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Fully Dynamic Algorithms for Transitive Reduction

Authors Gramoz Goranci , Adam Karczmarz , Ali Momeni , Nikos Parotsidis

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ACM Subject Classification
  • Theory of computation → Sparsification and spanners
Keywords
  • Spectral sparsification
  • Dynamic algorithms
  • (Directed) hypergraphs
  • Data structures

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Abstract

Given a directed graph G, a transitive reduction G^t of G (first studied by Aho, Garey, Ullman [SICOMP `72]) is a minimal subgraph of G that preserves the reachability relation between every two vertices in G.
In this paper, we study the computational complexity of transitive reduction in the dynamic setting. We obtain the first fully dynamic algorithms for maintaining a transitive reduction of a general directed graph undergoing updates such as edge insertions or deletions. Our first algorithm achieves O(m+n log n) amortized update time, which is near-optimal for sparse directed graphs, and can even support extended update operations such as inserting a set of edges all incident to the same vertex, or deleting an arbitrary set of edges. Our second algorithm relies on fast matrix multiplication and achieves O(m+ n^{1.585}) worst-case update time.

Cite As Get BibTex

Gramoz Goranci, Adam Karczmarz, Ali Momeni, and Nikos Parotsidis. Fully Dynamic Algorithms for Transitive Reduction. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 92:1-92:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.92

Author Details

Gramoz Goranci
  • Faculty of Computer Science, University of Vienna, Austria
Adam Karczmarz
  • University of Warsaw, Poland
Ali Momeni
  • Faculty of Computer Science, UniVie Doctoral School Computer Science DoCS, University of Vienna, Austria
Nikos Parotsidis
  • Google Research, Zürich, Switzerland

Funding

  • Karczmarz, Adam: Supported by the National Science Centre (NCN) grant number 2022/47/D/ST6/02184. Work partially done at IDEAS NCBR, Poland.

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