Local Computation Algorithms for (Minimum) Spanning Trees on Expander Graphs
Pan Peng, Yuyang Wang
TL;DR
This work investigates the local computation paradigm for constructing spanning trees and minimum spanning trees on graphs with strong expansion properties. It develops a sublinear LCA for spanning trees on expander graphs and an average-case LCA for Erdős–Rényi graphs, leveraging random walks, anchors, and a core-tree structure to enable on-demand, consistent edge queries. The MST extension uses a percolation-based, layered approach on weighted expanders with random edge weights, achieving exact MST consistency under a non-adversarial weight model with sublinear probes. Together, these results show that strong graph expansion and random weight randomness enable efficient, locally computable global structures with significant implications for scalable network design and distributed computation.
Abstract
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}{φ^2} + d\right)\right)$ for graphs with conductance at least $ φ$ and maximum degree at most $ d $ (not necessarily constant), which is nearly optimal when $φ$ and $d$ are constants, since $Ω(\sqrt{n})$ probes are necessary even for expanders. Next, we show that for the natural class of \emph{\ER graphs} $ G(n, p) $ with $ np = n^δ $ for any constant $ δ> 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 - δ}})$. 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 $\tilde{O}(\sqrt{n}d^2)$.