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 v ∈ V , the algorithm’s goal is to output v’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 for graphs with good mixing time, there exists a randomized algorithm that replies to queries using Õ( √ n) probes and no additional memory. In contrast, we show that any deterministic algorithm for such graphs that uses no memory augmentation requires a linear number of probes. We give an algorithm for grids and tori that uses a sublinear number of probes and no memory. On the negative side, we show that even with unbounded preprocessing, a natural family of algorithms, probe-first local computation algorithms, requires Ω(n/α) probes if the graph is augmented with α words of 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.