Succinct encoding of arbitrary graphs ?
We consider the problem of encoding graphs with n vertices and m  edges compactly supporting adjacency, neighborhood and degree queries in constant time in the ? ( log n ) -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 n and m , to within lower order terms. We prove a lower bound in the cell probe model indicating it is impossible to achieve the information-theory lower bound up to lower order terms unless the graph is either too sparse (namely, m=o(n?) m = o ( n ? ) for any constant ?>0 ? > 0 ) or too dense (namely m=?(n2??) m = ? ( n 2 ? ? ) for any constant ?>0 ? > 0 ). Furthermore, we present a succinct encoding of graphs supporting aforementioned queries in constant time. The space requirement of the encoding is within a multiplicative 1+? 1 + ? factor of the information-theory lower bound for any arbitrarily small constant ?>0 ? > 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 where the graph is very sparse (m=o(n?) m = o ( n ? ) for any constant ?>0 ? > 0 ), or very dense ( m > n 2 / lg 1 ? ? n for an arbitrarily small constant ?>0 ? > 0 ).
succinct representation compression graph encoding