Embedding Ray Intersection Graphs and Global Curve Simplification

We prove that circle graphs (intersection graphs of circle chords) can be embedded as intersection graphs of rays in the plane with polynomial-size bit complexity. We use this embedding to show that the global curve simplification problem for the directed Hausdorff distance is NP-hard. In this problem, we are given a polygonal curve P and the goal is to find a second polygonal curve P ′ such that the directed Hausdorff distance from P ′ to P is at most a given constant, and the complexity of P ′ is as small as possible.

[1]  Peter Damaschke,et al.  The Hamiltonian Circuit Problem for Circle Graphs is NP-Complete , 1989, Inf. Process. Lett..

[2]  Michael Godau,et al.  A Natural Metric for Curves - Computing the Distance for Polygonal Chains and Approximation Algorithms , 1991, STACS.

[3]  Maarten Löffler,et al.  On the complexity of minimum-link path problems , 2016, J. Comput. Geom..

[4]  M. V. Kreveld,et al.  Topologically correct subdivision simplification using the bandwidth criterion , 1998 .

[5]  John Hershberger,et al.  Polygonal path simplification with angle constraints , 2005, Comput. Geom..

[6]  Mark de Berg,et al.  Streaming Algorithms for Line Simplification , 2010, Discret. Comput. Geom..

[7]  Hiroshi Imai,et al.  An optimal algorithm for approximating a piecewise linear function , 1986 .

[8]  Nabil H. Mustafa,et al.  Near-Linear Time Approximation Algorithms for Curve Simplification , 2005, Algorithmica.

[9]  Yuanjinjiang,et al.  OPTIMAL LABELLING OF UNIT INTERVAL GRAPHS , 1995 .

[10]  David H. Douglas,et al.  ALGORITHMS FOR THE REDUCTION OF THE NUMBER OF POINTS REQUIRED TO REPRESENT A DIGITIZED LINE OR ITS CARICATURE , 1973 .

[11]  Sergio Cabello,et al.  Refining the Hierarchies of Classes of Geometric Intersection Graphs , 2017 .

[12]  Michael T. Goodrich,et al.  Efficiently Approximating Polygonal Paths in Three and Higher Dimensions , 1998, SCG '98.

[13]  Robert E. Tarjan,et al.  Intersection graphs of curves in the plane , 1976, J. Comb. Theory, Ser. B.

[14]  Colin McDiarmid,et al.  Integer realizations of disk and segment graphs , 2011, J. Comb. Theory, Ser. B.

[15]  J. Kratochvil,et al.  Intersection Graphs of Segments , 1994, J. Comb. Theory, Ser. B.

[16]  Jean Cardinal,et al.  Intersection Graphs of Rays and Grounded Segments , 2016, WG.

[17]  Ilan Newman,et al.  On grid intersection graphs , 1991, Discret. Math..

[18]  R. Möhring.  Algorithmic graph theory and perfect graphs , 1986 .

[19]  Irina Mustata.  On subclasses of grid intersection graphs , 2014 .

[20]  M. Iri,et al.  Polygonal Approximations of a Curve — Formulations and Algorithms , 1988 .

[21]  F. McMorris,et al.  Topics in Intersection Graph Theory , 1987 .

[22]  Jérémie Chalopin,et al.  Planar graphs are in 1-STRING , 2007, SODA '07.

[23]  Lilian Buzer.  Optimal simplification of polygonal chain for rendering , 2007, SCG '07.

[24]  W. S. Chan,et al.  Approximation of Polygonal Curves with Minimum Number of Line Segments or Minimum error , 1996, Int. J. Comput. Geom. Appl..

[25]  Maarten Löffler,et al.  On Optimal Polyline Simplification using the Hausdorff and Fréchet Distance , 2018, SoCG.