Edgebreaker is a simple scheme for compressing the triangle/vertex incidence graphs (sometimes called connectivity or topology) of three-dimensional triangle meshes. Edgebreaker improves upon the worst case storage required by previously reported schemes, most of which require O(nlogn) bits to store the incidence graph of a mesh of n triangles. Edgebreaker requires only 2n bits or less for simple meshes and can also support fully general meshes by using additional storage per handle and hole. Edgebreaker's compression and decompression processes perform the same traversal of the mesh from one triangle to an adjacent one. At each stage, compression produces an op-code describing the topological relation between the current triangle and the boundary of the remaining part of the mesh. Decompression uses these op-codes to reconstruct the entire incidence graph. Because Edgebreaker's compression and decompression are independent of the vertex locations, they may be combined with a variety of vertex-compressing techniques that exploit topological information about the mesh to better estimate vertex locations. Edgebreaker may be used to compress the connectivity of an entire mesh bounding a 3D polyhedron or the connectivity of a triangulated surface patch whose boundary needs not be encoded. Its superior compression capabilities, the simplicity of its implementation, and its versatility make Edgebreaker particularly suitable for the emerging 3D data exchange standards for interactive graphic applications. The paper also offers a comparative survey of the rapidly growing field of geometric compression.

. We consider the problem of coding planar graphs by binary strings. Depending on whether O(1)-time queries for adjacency and degree are supported, we present three sets of coding schemes which all take linear time for encoding and decoding. The encoding lengths are significantly shorter than the previously known results in each case. 1 Introduction This paper investigates the problem of encoding a graph G with n nodes and m edges into a binary string S. This problem has been extensively studied with three objectives: (1) minimizing the length of S, (2) minimizing the time needed to compute and decode S, and (3) supporting queries efficiently. A number of coding schemes with different trade-offs have been proposed. The adjacency-list encoding of a graph is widely useful but requires 2mdlog ne bits. (All logarithms are of base 2.) A folklore scheme uses 2n bits to encode a rooted n-node tree into a string of n pairs of balanced parentheses. Since the total number of such trees is...

We discuss space-efficient encoding schemes for planar graphs and maps. Our results improve on the constants of previous schemes and can be achieved with simple encoding algorithms. They are near-optimal in number of bits per edge. 1 Introduction In this paper we discuss space-efficient binary encoding schemes for several classes of unlabeled connected planar graphs and maps. In encoding a graph we must encode the incidences among vertexes and edges. By maps we understand topological equivalence classes of planar embeddings of planar graphs. In encoding a map we are required to encode the topology of the embedding i.e., incidences among faces, edges, and vertexes, as well as the graph. Each map is an embedding of a unique graph, but a given graph may have multiple embeddings. Hence maps must require more bits to encode than graphs in some average sense. There are a number of recent results on space-efficient encoding. A standard adjacency list encoding of an unlabeled graph G requires...

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...

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 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 ...

Abstract. Let G be an embedded planar undirected graph that has n vertices, m edges, and f faces but has no self-loop or multiple edge. If G is triangulated, we can encode it using 4 m − 1 bits, improving on the best previous bound of about 1.53m bits. In case exponential time 3 is acceptable, roughly 1.08m bits have been known to suffice. If G is triconnected, we use at most (2.5 + 2 log 3) min{n, f} −7 bits, which is at most 2.835m bits and smaller than the best previous bound of 3m bits. Both of our schemes take O(n) time for encoding and decoding.

The Edgebreaker compression technique, introduced in [11], encodes any unlabeled triangulated planar graph of t triangles using a string of 2t bits. The string contains a sequence of t letters from the set {C, L, E, R, S} and 50% of these letters are C. Exploiting constraints on the sequence, we show that the string may in practice be further compressed to 1.6t bits using model independent codes and even more using model specific entropy codes. These results improve over the 2.3t bits needed by Keeler and Westbrook [6] and over the various 3D triangle mesh compression techniques published recently, which all exhibit larger constants or non-linear worst case storage costs. As in [11], we compress the mesh using a spiraling triangle-spanning tree and generate the same sequence of letters. Edgebreaker's decompression uses a look-ahead procedure to identify the third vertex of split triangles (S letter) by counting letter occurrences in the remaining part of the sequences. We introduce her...

We consider graph compression in terms of graph families. In particular, we show that graphs of bounded genus can be compressed to O(n) bits, where n is the number of vertices. We identify a property based on separators that makes O(n)-bit compression possible for some graphs of bounded arboricity. 1 Introduction Graph representation as a data compression problem Lossless data compression is a process of representing a body of data by another body of data of smaller size from which the original data can be completely reconstructed. In the past thirty years a great deal of work has been done on the theory and practice of text compression (e.g., printed text or program source code) and of digitized data (e.g., voice or images). In fact, data compression has become a well-established subject in computer science, information theory, and communication theory. In contrast, very little has been done on compressing graphs. Since graphs are encountered everywhere and are often of very la...

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.