A Compact Encoding of Plane Triangulations with Efficient Query Supports

In this paper we give a coding scheme for plane triangulations. The coding scheme is very simple, and needs only 6n bits for each plane triangulation with n vertices. Also with additional o(n) bits it supports adjacency, degree and clockwise neighbour queries in constant time. Our scheme is based on a realizer of a plane triangulation. The best known algorithm needs only 4.35n+o(n) bits for each plane triangulation, however, within o(n) bits it needs to store a complete list of all possible triangulations having at most (log n)/4 nodes, while our algorithm is simple and does not need such a list. The second best known algorithm needs 2m+(5+1/k)n+o(m+n) bits for each (general) plane graph with m edges and 7n+o(n) bits for each plane triangulation, while our algorithm needs only 6n+o(n) bits for each plane triangulation.

[1]  György Turán,et al.  On the succinct representation of graphs , 1984, Discret. Appl. Math..

[2]  J. Ian Munro,et al.  Succinct Representation of Balanced Parentheses and Static Trees , 2002, SIAM J. Comput..

[3]  Yi-Ting Chiang,et al.  Orderly Spanning Trees with Applications , 2001, SIAM J. Comput..

[4]  Walter Schnyder,et al.  Embedding planar graphs on the grid , 1990, SODA '90.

[5]  Guy Jacobson,et al.  Space-efficient static trees and graphs , 1989, 30th Annual Symposium on Foundations of Computer Science.

[6]  Olivier Devillers,et al.  Optimal succinct representations of planar maps , 2006, SCG '06.

[7]  Xin He,et al.  A Fast General Methodology for Information-Theoretically Optimal Encodings of Graphs , 2000, SIAM J. Comput..

[8]  Avi Wigderson,et al.  Succinct Representations of Graphs , 1984, Inf. Control..

[9]  Olivier Devillers,et al.  Succinct Representation of Triangulations with a Boundary , 2005, WADS.

[10]  Jarek Rossignac,et al.  Edgebreaker: Connectivity Compression for Triangle Meshes , 1999, IEEE Trans. Vis. Comput. Graph..

[11]  Jeffery R. Westbrook,et al.  Short Encodings of Planar Graphs and Maps , 1995, Discret. Appl. Math..

[12]  Xin He,et al.  Compact Encodings of Planar Graphs via Canonical Orderings and Multiple Parentheses , 1998, ICALP.

[13]  W. T. Tutte,et al.  A Census of Planar Triangulations , 1962, Canadian Journal of Mathematics.

[14]  Hsueh-I Lu,et al.  Balanced parentheses strike back , 2008, TALG.

[15]  Xin He,et al.  Linear-Time Succinct Encodings of Planar Graphs via Canonical Orderings , 1999, SIAM J. Discret. Math..

[16]  Nicolas Bonichon,et al.  An Information-Theoretic Upper Bound of Planar Graphs Using Triangulation , 2003, STACS.

[17]  David Richard Clark,et al.  Compact pat trees , 1998 .

[18]  Yi-Ting Chiang,et al.  Orderly spanning trees with applications to graph encoding and graph drawing , 2001, SODA '01.

[19]  Dominique Poulalhon,et al.  Optimal Coding and Sampling of Triangulations , 2003, ICALP.

[20]  Venkatesh Raman,et al.  Succinct representation of balanced parentheses, static trees and planar graphs , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.