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

Authors Pan Peng , Yuyang Wang

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LIPIcs.ICALP.2026.147.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Graph algorithms
Keywords
  • Local Computation Algorithms
  • (Minimum) Spanning Trees
  • Expander Graphs
  • Random Graphs

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Abstract

We study local computation algorithms (LCAs) for constructing spanning trees. In this setting, the goal is to determine locally, for each edge e ∈ 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 sublinear-probe 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 - expander graphs - that do admit sublinear-time LCAs for spanning trees. This is perhaps surprising, as previous work on expanders only succeeded in designing LCAs for sparse spanning subgraphs, rather than full spanning trees. We design an LCA with probe complexity O(√n ((log²n)/ϕ² + d)) 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(√n) probes are necessary even for expanders. Next, we show that for the natural class of Erdős-Rényi graphs G(n, p) with np = n^δ for any constant δ > 0 (which are expanders with high probability), the √n lower bound can be bypassed. Specifically, we give an average-case LCA for such graphs with probe complexity Õ(√{n^{1 - δ}}). 
Finally, we extend our techniques to design LCAs for the minimum spanning tree (MST) problem on weighted expander graphs. Specifically, given a d-regular unweighted graph ̄{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,…,W}, where W ≤ d/2 and W = o(log n). We show that there exists an LCA that is consistent with an exact MST of G, with probe complexity Õ(√nd²).

Cite As Get BibTex

Pan Peng and Yuyang Wang. Local Computation Algorithms for (Minimum) Spanning Trees on Expander Graphs. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 147:1-147:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.147

Author Details

Pan Peng
  • School of Computer Science and Technology, University of Science and Technology of China, Hefei, China
Yuyang Wang
  • School of Computer Science and Technology, University of Science and Technology of China, Hefei, China

Funding

Supported in part by NSFC Grant 62272431 and Quantum Science and Technology - National Science and Technology Major Project (Grant No. 2021ZD0302901).

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