Local algorithms for sparse spanning graphs

We initiate the study of the problem of designing sublinear-time (local) algorithms that, given an edge (u, v) in a connected graph G = (V,E), decide whether (u, v) belongs to a sparse spanning graph G0 = (V,E0) of G. Namely, G0 should be connected and |E0| should be upper bounded by (1 + ∈)|V | for a given parameter ∈ > 0. To this end the algorithms may query the incidence relation of the graph G, and we seek algorithms whose query complexity and running time (per given edge (u, v)) is as small as possible. Such an algorithm may be randomized but (for a fixed choice of its random coins) its decision on different edges in the graph should be consistent with the same spanning graph G0 and independent of the order of queries. We first show that for general (bounded-degree) graphs, the query complexity of any such algorithm must be Ω (√ |V |). This lower bound holds for graphs that have high expansion. We then turn to design and analyze algorithms both for graphs with high expansion (obtaining a result that roughly matches the lower bound) and for graphs that are (strongly) non-expanding (obtaining results in which the complexity does not depend on |V |). The complexity of the problem for graphs that do not fall into these two categories is left as an open question.