Local Computation Algorithms for (Minimum) Spanning Trees on Expander Graphs

We study \emph{local computation algorithms (LCAs)} for constructing spanning trees. In this setting, the goal is to locally determine, for each edge $e \in E$, whether it belongs to a spanning tree $T$ of the input graph $G$, where $T$ is defined implicitly by $G$ and the randomness of the algorithm. It is known that LCAs for spanning trees do not exist in general graphs, even for simple graph families. We identify a natural and well-studied class of graphs -- \emph{expander graphs} -- that do admit \emph{sublinear-time} LCAs for spanning trees. This is perhaps surprising, as previous work on expanders only succeeded in designing LCAs for \emph{sparse spanning subgraphs}, rather than full spanning trees. We design an LCA with probe complexity $O\left(\sqrt{n}\left(\frac{\log^2 n}{\phi^2} + d\right)\right)$ for graphs with conductance at least $\phi$ and maximum degree at most $d$ (not necessarily constant), which is nearly optimal when $\phi$ and $d$ are constants, since $\Omega(\sqrt{n})$ probes are necessary even for expanders. Next, we show that for the natural class of \emph{\ER graphs} $G(n, p)$ with $np = n^{\delta}$ for any constant $\delta > 0$ (which are expanders with high probability), the $\sqrt{n}$ lower bound can be bypassed. Specifically, we give an \emph{average-case} LCA for such graphs with probe complexity $\tilde{O}(\sqrt{n^{1 - \delta}})$. Finally, we extend our techniques to design LCAs for the \emph{minimum spanning tree (MST)} problem on weighted expander graphs. Specifically, given a $d$-regular unweighted graph $\bar{G}$ with sufficiently strong expansion, we consider the weighted graph $G$ obtained by assigning to each edge an independent and uniform random weight from $\{1,\ldots,W\}$, where $W = O(d)$. We show that there exists an LCA that is consistent with an exact MST of $G$, with probe complexity $\tilde{O}(\sqrt{n}d^2)$.