On the Reflexivity of Point Sets

We introduce a new measure for planar point sets S. Intuitively, it describes the combinatorial distance from a convex set: The reflexivity ρ(S) of S is given by the smallest number of reflex vertices in a simple polygonalization of S. We prove various combinatorial bounds and provide efficient algorithms to compute reflexivity, both exactly (in special cases) and approximately (in general). Our study naturally takes us into the examination of some closely related quantities, such as the convex cover number ϰ1(S) of a planar point set, which is the smallest number of convex chains that cover S, and the convex partition number ϰ2(S), which is given by the smallest number of disjoint convex chains that cover S. We prove that it is NP-complete to determine the convex cover or the convex partition number, and we give logarithmic-approximation algorithms for determining each.

[1]  Tamal K. Dey,et al.  A simple provable algorithm for curve reconstruction , 1999, SODA '99.

[2]  Subhash Suri,et al.  Applications of a semi-dynamic convex hull algorithm , 1992, BIT Comput. Sci. Sect..

[3]  Bernard Chazelle,et al.  On the convex layers of a planar set , 1985, IEEE Trans. Inf. Theory.

[4]  J. Mark Keil,et al.  Polygon Decomposition , 2000, Handbook of Computational Geometry.

[5]  Joseph S. B. Mitchell,et al.  Counting Convex Polygons in Planar Point Sets , 1995, Inf. Process. Lett..

[6]  Marc Noy,et al.  Triangulations, Visibility Graph and Reflex Vertices of a Simple Polygon , 1996, Comput. Geom..

[7]  David Eppstein,et al.  The Crust and the beta-Skeleton: Combinatorial Curve Reconstruction , 1998, Graph. Model. Image Process..

[8]  Xiaojun Shen,et al.  Covering convex sets with non-overlapping polygons , 1990, Discret. Math..

[9]  Oswin Aichholzer,et al.  The point set order type data base: A collection of applications and results , 2001, CCCG.

[10]  David Rappaport,et al.  On Convex Decompositions of Points , 2000, JCDCG.

[11]  Gerhard J. Woeginger,et al.  Angle-Restricted Tours in the Plane , 1997, Comput. Geom..

[12]  Franz Aurenhammer,et al.  Enumerating order types for small sets with applications , 2001, SCG '01.

[13]  Esther M. Arkin,et al.  On the Reflexivity of Point Sets , 2001, WADS.

[14]  Masatsugu Urabe.  On a partition of point sets into convex polygons , 1997, CCCG.

[15]  János Pach,et al.  A Generalization of the Erdos - Szekeres Theorem to Disjoint Convex Sets , 1998, Discret. Comput. Geom..

[16]  Pankaj K. Agarwal,et al.  Polygon decomposition for efficient construction of Minkowski sums , 2000, Comput. Geom..

[17]  Masatsugu Urabe.  on a Partition into Convex Polygons , 1996, Discret. Appl. Math..

[18]  J. O´Rourke,et al.  Computational Geometry in C: Arrangements , 1998 .

[19]  Marc Noy,et al.  Lower bounds on the number of crossing-free subgraphs of KN , 2000, Comput. Geom..

[20]  Giovanni Manzini,et al.  Algebraic Techniques in Communication Complexity , 1995, Inf. Process. Lett..

[21]  Kiyoshi Hosono,et al.  On the number of disjoint convex quadrilaterals for a planar point set , 2001, Comput. Geom..

[22]  B. Chazelle.  Computational geometry and convexity , 1980 .

[23]  Masatsugu Urabe.  Partitioning point sets in space into disjoint convex polytopes , 1999, Comput. Geom..

[24]  Franz Aurenhammer,et al.  Enumerating Order Types for Small Point Sets with Applications , 2002, Order.

[25]  Kurt Mehlhorn,et al.  Curve reconstruction: connecting dots with good reason , 1999, SCG '99.

[26]  Kurt Mehlhorn,et al.  Fast Triangulation of the Plane with Respect to Simple Polygons , 1985, Inf. Control..

[27]  J. Horton.  Sets with No Empty Convex 7-Gons , 1983, Canadian Mathematical Bulletin.

[28]  Joseph S. B. Mitchell,et al.  Generating Random Polygons with Given Vertices , 1996, Comput. Geom..

[29]  Subhash Suri,et al.  Surface approximation and geometric partitions , 1994, SODA '94.

[30]  Pankaj K. Agarwal,et al.  Ray shooting and other applications of spanning trees with low stabbing number , 1992, SCG '89.

[31]  Martin Held,et al.  Heuristics for the Generation of Random Polygons , 1996, CCCG.

[32]  Joseph O'Rourke,et al.  Computational Geometry in C. , 1995 .

[33]  David Lichtenstein,et al.  Planar Formulae and Their Uses , 1982, SIAM J. Comput..

[34]  Alok Aggarwal,et al.  The angular-metric traveling salesman problem , 1997, SODA '97.

[35]  Subhash Suri,et al.  A pedestrian approach to ray shooting: shoot a ray, take a walk , 1993, SODA '93.