Grammar-based graph compression

Abstract

We present a new graph compressor that works by recursively detecting repeated substructures and representing them through grammar rules. We show that for a large number of graphs the compressor obtains smaller representations than other approaches. Specific queries such as reachability between two nodes or regular path queries can be evaluated in linear time (or quadratic times, respectively), over the grammar, thus allowing speed-ups proportional to the compression ratio.Locality: most links lead to pages within the same host (i.e., the URLs have the same prefix) andSimilarity: pages on the same host often share the same links.Due to these properties, ordering the nodes lexicographically by their URL provides an order in which similar nodes are close to each other. The WebGraph framework [4] by Boldi and Vigna is originally based on this order, but was later improved with a different order [3]. It represents the adjacency list of a graph using several layers of encodings, while retaining the ability to answer out-neighborhood queries. An out-neighborhood (in-neighborhood) query applied to a node u retrieves all nodes v such that there is an edge from u to v (from v to u). As not every graph is a web graph, a lexicographical order 4 of the node-names is not always possible or useful. Apostolico and Drovandi therefore propose to use a BFS-order [2] combined with another encoding. A different approach is proposed by Grabowski and Bieniecki [24], where contiguous blocks of the adjacency list are merged into a single ordered list, and a list of flags which are used to recover the original lists. They then encode the ordered list and use the deflate-compressor to compress both lists. To our knowledge, their method is the current state-of-the-art in compression/query trade-off, when only out-neighborhood queries are considered. The methods above have in common that they encode the adjacency list of a graph and natively only support out-neighborhood queries. The k 2 -trees of Brisaboa et al.[5] on the other hand compress the adjacency matrix of the graph. They do this by recursively partitioning it into k 2 many squares. If one of these includes only 0-values, then it is represented by a 0-leaf in the tree, and otherwise the square is partitioned further. This Quadtree-like representation is well known (see, e.g., [55]), but their succinct binary encoding is a clever new approach. The method provides access to both, in-and outneighborhood queries, and can be applied to any binary relation. We use k 2 -trees to represent the start graph of our grammars. The k 2 -tree-method was combined by Hernández and Navarro [27] with dense substructure removal, originally proposed by Buehrer and Chellapilla [6]. A dense substructure is defined by two sets of nodes U, S such that they induce a complete bipartite graph. Note that U and S need not be disjoint. The edges in these bicliques are replaced by a single "virtual node". To our knowledge, the method of [27] is the current state-of-the-art in compression/query trade-off, when in...

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