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

2026-02-07

AI Summary

This work addresses the challenge of efficiently constructing spanning trees via local computation algorithms (LCAs) in general graphs, where sublinear-time methods—particularly for expander graphs—have been lacking. The authors present the first sublinear-time LCA for high-conductance expanders that locally determines, for any given edge, whether it belongs to a (minimum) spanning tree without globally constructing the entire tree. Their approach combines local probing, conductance-based graph analysis, and random edge weighting, thereby overcoming prior limitations restricted to sparse subgraphs and extending to the weighted minimum spanning tree setting. For an expander with conductance at least φ and maximum degree at most d, the query complexity is O(√n (log²n/φ² + d)); on G(n,p) random graphs, it achieves Õ(√n^{1−δ}), and for minimum spanning trees, the complexity is Õ(√n d²).

Technology Category

• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

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}{\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 $\tilde{O}(\sqrt{n}d^2)$.

Problem

• Local Computation Algorithms
• Spanning Trees
• Expander Graphs
• Minimum Spanning Tree
• Sublinear-time Algorithms

Innovation

• Local Computation Algorithms
• Expander Graphs
• Spanning Trees
• Minimum Spanning Tree
• Sublinear-time Algorithms