Approximately Counting Subgraphs in Data Streams

Abstract: 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.