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∈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 α, uses O(m/α) probes and Õ(α) 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 (formula presented) probes with no additional memory on regular graphs with conductance Φ (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.