Succinct Representation of Labeled Graphs

Abstract

These results have been published in extended form in "Algorithmica 62(1-2): 224-257 (2012)"International audienceIn many applications, the properties of an object being modeled are stored as labels on vertices or edges of a graph. In this paper, we consider succinct representation of labeled graphs. Our main results are the succinct representations of labeled and multi-labeled graphs (we consider vertex labeled planar triangulations, as well as edge labeled planar graphs and the more general $k$-page graphs) to support various label queries efficiently. The additional space cost to store the labels is essentially the information-theoretic minimum. As far as we know, our representations are the first succinct representations of labeled graphs. We also have two preliminary results to achieve the main results. First, we design a succinct representation of unlabeled planar triangulations to support the rank/select of edges in ccw (counter clockwise) order in addition to the other operations supported in previous work. Second, we design a succinct representation for a $k$-page graph when $k$ is large to support various navigational operations more efficiently. In particular, we can test the adjacency of two vertices in $O(\lg k\lg\lg k)$ time, while previous work uses $O(k)$ time

Schnyder Woods for Higher Genus Triangulated Surfaces, with Applications to Encoding

“…One of our main motivations for generalizing Schnyder woods to higher genus is the great number of existing works in the domain of graph encoding and mesh compression that take advantage of spanning tree decompositions [23,30,33], and in particular of the ones underlying Schnyder woods (and related extensions) for planar graphs [1,11,12,20,21,29]. The combinatorial properties of Schnyder woods and the related characterizations (canonical orderings [22]) for planar graphs yield efficient procedures for encoding tree structures based on multiple parenthesis words.…”

“…In this context a number of works have been proposed for the simple compression [21] or the succinct encoding [12,11] of several classes of planar graphs. More recently, this approach based on spanning tree decompositions has been further extended to design a new succinct encoding of labeled planar graphs [1]. Once again, the main ingredient is the definition of three traversal orders on the vertices of a triangulation, directly based on the properties of Schnyder woods.…”

“…We can also design a more sophisticated code that supports queries: this time we need to encode the three maps Ti using multiple parenthesis systems (as done in [12,1]). …”

“…Schnyder woods, and more generally α-orientations, received a great deal of attention [32,17,22,19]. From the combinatorial point of view, the set of Schnyder woods of a fixed triangulation has an interesting lattice structure [7,3,18,13,14], and the nice characterization in terms of spanning trees motivated a large number of applications in several domains such as graph drawing [32,22], graph coding and sampling [12,21,4,29,20,5,10,1]. Previous work focused mainly on the application and extension of the combinatorial properties of Schnyder woods to 3-connected plane graphs [17,22].…”

On the Succinct Representation of Equivalence Classes

“…Designing succinct data structures is an area of interest in theory and practice motivated by the need for storing large amount of data using the smallest space possible. For succinct representations of dictionaries, trees, arbitrary graphs and partially ordered sets see [5,20,11,19,4,10,8,3,7,18].…”