Succinct Representations of Arbitrary Graphs

• Authors: Arash Farzan;J. Ian Munro
• Venue: ESA '08 Proceedings of the 16th annual European symposium on Algorithms
• Year: 2008

We consider the problem of encoding a graph with nvertices and medges compactly supporting adjacency, neighborhood and degree queries in constant time in the logn-bit word RAM model. The adjacency query asks whether there is an edge between two vertices, the neighborhood query reports the neighbors of a given vertex in constant time per neighbor, and the degree query reports the number of incident edges to a given vertex.We study the problem in the context of succinctness, where the goal is to achieve the optimal space requirement as a function of nand m, to within lower order terms. We prove a lower bound in the cell probe model that it is impossible to achieve the information-theory lower bound within lower order terms unless the graph is too sparse (namely m= o(n茂戮驴) for any constant 茂戮驴 0) or too dense (namely m= 茂戮驴(n2 茂戮驴 茂戮驴) for any constant 茂戮驴 0).Furthermore, we present a succinct encoding for graphs for all values of n,msupporting queries in constant time. The space requirement of the representation is always within a multiplicative 1 + 茂戮驴factor of the information-theory lower bound for any arbitrarily small constant 茂戮驴 0. This is the best achievable space bound according to our lower bound where it applies. The space requirement of the representation achieves the information-theory lower bound tightly within lower order terms when the graph is sparse (m= o(n茂戮驴) for any constant 茂戮驴 0).