research-article

Incremental Exact Min-Cut in Polylogarithmic Amortized Update Time

Authors:

Gramoz Goranci,

Monika Henzinger,

Mikkel ThorupAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 14, Issue 2

Article No.: 17, Pages 1 - 21

https://doi.org/10.1145/3174803

Published: 12 March 2018 Publication History

Abstract

We present a deterministic incremental algorithm for exactly maintaining the size of a minimum cut with O(log³ n log log² n) amortized time per edge insertion and O(1) query time. This result partially answers an open question posed by Thorup (2007). It also stays in sharp contrast to a polynomial conditional lower bound for the fully dynamic weighted minimum cut problem. Our algorithm is obtained by combining a sparsification technique of Kawarabayashi and Thorup (2015) or its recent improvement by Henzinger, Rao, and Wang (2017), and an exact incremental algorithm of Henzinger (1997).

We also study space-efficient incremental algorithms for the minimum cut problem. Concretely, we show that there exists an O(nlog n/ε²) space Monte Carlo algorithm that can process a stream of edge insertions starting from an empty graph, and with high probability, the algorithm maintains a (1+ε)-approximation to the minimum cut. The algorithm has O((α (n) log³ n)/ε ²) amortized update time and constant query time, where α (n) stands for the inverse of Ackermann function.

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Cited By

Sun HZhou XZhang ZBaeza-Yates RBonchi F(2024)Fast Computation for the Forest Matrix of an Evolving GraphProceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining10.1145/3637528.3671822(2755-2764)Online publication date: 25-Aug-2024
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Kyng RMeierhans SProbst Gutenberg MMohar BShinkar IO'Donnell R(2024)A Dynamic Shortest Paths Toolbox: Low-Congestion Vertex Sparsifiers and Their ApplicationsProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649767(1174-1183)Online publication date: 10-Jun-2024
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de Berg MLópez Martínez ASpieksma F(2024)Stable and Dynamic Minimum CutsWALCOM: Algorithms and Computation10.1007/978-981-97-0566-5_20(273-287)Online publication date: 18-Mar-2024
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Index Terms

Incremental Exact Min-Cut in Polylogarithmic Amortized Update Time
1. Theory of computation
  1. Design and analysis of algorithms
    1. Graph algorithms analysis
      1. Dynamic graph algorithms
    2. Streaming, sublinear and near linear time algorithms

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Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 14, Issue 2

April 2018

339 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/3196491

Editor:
Aravind Srinivasan
University of Maryland, USA

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Copyright © 2018 ACM.

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Publication History

Published: 12 March 2018

Accepted: 01 December 2017

Revised: 01 September 2017

Received: 01 November 2016

Published in TALG Volume 14, Issue 2

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Danish Council for Independent Research under the Sapere Aude research career programme
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Cited By

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https://dl.acm.org/doi/10.1145/3637528.3671822
Kyng RMeierhans SProbst Gutenberg MMohar BShinkar IO'Donnell R(2024)A Dynamic Shortest Paths Toolbox: Low-Congestion Vertex Sparsifiers and Their ApplicationsProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649767(1174-1183)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649767
de Berg MLópez Martínez ASpieksma F(2024)Stable and Dynamic Minimum CutsWALCOM: Algorithms and Computation10.1007/978-981-97-0566-5_20(273-287)Online publication date: 18-Mar-2024
https://dl.acm.org/doi/10.1007/978-981-97-0566-5_20
Baswana SBhanja KPandey A(2023)Minimum+1 (s, t)-cuts and Dual-edge Sensitivity OracleACM Transactions on Algorithms10.1145/362327119:4(1-41)Online publication date: 7-Sep-2023
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Malik VKarmakar S(2023)Some Insights on Dynamic Maintenance of Gomory-Hu Tree in Cactus Graphs and General GraphsAlgorithms and Discrete Applied Mathematics10.1007/978-3-031-25211-2_18(231-244)Online publication date: 26-Jan-2023
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Hanauer KHenzinger MSchulz C(2022)Recent Advances in Fully Dynamic Graph Algorithms – A Quick Reference GuideACM Journal of Experimental Algorithmics10.1145/355580627(1-45)Online publication date: 12-Aug-2022
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Baswana SGupta SKnollmann T(2022)Mincut Sensitivity Data Structures for the Insertion of an EdgeAlgorithmica10.1007/s00453-022-00978-084:9(2702-2734)Online publication date: 1-Sep-2022
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Law JAkers KTasnina NSantina CDeutsch SKshirsagar MKlein-Seetharaman JCrovella MRajagopalan PKasif SMurali T(2021)Interpretable network propagation with application to expanding the repertoire of human proteins that interact with SARS-CoV-2GigaScience10.1093/gigascience/giab08210:12Online publication date: 29-Dec-2021
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Chen LGoranci GHenzinger MPeng RSaranurak T(2020)Fast Dynamic Cuts, Distances and Effective Resistances via Vertex Sparsifiers2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS46700.2020.00109(1135-1146)Online publication date: Nov-2020
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Durfee DGao YGoranci GPeng RCharikar MCohen E(2019)Fully dynamic spectral vertex sparsifiers and applicationsProceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing10.1145/3313276.3316379(914-925)Online publication date: 23-Jun-2019
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