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...

We present a new representation that is guaranteed to encode any planar triangle graph of V vertices in less than 3.67V bits. Our code improves on all prior solutions to this well studied problem and lies within 13% of the theoretical lower limit of the worst case guaranteed bound. It is based on a new encoding of the CLERS string produced by Rossignacs Edgebreaker compression [Rossignac99]. The elegance and simplicity of this technique makes it suitable for a variety of 2D and 3D triangle mesh compression applications. Simple and fast compression/decompression algorithms with linear time and space complexity are available.

The Edgebreaker compression (Rossignac, 1999; King and Rossignac, 1999) is guaranteed to encode any unlabeled triangulated planar graph of t triangles with at most 1.84t bits. It stores the graph as a CLERS string--- a sequence of t symbols from the set {C, L,E,R,S}, each represented by a 1, 2 or 3 bit code. We show here that, in practice, the string can be further compressed to between 0.91t and 1.26t bits using an entropy code. These results improve over the 2.3t bits code proposed by Keeler and Westbrook (1995) and over the various 3D triangle mesh compression techniques published recently (Gumhold and Strasser, 1998; Itai and Rodeh, 1982; Naor, 1990; Touma and Gotsman, 1988; Turan, 1984), which exhibit either larger constants or cannot guarantee a linear worst case storage complexity. The decompression proposed by Rossignac (1999) is complicated and exhibits a non-linear time complexity. The main contribution reported here is a simpler and efficient decompression algorithm, calle...

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

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 fast methodology for encoding graphs with information-theoretically minimum numbers of bits. Specifically, a graph with property is called a -graph. If satisfies certain properties, then an n-node m-edge -graph G can be encoded by a binary string X such that (1) G and X can be obtained from each other in O(n log n) time, and (2) X has at most fi(n)+o(fi(n)) bits for any continuous super-additive function fi(n) so that there are at most 2 fi(n)+o(fi(n)) distinct n-node -graphs. The methodology is applicable to general classes of graphs; this paper focuses on planar graphs. Examples of such include all conjunctions over the following groups of properties: (1) G is a planar graph or a plane graph; (2) G is directed or undirected; (3) G is triangulated, triconnected, biconnected, merely connected, or not required to be connected; (4) the nodes of G are labeled with labels from f1; : : : ; ` 1 g for ` 1 n; (5) the edges of G are labeled with labels from f1; : : : ; ` 2 ...

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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.