Succinct Representations of Arbitrary Graphs

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(ni¾?) for any constant i¾?> 0) or too dense (namely m= i¾?(n2 i¾? i¾?) for any constant i¾?> 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 + i¾?factor of the information-theory lower bound for any arbitrarily small constant i¾?> 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(ni¾?) for any constant i¾?> 0).