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DOI: 10.4230/LIPIcs.ESA.2025.113

Bootstrapping Dynamic APSP via Sparsification

Authors: Rasmus Kyng, Simon Meierhans, and Gernot Zöcklein

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract

We give a simple algorithm for the dynamic approximate All-Pairs Shortest Paths (APSP) problem. Given a graph G = (V, E, l) with polynomially bounded edge lengths, our data structure processes |E| edge insertions and deletions in total time |E|^{1+o(1)} and provides query access to |E|^o(1)-approximate distances in time Õ(1) per query. We produce a data structure that mimics Thorup-Zwick distance oracles [Thorup and Zwick, 2005], but is dynamic and deterministic. Our algorithm selects a small number of pivot vertices. Then, for every other vertex, it reduces distance computation to maintaining distances to a small neighborhood around that vertex and to the nearest pivot. We maintain distances between pivots efficiently by representing them in a smaller graph and recursing. We maintain these smaller graphs by (a) reducing vertex count using the dynamic distance-preserving core graphs of Kyng-Meierhans-Probst Gutenberg [Kyng et al., 2024] in a black-box manner and (b) reducing edge-count using a dynamic spanner akin to Chen-Kyng-Liu-Meierhans-Probst Gutenberg [Chen et al., 2024]. Our dynamic spanner internally uses an APSP data structure. Choosing a large enough size reduction factor in the first step allows us to simultaneously bootstrap a spanner and a dynamic APSP data structure. Notably, our approach does not need expander graphs, an otherwise ubiquitous tool in derandomization.

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Rasmus Kyng, Simon Meierhans, and Gernot Zöcklein. Bootstrapping Dynamic APSP via Sparsification. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 113:1-113:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kyng_et_al:LIPIcs.ESA.2025.113,
  author =	{Kyng, Rasmus and Meierhans, Simon and Z\"{o}cklein, Gernot},
  title =	{{Bootstrapping Dynamic APSP via Sparsification}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{113:1--113:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.113},
  URN =		{urn:nbn:de:0030-drops-245826},
  doi =		{10.4230/LIPIcs.ESA.2025.113},
  annote =	{Keywords: Dynamic Graph Algorithms, Spanners, Vertex Sparsification, Bootstrapping}
}

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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2025.111

A Simple Dynamic Spanner via APSP

Authors: Rasmus Kyng, Simon Meierhans, and Gernot Zöcklein

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract

We give a simple algorithm for maintaining a n^{o(1)}-approximate spanner H of a graph G with n vertices as G receives edge updates by reduction to the dynamic All-Pairs Shortest Paths (APSP) problem. Given an initially empty graph G, our algorithm processes m insertions and n deletions in total time m^{1 + o(1)} and maintains an initially empty spanner H with total recourse n^{1 + o(1)}. When the number of insertions is much larger than the number of deletions, this notably yields recourse sub-linear in the total number of updates. Our simple algorithm can be extended to maintain a δ ≥ ω(1)-approximate spanner with n^{1+o(1)} edges throughout a sequence of m insertions and D deletions with amortized update time n^{o(1)} and total recourse n^{1 + o(1)} + n^{o(1)} ⋅ D via batching.

Cite as

Rasmus Kyng, Simon Meierhans, and Gernot Zöcklein. A Simple Dynamic Spanner via APSP. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 111:1-111:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kyng_et_al:LIPIcs.ICALP.2025.111,
  author =	{Kyng, Rasmus and Meierhans, Simon and Z\"{o}cklein, Gernot},
  title =	{{A Simple Dynamic Spanner via APSP}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{111:1--111:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l 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.2025.111},
  URN =		{urn:nbn:de:0030-drops-234886},
  doi =		{10.4230/LIPIcs.ICALP.2025.111},
  annote =	{Keywords: Dynamic graph algorithms, Spanner, Dynamic Greedy Spanner}
}

Document

DOI: 10.4230/LIPIcs.ESA.2019.70

Resilient Dictionaries for Randomly Unreliable Memory

Authors: Stefano Leucci, Chih-Hung Liu, and Simon Meierhans

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Abstract

We study the problem of designing a dictionary data structure that is resilient to memory corruptions. Our error model is a variation of the faulty RAM model in which, except for constant amount of definitely reliable memory, each memory word is randomly unreliable with a probability p < 1/2, and the locations of the unreliable words are unknown to the algorithm. An adversary observes the whole memory and can, at any time, arbitrarily corrupt (i.e., modify) the contents of one or more unreliable words. Our dictionary has capacity n, stores N<n keys in the optimal O(N) amount of space, supports insertions and deletions in O(log n) amortized time, and allows to search for a key in O(log n) worst-case time. With a global probability of at least 1-1/n, all possible search operations are guaranteed to return the correct answer w.r.t. the set of uncorrupted keys. The closest related results are the ones of Finocchi et al. [Irene Finocchi et al., 2009] and Brodal et al. [Brodal et al., 2007] on the faulty RAM model, in which all but O(1) memory is unreliable. There, if an upper bound delta on the number of corruptions is known in advance, all dictionary operations can be implemented in Theta(log n + delta) amortized time, thus trading resiliency for speed as soon as delta = omega(log n). Our construction does not need to know the value of delta in advance and remains fast and effective even when up to a constant fraction of the available memory is corrupted. Our techniques can be immediately extended to implement other data types (e.g., associative containers and priority queues), which can then be used as a building block in the design of other resilient algorithms. For example, we are able to solve the resilient sorting problem in our model using O(n log n) time.

Cite as

Stefano Leucci, Chih-Hung Liu, and Simon Meierhans. Resilient Dictionaries for Randomly Unreliable Memory. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 70:1-70:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{leucci_et_al:LIPIcs.ESA.2019.70,
  author =	{Leucci, Stefano and Liu, Chih-Hung and Meierhans, Simon},
  title =	{{Resilient Dictionaries for Randomly Unreliable Memory}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{70:1--70:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.70},
  URN =		{urn:nbn:de:0030-drops-111911},
  doi =		{10.4230/LIPIcs.ESA.2019.70},
  annote =	{Keywords: resilient dictionary, unreliable memory, faulty RAM}
}

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Documents authored by Meierhans, Simon

Bootstrapping Dynamic APSP via Sparsification

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A Simple Dynamic Spanner via APSP

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Resilient Dictionaries for Randomly Unreliable Memory

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