We consider the implementation of abstract data types for the static objects: binary tree, rooted ordered tree and balanced parenthesis expression. Our representations use an amount of space within a lower order term of the information theoretic minimum and support, in constant time, a richer set of navigational operations than has previously been considered in similar work. In the case of binary trees, for instance, we can move from a node to its left or right child or to the parent in constant time while retaining knowledge of the size of the subtree at which we are positioned. The approach is applied to produce succinct representation of planar graphs in which one can test adjacency in constant time. Keywords: abstract data type, succinct representation, binary trees, balanced parenthesis, rooted ordered trees, planar graphs. AMS subject classifications: 68P05, 68Q65 1 Introduction The binary tree is among the most fundamental of data structures. While it is often the c...

One of the most natural measures of regularity of a triangular mesh homeomorphic to the two-dimensional sphere is the fraction of its vertices having degree 6. We construct a linear-time connectivity compression scheme build upon Edgebreaker which explicitly takes advantage of regularity and prove rigorously that, for suciently large and regular meshes, it produces encodings not longer than 0:811 bits per triangle: 50% below the information-theoretic lower bound for the class of all meshes. Our method uses predictive techniques enabled by the Spirale Reversi decoding algorithm. 1 Introduction Geometric data is typically represented by meshes, often triangular. Frequently, there is need to access such data via a network connection and, in such cases, bandwidth tends to become a serious obstacle to interactivity. An obvious way out of this problem is to use compressed representations. The standard representation of a triangular mesh consists of two parts: connectivity and vertex coord...

Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a byproduct we derive: (i) a simple interpretation of the formula for the number of plane triangulations with n vertices, (ii) a linear random sampling algorithm, (iii) an explicit and simple information theory optimal encoding. 1

The canonical ordering for triconnected planar graphs is a powerful method for designing graph algorithms. This paper introduces the orderly pair of connected planar graphs, which extends the concept of canonical ordering to planar graphs not required to be triconnected. Let G be a connected planar graph. We give a linear-time algorithm that obtains an orderly pair (H

We consider the problem of representing graphs compactly while supporting queries e#ciently. In particular we describe a data structure for representing n-vertex unlabeled graphs that satisfy an O(n )-separator theorem, c < 1. The structure uses O(n) bits, and supports adjacency and degree queries in constant time, and neighbor listing in constant time per neighbor. This generalizes previous results for graphs with constant genus, such as planar graphs.

We propose a new linear time algorithm to represent a planar graph. Based on a specific triangulation of the graph, our coding takes on average 5.03 bits per node, and 3.37 bits per node if the graph is maximal. We derive from this representation that the number of unlabeled planar graphs with n nodes is at most 2 n+O(log n) where 5.007. The current lower bound is 2 n+(log n) for 4.71. We also show that almost all unlabeled and almost all labeled n-node planar graphs have at least 1.70n edges and at most 2.54n edges.

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In previous work we described a method for compactly representing graphs with small separators, which makes use of small separators, and presented preliminary experimental results. In this paper we extend the experimental results in several ways, including extensions for dynamic insertion and deletion of edges, a comparison of a variety of coding schemes, and an implementation of two applications using the representation.

Compressed graphs representation has become an attractive research topic because of its applications in the manipulation of huge Web graphs in main memory. By far the best current result is the technique by Boldi and Vigna, which takes advantage of several particular properties of Web graphs. In this paper we show that the same properties can be exploited with a different and elegant technique, built on Re-Pair compression, which achieves about the same space but much faster navigation of the graph. Moreover, the technique has the potential of adapting well to secondary memory. In addition, we introduce an approximate Re-Pair version that works efficiently with limited main memory.