Near-Optimal Four-Cycle Counting in Graph Streams

2026-04-01

AI Summary

This work addresses the challenging problem of efficiently approximating the number of 4-cycles in graph streams presented in arbitrary order. The authors propose a three-pass streaming algorithm that estimates the total number of 4-cycles, denoted $T$, within a relative error of $(1+\varepsilon)$ using $\widetilde{O}(m/\sqrt{T})$ space, where $m$ is the number of edges. This result matches the known $\Omega(m/\sqrt{T})$ space lower bound in the three-pass graph streaming model—the first algorithm to do so—and significantly improves upon the previous best-known bound of $\widetilde{O}(m/T^{1/3})$. By integrating multi-pass streaming techniques with refined sampling and subgraph counting strategies, the proposed method achieves nearly space-optimal 4-cycle estimation.

Abstract

We study four-cycle counting in arbitrary order graph streams. We present a 3-pass algorithm for $(1+\varepsilon)$-approximating the number of four-cycles using $\widetilde{O}(m/\sqrt{T})$ space, where $m$ is the number of edges and $T$ the number of four-cycles in the graph. This improves upon a 3-pass algorithm by Vorotnikova using space $\widetilde{O}(m/T^{1/3})$ and matches a multi-pass lower bound of $Ω(m/\sqrt{T})$ by McGregor and Vorotnikova.

Problem

• four-cycle counting
• graph streams
• streaming algorithms
• subgraph counting

Innovation

• four-cycle counting
• graph streams
• space complexity
• multi-pass algorithm
• near-optimal approximation