发表机构
Yale University; University of Massachusetts Amherst（耶鲁大学; 马萨诸塞大学阿默斯特分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出无参数流式三角形计数算法，无需先验知识即可近似三角形数，使用O(p)遍和期望次线性空间，并证明任何无参数乘法近似算法需线性空间的下界。
AI 中文摘要
给定一个无向无权图$G=(V,E)$，其中包含$n$个顶点和$m$条边，三角形计数问题旨在计算图中三元环的数量。三角形和子图计数是图算法中的经典问题，在社区检测、聚类系数计算、蛋白质网络中的模体发现以及社交网络分析等应用中至关重要。在许多此类应用中，图数据集规模庞大，以至于我们将其建模为底层图上的更新流。对于流式三角形计数，已有许多基础性成果，既有理论上的也有实践上的。然而，所有先前的次线性空间算法都有一个主要缺陷：为了实现常数因子近似和次线性空间保证，需要预先知道三角形计数$T$的常数因子近似值，这本质上是一个循环要求。我们开创了无参数流式三角形计数的研究，即在没有$T$或任何依赖于$T$的量的先验知识的情况下（仅给定流长度$m$），进行三角形计数。我们描述了一族$O(p)$遍无参数三角形计数算法，该算法保证对$T$的混合乘法和加法近似，并使用$\tilde{O}(\frac{m+T}{\sqrt{T}})$的期望空间。此外，该族算法导出了一个$O(\log\log(n))$遍算法，该算法以相同的空间复杂度给出$T$的$(1+\epsilon)$乘法近似。这些算法依赖于“已验证”参数化算法的概念：一种以$\tau$为参数的算法，当$\tau \le T$时提供$T$的近似值，或者声明$T<\tau$。此外，我们证明了一个下界：任何为所有$T$值提供乘法近似的无参数算法必须使用$\Theta(m)$空间，即使在三角形计数中等规模的流上也是如此。
英文摘要
Given an undirected, unweighted graph $G = (V,E)$ with $n$ vertices and $m$ edges, the triangle counting problem seeks the number of three-cycles in it. Triangle and subgraph counting are classical problems in graph algorithms, central to applications such as community detection, computing the clustering coefficient, motif discovery in protein networks, and social network analysis. In many of these applications, the graph datasets are so voluminous that we model them as streams of updates to an underlying graph. There are a number of foundational results for streaming triangle counting, both theoretical and practical. There is, however, one major drawback to all previous sublinear-space algorithms: to achieve both a constant factor approximation and the sublinear space guarantees, one needs to know a priori a constant factor approximation of the triangle count $T$, an inherently circular requirement. We initiate the study of parameter-free streaming triangle counting, without any a priori knowledge of $T$ or any quantities depending on $T$, provided $m$, the length of the stream. We describe a family of $O(p)$ pass parameter-free triangle counting algorithms that guarantee a mixed multiplicative and additive approximation of $T$ and use $\widetilde{O}(\frac{m+T}{\sqrt{T}})$ expected space. Moreover, this family leads to an $O(\log\log(n))$ pass algorithm that gives a $(1+\eps)$ multiplicative approximation of $T$ with the same space complexity. These algorithms rely on the notion of a \emph{verified} parametrized algorithm: an algorithm parametrized by $τ$ that either provides an approximation of $T$ when $τ\le T$, or declares that $T < τ$. Furthermore, we prove a lower bound: any parameter-free algorithm that provides a multiplicative approximation for all values of $T$ must use $Θ(m)$ space, even on streams where the triangle count is moderately large.
