可扩展三角形计数：阈值算法

中文摘要 AI 辅助

我们研究随机顺序边流上的单遍三角形计数。我们提出一种极其简单的算法——从流中读取边，直到前缀中观察到 $Q$ 个三角形，然后输出 $Q\\,(m/S)^3$，其中 $S$ 是停止长度——并证明，当与任何边关联的最大三角形数满足 $\eta \le T^{2/3}$ 时，这是 $T$ 的 $(1\pm\varepsilon)$ 近似，概率为 $1-\delta$，使用 $O(\varepsilon^{-2}\log(1/\delta)\\, m/T^{1/3})$ 内存。关键在于，该算法不需要任何 $T$ 的先验估计，这与最先进的基于采样率的算法（McGregor 和 Vorotnikova，PODS 2020；Tsourakakis 等人，KDD 2009）形成鲜明对比。它也不需要预设的内存预算：停止规则自行选择前缀长度，并且可以在读取整个流之前返回估计值。证明基于 Schudy--Sviridenko 集中不等式，用于独立边采样估计器，并结合算法产生的无放回前缀。在六个真实时间流上，该算法的停止前缀遵循预测的立方根缩放，在 $10\\%$ 前缀处实现至多 $6\\%$ 的误差，而无需使用 $T$。在固定的存储边预算下，方差缩减的蓄水池采样器通常更准确，但仅在读取整个流之后。在一个单独的、更大的 $1.8\times10^9$ 边图上，阈值算法读取流的 $0.46\\%$ 并返回 $3.8\\%$ 的误差，而最强的蓄水池基线在墙钟时间上限内未能完成一遍遍历。

英文摘要

We study one-pass triangle counting on random-order edge streams. We present a remarkably simple algorithm---read edges from the stream until $Q$ triangles are observed in the prefix, then output $Q\,(m/S)^3$ where $S$ is the stopping length---and prove that, when the maximum number of triangles incident to any edge satisfies $η\le T^{2/3}$, this is a $(1\pm\varepsilon)$-approximation of $T$ with probability $1-δ$ using $O(\varepsilon^{-2}\log(1/δ)\, m/T^{1/3})$ memory. Crucially, the algorithm does not need any a priori estimate of $T$, in sharp contrast with state-of-the-art sampling-rate based algorithms (McGregor and Vorotnikova, PODS 2020; Tsourakakis et al., KDD 2009). It also does not need a prescribed memory budget: the stopping rule self-selects the prefix length and can return an estimate before reading the entire stream. The proof rests on a Schudy--Sviridenko concentration argument for an independent-edge-sampling estimator, coupled to the without-replacement prefix produced by the algorithm. On six real temporal streams, the algorithm's stopping prefix follows the predicted cube-root scaling and achieves at most $6\%$ error at a $10\%$ prefix, without using $T$. At a fixed stored-edge budget, variance-reduced reservoir samplers are often more accurate, but only after reading the entire stream. On a separate, much larger, $1.8\times10^9$-edge graph, the threshold algorithm reads $0.46\%$ of the stream and returns $3.8\%$ error, while the strongest reservoir baselines do not finish a pass within the wall-clock cap.

发表机构

• Yale University（耶鲁大学）
• University of Massachusetts（马萨诸塞大学）

机构由 AI 辅助整理，请以论文原文为准。