Bipartite grammar-based representations of large sparse binary matrices: Framework and transforms

In this paper, we introduce a new concept called context-free bipartite grammar (CFBG) and present a framework wherein large sparse binary matrices can be compactly represented by CFBGs. Similar to the traditional concept of context-free grammar (CFG), a CFBG consists of a set of production rules. Unlike CFGs, however, the right member of each production rule in a CFBG is a labeled bipartite graph with each edge labeled either as a variable or terminal symbol. Given a CFBG, start with its initial variable and repeatedly expand each variable labeled edge by first deleting that edge and then inserting in some manner all edges contained in the right member of that variable. The CFBG is admissible if the edge expansion process leads to a unique bipartite graph containing only terminal symbol labeled edges, in which case the CFBG is said to represent the matrix equal to the biadjacency matrix of the unique graph. Two bipartite grammar transforms, a sequential D-neighborhood pairing transform and an iterative pairing transform (IPT), are further presented to convert any binary matrix into a CFBG representing it. Experiments show that compared with popular sparse matrix storage methods such as compressed row storage and quadtree, CFBGs obtained by IPT can reduce the storage of sparse matrices significantly (by a factor of as much as 68).