problem in computational geometry, on planar triangulations in O(lg n) time 1 .  ...  in the data set permuted and stored elsewhere as a sequence.  ...  Denny and Sohler [15] showed how to encode the connectivity information of a planar triangulation by permuting its vertex set: Point Location in Planar Triangulations In this section, we show how  ... 
doi:10.1145/2151171.2151173 fatcat:aop6jhrcqrfd3lklm62oewp35u

A Note On Universal Point Sets for Planar Graphs [article]

Manfred Scheucher and Hendrik Schrezenmaier and Raphael Steiner
2019 arXiv   pre-print
We investigate which planar point sets allow simultaneous straight-line embeddings of all planar graphs on a fixed number of vertices.  ...  Moreover, we provide a set of 49 planar 11-vertex graphs which cannot be simultaneously drawn on any set of 11 points.  ...  We denote by f p (n) the size of a minimal n-universal set (for planar graphs), and by f s (n) the size of a minimal n-universal set for stacked triangulations, where stacked triangulations (a.k.a. planar  ... 
arXiv:1811.06482v3 fatcat:s7x2wkd5w5ggpjdkws4szxloqy

Flips in planar graphs

Prosenjit Bose, Ferran Hurtado
2009 Computational geometry  
We review results concerning edge flips in planar graphs concentrating mainly on various aspects of the following problem: Given two different planar graphs of the same size, how many edge flips are necessary  ...  We overview both the combinatorial perspective (where only a combinatorial embedding of the graph is specified) and the geometric perspective (where the graph is embedded in the plane, vertices are points  ...  [53] show that given two near-triangulations defined on the same set of n points, the number of flips sufficient to convert one to the other is bounded by the number of intersections between the edges  ... 
doi:10.1016/j.comgeo.2008.04.001 fatcat:fxq5ql3imzftpoh5naidvy5e3q
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