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On Incremental Approximate Shortest Paths in Directed Graphs

Authors Adam Górkiewicz , Adam Karczmarz

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  • Theory of computation → Dynamic graph algorithms
Keywords
  • dynamic shortest paths
  • incremental shortest paths
  • offline dynamic algorithms

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Abstract

In this paper, we show new data structures maintaining approximate shortest paths in sparse directed graphs with polynomially bounded non-negative edge weights under edge insertions.
We give more efficient incremental (1+ε)-approximate APSP data structures that work against an adaptive adversary: a deterministic one with Õ(m^{3/2}n^{3/4}) total update time and a randomized one with Õ(m^{4/3}n^{5/6}) total update time. For sparse graphs, these both improve polynomially upon the best-known bound against an adaptive adversary [Karczmarz and Łącki, ESA 2019]. To achieve that, building on the ideas of [Chechik and Zhang, SODA 2021] and [Kyng, Meierhans and Probst Gutenberg, SODA 2022], we show a near-optimal (1+ε)-approximate incremental SSSP data structure for a special case when all edge updates are adjacent to the source, that might be of independent interest.
We also describe a very simple and near-optimal offline incremental (1+ε)-approximate SSSP data structure. While online near-linear partially dynamic SSSP data structures have been elusive so far (except for dense instances), our result excludes using certain types of impossibility arguments to rule them out. Additionally, our offline solution leads to near-optimal and deterministic all-pairs bounded-leg shortest paths data structure for sparse graphs.

Cite As Get BibTex

Adam Górkiewicz and Adam Karczmarz. On Incremental Approximate Shortest Paths in Directed Graphs. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 93:1-93:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.93

Author Details

Adam Górkiewicz
  • University of Wrocław, Poland
Adam Karczmarz
  • University of Warsaw, Poland

Funding

  • Górkiewicz, Adam: Work partially done when the author was a student scholarship recipient at IDEAS NCBR supported by the National Science Centre (NCN) grant no. 2022/47/D/ST6/02184.
  • Karczmarz, Adam: Supported by the National Science Centre (NCN) grant no. 2022/47/D/ST6/02184. Work partially done at IDEAS NCBR, Poland.

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