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 $G' = (V,E')$ of $G$. Namely, $G'$ should be connected and $|E'|$ should be upper bounded by $(1+\epsilon)|V|$ for a given parameter $\epsilon > 0$. 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 $G'$ 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 $\Omega(\sqrt{|V|})$. 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.