Sketching Distances in Monotone Graph Classes

We study the problems of adjacency sketching, small-distance sketching, and approximate distance threshold sketching for monotone classes of graphs. The problem is to obtain randomized sketches of the vertices of any graph G in the class, so that adjacency, exact distance thresholds, or approximate distance thresholds of two vertices u, v can be decided (with high probability) from the sketches of u and v, by a decoder that does not know the graph. The goal is to determine when sketches of constant size exist. We show that, for monotone classes of graphs, there is a strict hierarchy: approximate distance threshold sketches imply small-distance sketches, which imply adjacency sketches, whereas the reverse implications are each false. The existence of an adjacency sketch is equivalent to the condition of bounded arboricity, while the existence of small-distance sketches is equivalent to the condition of bounded expansion. Classes of constant expansion admit approximate distance threshold sketches, while a monotone graph class can have arbitrarily small non-constant expansion without admitting an approximate distance threshold sketch.