Amortized efficiency of generating planar paths in convex position

Let S be a set of n>=3 points arranged in convex position in the plane and suppose that all points of S are labeled from 1 to n in clockwise direction. A planar path P on S is a path whose edges connect all points of S with straight line segments such that no two edges of P cross. Flipping an edge on P means that a new path P^' is obtained from P by a single edge replacement. In this paper, we provide efficient algorithms to generate all planar paths. With the help of a loopless algorithm to produce a set of 2-way binary-reflected Gray codes, the proposed algorithms generate the next planar path by means of a flip and such that the number of position changes for points in the path has a constant amortized upper bound.

[1]  Carla D. Savage,et al.  Balanced Gray Codes , 1996, Electron. J. Comb..

[2]  Edward M. Reingold,et al.  Efficient generation of the binary reflected gray code and its applications , 1976, CACM.

[3]  Franz Aurenhammer,et al.  Gray Code Enumeration of Plane Straight-Line Graphs , 2007, Graphs Comb..

[4]  Shin-ichi Tanigawa,et al.  Fast enumeration algorithms for non-crossing geometric graphs , 2008, SCG '08.

[5]  Micha Sharir,et al.  Off-line dynamic maintenance of the width of a planar point set , 1991 .

[6]  J. Ludman,et al.  Gray Code Generation for MPSK Signals , 1981, IEEE Trans. Commun..

[7]  Selim G. Akl,et al.  On Planar Path Transformation , 2006, CCCG.

[8]  Eduardo Rivera-Campo,et al.  Hamilton cycles in the path graph of a set of points in convex position , 2001, Comput. Geom..

[9]  Makoto Ohsaki,et al.  Enumerating Constrained Non-crossing Minimally Rigid Frameworks , 2008, Discret. Comput. Geom..

[10]  Prosenjit Bose,et al.  Flips in planar graphs , 2009, Comput. Geom..

[11]  Carla Savage,et al.  A Survey of Combinatorial Gray Codes , 1997, SIAM Rev..

[12]  M. C. Hernandoa,et al.  Geometric tree graphs of points in convex position + , 2003 .

[13]  Franz Aurenhammer,et al.  Transforming spanning trees and pseudo-triangulations , 2005, EuroCG.

[14]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[15]  Shin-ichi Tanigawa,et al.  Enumerating edge-constrained triangulations and edge-constrained non-crossing geometric spanning trees , 2009, Discret. Appl. Math..

[16]  Oswin Aichholzer,et al.  A quadratic distance bound on sliding between crossing-free spanning trees , 2007, Comput. Geom..

[17]  David Avis,et al.  Reverse Search for Enumeration , 1996, Discret. Appl. Math..

[18]  Takeaki Uno,et al.  Transforming spanning trees: A lower bound , 2009, Comput. Geom..

[19]  Donald E. Knuth,et al.  The Art of Computer Programming, Volume 4, Fascicle 2: Generating All Tuples and Permutations (Art of Computer Programming) , 2005 .

[20]  Franz Aurenhammer,et al.  Sequences of spanning trees and a fixed tree theorem , 2002, Comput. Geom..

[21]  Jou-Ming Chang,et al.  On the diameter of geometric path graphs of points in convex position , 2009, Inf. Process. Lett..

[22]  Selim G. Akl,et al.  Planar tree transformation: Results and counterexample , 2008, Inf. Process. Lett..

[23]  V. E. Vickers,et al.  A Technique for Generating Specialized Gray Codes , 1980, IEEE Transactions on Computers.

[24]  J. E. Ludman,et al.  A technique for generating Gray codes , 1981 .

[25]  Makoto Ohsaki,et al.  Enumerating Non-crossing Minimally Rigid Frameworks , 2007, Graphs Comb..