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Sublinear Space Graph Algorithms in the Continual Release Model

Authors Alessandro Epasto , Quanquan C. Liu , Tamalika Mukherjee , Felix Zhou

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Subject Classification

ACM Subject Classification
  • Theory of computation → Theory of database privacy and security
  • Theory of computation → Dynamic graph algorithms
  • Theory of computation → Sketching and sampling
Keywords
  • Differential Privacy
  • Continual Release
  • Densest Subgraph
  • k-Core Decomposition
  • Maximum Matching

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Abstract

The graph continual release model of differential privacy seeks to produce differentially private solutions to graph problems under a stream of edge updates where new private solutions are released after each update. Thus far, previously known edge-differentially private algorithms for most graph problems including densest subgraph and matchings in the continual release setting only output real-value estimates (not vertex subset solutions) and do not use sublinear space. Instead, they rely on computing exact graph statistics on the input [Hendrik Fichtenberger et al., 2021; Shuang Song et al., 2018]. In this paper, we leverage sparsification to address the above shortcomings for edge-insertion streams. Our edge-differentially private algorithms use sublinear space with respect to the number of edges in the graph while some also achieve sublinear space in the number of vertices in the graph. In addition, for the densest subgraph problem, we also output edge-differentially private vertex subset solutions; no previous graph algorithms in the continual release model output such subsets.
We make novel use of assorted sparsification techniques from the non-private streaming and static graph algorithms literature to achieve new results in the sublinear space, continual release setting. This includes algorithms for densest subgraph, maximum matching, as well as the first continual release k-core decomposition algorithm. We also develop a novel sparse level data structure for k-core decomposition that may be of independent interest. To complement our insertion-only algorithms, we conclude with polynomial additive error lower bounds for edge-privacy in the fully dynamic setting, where only logarithmic lower bounds were previously known.

Cite As Get BibTex

Alessandro Epasto, Quanquan C. Liu, Tamalika Mukherjee, and Felix Zhou. Sublinear Space Graph Algorithms in the Continual Release Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 40:1-40:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.40

Author Details

Alessandro Epasto
  • Google Research, New York, NY, USA
Quanquan C. Liu
  • Yale University, New Haven, CT, USA
Tamalika Mukherjee
  • Columbia University, New York, NY, USA
Felix Zhou
  • Yale University, New Haven, CT, USA

Funding

Quanquan C. Liu and Felix Zhou are supported by a Google Academic Research Award and by NSF Grant #2453323. Tamalika Mukherjee is supported in part by NSF grant CNS-2138834, a Google Cyber NYC Award, and the Center for Smart Streetscapes, an NSF Engineering Research Center, under grant agreement EEC-2133516. Felix Zhou acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC).

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