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Nearly Space-Optimal Graph and Hypergraph Sparsification in Insertion-Only Data Streams

Vincent Cohen-Addad, David P. Woodruff, Shenghao Xie, Samson Zhou

TL;DR

The paper addresses sparsification of graphs and hypergraphs in insertion-only data streams, aiming to preserve energy-like quadratic forms for all vectors while using near-minimal space. It develops online and streaming algorithms that yield $(1+\varepsilon)$-spectral sparsifiers for graphs and hypergraphs, along with min-cut approximations, adversarially robust variants, and sliding-window adaptations; all achieve space bounds essentially matching offline sample complexities up to poly-logarithmic factors. The approach hinges on online leverage-score based sampling, gamma-balanced weight assignments, and a merge-and-reduce coreset framework to control error accumulation and memory usage, with update times polynomial in the graph size. These results significantly reduce space overhead compared to prior streaming and online work, enabling practical sparsification on massive, rapidly arriving data while supporting robust and time-sensitive streaming settings. The work also connects to concurrent results in online/dynamic sparsification, and provides a unified framework for extending sparsification guarantees to adversarial and sliding-window models, broadening applicability in real-time analytics and scalable graph-based learning.

Abstract

We study the problem of graph and hypergraph sparsification in insertion-only data streams. The input is a hypergraph $H=(V, E, w)$ with $n$ nodes, $m$ hyperedges, and rank $r$, and the goal is to compute a hypergraph $\widehat{H}$ that preserves the energy of each vector $x \in \mathbb{R}^n$ in $H$, up to a small multiplicative error. In this paper, we give a streaming algorithm that achieves a $(1+\varepsilon)$-approximation, using $\frac{rn}{\varepsilon^2} \log^2 n \log r \cdot\text{poly}(\log \log m)$ bits of space, matching the sample complexity of the best known offline algorithm up to $\text{poly}(\log \log m)$ factors. Our approach also provides a streaming algorithm for graph sparsification that achieves a $(1+\varepsilon)$-approximation, using $\frac{n}{\varepsilon^2} \log n \cdot\text{poly}(\log\log n)$ bits of space, improving the current bound by $\log n$ factors. Furthermore, we give a space-efficient streaming algorithm for min-cut approximation. Along the way, we present an online algorithm for $(1+\varepsilon)$-hypergraph sparsification, which is optimal up to poly-logarithmic factors. As a result, we achieve $(1+\varepsilon)$-hypergraph sparsification in the sliding window model, with space optimal up to poly-logarithmic factors. Lastly, we give an adversarially robust algorithm for hypergraph sparsification using $\frac{n}{\varepsilon^2} \cdot\text{poly}(r, \log n, \log r, \log \log m)$ bits of space.

Nearly Space-Optimal Graph and Hypergraph Sparsification in Insertion-Only Data Streams

TL;DR

The paper addresses sparsification of graphs and hypergraphs in insertion-only data streams, aiming to preserve energy-like quadratic forms for all vectors while using near-minimal space. It develops online and streaming algorithms that yield -spectral sparsifiers for graphs and hypergraphs, along with min-cut approximations, adversarially robust variants, and sliding-window adaptations; all achieve space bounds essentially matching offline sample complexities up to poly-logarithmic factors. The approach hinges on online leverage-score based sampling, gamma-balanced weight assignments, and a merge-and-reduce coreset framework to control error accumulation and memory usage, with update times polynomial in the graph size. These results significantly reduce space overhead compared to prior streaming and online work, enabling practical sparsification on massive, rapidly arriving data while supporting robust and time-sensitive streaming settings. The work also connects to concurrent results in online/dynamic sparsification, and provides a unified framework for extending sparsification guarantees to adversarial and sliding-window models, broadening applicability in real-time analytics and scalable graph-based learning.

Abstract

We study the problem of graph and hypergraph sparsification in insertion-only data streams. The input is a hypergraph with nodes, hyperedges, and rank , and the goal is to compute a hypergraph that preserves the energy of each vector in , up to a small multiplicative error. In this paper, we give a streaming algorithm that achieves a -approximation, using bits of space, matching the sample complexity of the best known offline algorithm up to factors. Our approach also provides a streaming algorithm for graph sparsification that achieves a -approximation, using bits of space, improving the current bound by factors. Furthermore, we give a space-efficient streaming algorithm for min-cut approximation. Along the way, we present an online algorithm for -hypergraph sparsification, which is optimal up to poly-logarithmic factors. As a result, we achieve -hypergraph sparsification in the sliding window model, with space optimal up to poly-logarithmic factors. Lastly, we give an adversarially robust algorithm for hypergraph sparsification using bits of space.
Paper Structure (36 sections, 44 theorems, 39 equations, 4 figures, 2 tables, 5 algorithms)

This paper contains 36 sections, 44 theorems, 39 equations, 4 figures, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Data streams.
  3. Our Contributions
  4. Graph Sparsification.
  5. Graph min-cut approximation.
  6. Hypergraph sparsification.
  7. Adversarial robustness.
  8. The sliding window model.
  9. Concurrent and independent work.
  10. Online algorithms.
  11. Dynamic algorithms.
  12. Hypergraph cut sparsifier.
  13. Preliminaries
  14. Hypergraphs.
  15. Graph Laplacian.
  16. ...and 21 more sections

Key Result

Theorem 1.1

Given a graph $G=(V,E, w)$ with $n$ vertices defined by an insertion-only stream, there exists an algorithm that gives a $(1+\varepsilon)$-spectral sparsifier with probability $1-\frac{1}{\mathop{\mathrm{poly}}\limits(n)}$, storing $\frac{n}{\varepsilon^2} \mathop{\mathrm{poly}}\limits(\log \log n)$

Figures (4)

  • Figure 1: Comparison of our results and the optimal results in offline, online and streaming settings. The space is measured in number of bits.
  • Figure 2: Merge and reduce framework on a stream of length $m$. The coresets at level 1 are precisely the hyperedges in the block, while the coresets at level $\ell>1$ are $\left(1+\mathcal{O}\left(\frac{\varepsilon}{2\log(mn)}\right)\right)$-coresets of the hyperedges contained in the coresets of the children nodes in level $\ell-1$.
  • Figure 3: Comparison under different budgets. The $x$-axis shows the space budget and the $y$-axis shows the multiplicative error. The left is the synthetic graph and the right is the Facebook graph. We include the result of the online algorithm as a baseline.
  • Figure 4: Comparison under different numbers of edges

Theorems & Definitions (73)

  • Theorem 1.1: Streaming graph sparsification
  • Theorem 1.2: Streaming min-cut approximation
  • Theorem 1.3: Streaming hypergraph sparsification
  • Theorem 1.4: Streaming hypergraph sparsification, high probability
  • Theorem 1.5: Online hypergraph sparsification
  • Theorem 1.6: Robust hypergraph sparsification
  • Theorem 1.7: Streaming hypergraph sparsification in the sliding window model
  • Definition 2.1: Energy
  • Definition 2.2: Hypergraph spectral sparsifier
  • Definition 2.3: Associated graph
  • ...and 63 more