Sublinear Graph Augmentation For Fast Query Implementation

We introduce the problem of augmenting graphs with sublinear memory in order to speed up replies to queries. As a concrete example, we focus on the following problem: the input is an (unpartitioned) bipartite graph G = (V, E). Given a query q is an element of V, the algorithm's goal is to output q's color in some legal 2-coloring of G, using few probes to the graph. All replies have to be consistent with the same 2-coloring. We show that if a linear amount of preprocessing is allowed, there is a randomized algorithm that, for any alpha, uses (m/alpha) probes and (O) over tilde(alpha) memory, where m is the number of edges in the graph. On the negative side, we show that for a natural family of algorithms that we call probe-first local computation algorithms, this trade-off is optimal even with unbounded preprocessing.We describe a randomized algorithm that replies to queries using (O) over tilde (root n/Phi(2)) probes with no additional memory on regular graphs with conductance Phi (n is the number of vertices in G). In contrast, we show that any deterministic algorithm for regular graphs that uses no memory augmentation requires a linear (in n) number of probes, even if the conductance is the largest possible. We give an algorithm for grids and tori that uses a sublinear number of probes and no memory. Last, we give an algorithm for trees that errs on a sublinear number of edges (i.e., a sublinear number of edges are monochromatic under this coloring) that uses sublinear preprocessing, memory and probes per query.