Estimating the number of subgraphs in data streams is a fundamental problem that has received great attention in the past decade. In this paper, we give improved streaming algorithms for approximately counting the number of occurrences of an arbitrary subgraph $H$, denoted $\# H$, when the input graph $G$ is represented as a stream of $m$ edges. To obtain our algorithms, we provide a generic transformation that converts constant-round sublinear-time graph algorithms in the query access model to constant-pass sublinear-space graph streaming algorithms. Using this transformation, we obtain the following results. 1. We give a $3$-pass turnstile streaming algorithm for $(1\pm \epsilon)$-approximating $\# H$ in $\tilde{O}(\frac{m^{\rho(H)}}{\epsilon^2\cdot \# H})$ space, where $\rho(H)$ is the fractional edge-cover of $H$. This improves upon and generalizes a result of McGregor et al. [PODS 2016], who gave a $3$-pass insertion-only streaming algorithm for $(1\pm \epsilon)$-approximating the number $\# T$ of triangles in $\tilde{O}(\frac{m^{3/2}}{\epsilon^2\cdot \# T})$ space if the algorithm is given additional oracle access to the degrees. 2. We provide a constant-pass streaming algorithm for $(1\pm \epsilon)$-approximating $\# K_r$ in $\tilde{O}(\frac{m\lambda^{r-2}}{\epsilon^2\cdot \# K_r})$ space for any $r\geq 3$, in a graph $G$ with degeneracy $\lambda$, where $K_r$ is a clique on $r$ vertices. This resolves a conjecture by Bera and Seshadhri [PODS 2020]. More generally, our reduction relates the adaptivity of a query algorithm to the pass complexity of a corresponding streaming algorithm, and it is applicable to all algorithms in standard sublinear-time graph query models, e.g., the (augmented) general model.
翻译:估算数据流中的亚线性图表数量是一个根本性问题, 过去十年来这个问题引起了人们的极大关注。 在本文中, 我们给出了改进的流算法, 以大致计算任意的子图的发生次数 $H$, 表示$$H$, 当输入图显示为 $ 边际的流。 为了获取我们的算法, 我们提供了一个通用转换, 将查询访问模型中的恒定的子线性图表算法转换为 恒定的子线性平面图流算法 。 使用这种转换, 我们获得了以下结果 。 我们给出了 $1 pm 的天线性流算算法, 以 $1 美元 美元 表示 美元 美元, 以平面平面 美元 平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面, 。