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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2026.147

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

Authors: Pan Peng and Yuyang Wang

Published in: LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

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

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)

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@InProceedings{peng_et_al:LIPIcs.ICALP.2026.147,
  author =	{Peng, Pan and Wang, Yuyang},
  title =	{{Local Computation Algorithms for (Minimum) Spanning Trees on Expander Graphs}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{147:1--147:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.147},
  URN =		{urn:nbn:de:0030-drops-265361},
  doi =		{10.4230/LIPIcs.ICALP.2026.147},
  annote =	{Keywords: Local Computation Algorithms, (Minimum) Spanning Trees, Expander Graphs, Random Graphs}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2023.96

An Optimal Separation Between Two Property Testing Models for Bounded Degree Directed Graphs

Authors: Pan Peng and Yuyang Wang

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract

We revisit the relation between two fundamental property testing models for bounded-degree directed graphs: the bidirectional model in which the algorithms are allowed to query both the outgoing edges and incoming edges of a vertex, and the unidirectional model in which only queries to the outgoing edges are allowed. Czumaj, Peng and Sohler [STOC 2016] showed that for directed graphs with both maximum indegree and maximum outdegree upper bounded by d, any property that can be tested with query complexity O_{ε,d}(1) in the bidirectional model can be tested with n^{1-Ω_{ε,d}(1)} queries in the unidirectional model. In particular, {if the proximity parameter ε approaches 0, then the query complexity of the transformed tester in the unidirectional model approaches n}. It was left open if this transformation can be further improved or there exists any property that exhibits such an extreme separation. We prove that testing subgraph-freeness in which the subgraph contains k source components, requires Ω(n^{1-1/k}) queries in the unidirectional model. This directly gives the first explicit properties that exhibit an O_{ε,d}(1) vs Ω(n^{1-f(ε,d)}) separation of the query complexities between the bidirectional model and unidirectional model, where f(ε,d) is a function that approaches 0 as ε approaches 0. Furthermore, our lower bound also resolves a conjecture by Hellweg and Sohler [ESA 2012] on the query complexity of testing k-star-freeness.

Cite as

Pan Peng and Yuyang Wang. An Optimal Separation Between Two Property Testing Models for Bounded Degree Directed Graphs. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 96:1-96:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{peng_et_al:LIPIcs.ICALP.2023.96,
  author =	{Peng, Pan and Wang, Yuyang},
  title =	{{An Optimal Separation Between Two Property Testing Models for Bounded Degree Directed Graphs}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{96:1--96:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.96},
  URN =		{urn:nbn:de:0030-drops-181480},
  doi =		{10.4230/LIPIcs.ICALP.2023.96},
  annote =	{Keywords: Graph property testing, Directed graphs, Lower bound, Subgraph-freeness}
}

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Documents authored by Wang, Yuyang

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

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An Optimal Separation Between Two Property Testing Models for Bounded Degree Directed Graphs

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