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Space Complexity of Minimum Cut Problems in Single-Pass Streams

Authors Matthew Ding , Alexandro Garces, Jason Li , Honghao Lin , Jelani Nelson , Vihan Shah , David P. Woodruff

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Author Details

Matthew Ding
  • University of California, Berkeley, CA, USA
Alexandro Garces
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Jason Li
  • Carnegie Mellon University, Pittsburgh, PA, USA
Honghao Lin
  • Carnegie Mellon University, Pittsburgh, PA, USA
Jelani Nelson
  • University of California, Berkeley, CA, USA
Vihan Shah
  • University of Waterloo, Canada
David P. Woodruff
  • Carnegie Mellon University, Pittsburgh, PA, USA

Funding

  • Ding, Matthew: Supported in part by NSF CCF-1951384.
  • Garces, Alexandro: Supported in part by NSF CNS-2150186.
  • Li, Jason: This work was done in part as a Research Fellow at the Simons Institute for the Theory of Computing.
  • Lin, Honghao: Supported in part by a Simons Investigator Award, NSF CCF-2335412, and a CMU Paul and James Wang Sercomm Presidential Graduate Fellowship.
  • Nelson, Jelani: Supported in part by NSF CCF-1951384 and NSF CCF-2427808.
  • Shah, Vihan: Supported in part by Sepehr Assadi’s Sloan Research Fellowship and NSERC Discovery Grant.
  • Woodruff, David P.: Supported in part by a Simons Investigator Award and NSF CCF-2335412.

Acknowledgements

The authors would like to thank the ITCS 2025 reviewers for their anonymous feedback. Alexandro Garces and Vihan Shah are extremely grateful to Sepehr Assadi for many helpful conversations throughout the project. They also thank the organizers of DIMACS REU in Summer 2023, in particular Lazaros Gallos, for initiating this collaboration and for all their help and encouragement along the way.

Cite As Get BibTex

Matthew Ding, Alexandro Garces, Jason Li, Honghao Lin, Jelani Nelson, Vihan Shah, and David P. Woodruff. Space Complexity of Minimum Cut Problems in Single-Pass Streams. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 43:1-43:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.43

Abstract

We consider the problem of finding a minimum cut of a weighted graph presented as a single-pass stream. While graph sparsification in streams has been intensively studied, the specific application of finding minimum cuts in streams is less well-studied. To this end, we show upper and lower bounds on minimum cut problems in insertion-only streams for a variety of settings, including for both randomized and deterministic algorithms, for both arbitrary and random order streams, and for both approximate and exact algorithms. One of our main results is an Õ(n/ε) space algorithm with fast update time for approximating a spectral cut query with high probability on a stream given in an arbitrary order. Our result breaks the Ω(n/ε²) space lower bound required of a sparsifier that approximates all cuts simultaneously. Using this result, we provide streaming algorithms with near optimal space of Õ(n/ε) for minimum cut and approximate all-pairs effective resistances, with matching space lower-bounds. The amortized update time of our algorithms is Õ(1), provided that the number of edges in the input graph is at least (n/ε²)^{1+o(1)}. We also give a generic way of incorporating sketching into a recursive contraction algorithm to improve the post-processing time of our algorithms. In addition to these results, we give a random-order streaming algorithm that computes the exact minimum cut on a simple, unweighted graph using Õ(n) space. Finally, we give an Ω(n/ε²) space lower bound for deterministic minimum cut algorithms which matches the best-known upper bound up to polylogarithmic factors.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • minimum cut
  • approximate
  • random order
  • lower bound

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