Counting Simplices in Hypergraph Streams

We consider the problem of space-efficiently estimating the number of simplices in a hypergraph stream. This is the most natural hypergraph generalization of the highly-studied problem of estimating the number of triangles in a graph stream. Our input is a $k$-uniform hypergraph $H$ with $n$ vertices and $m$ hyperedges. A $k$-simplex in $H$ is a subhypergraph on $k+1$ vertices $X$ such that all $k+1$ possible hyperedges among $X$ exist in $H$. The goal is to process a stream of hyperedges of $H$ and compute a good estimate of $T_k(H)$, the number of $k$-simplices in $H$. We design a suite of algorithms for this problem. Under a promise that $T_k(H) \ge T$, our algorithms use at most four passes and together imply a space bound of $O( \epsilon^{-2} \log\delta^{-1} \text{polylog} n \cdot \min\{ m^{1+1/k}/T, m/T^{2/(k+1)} \} )$ for each fixed $k \ge 3$, in order to guarantee an estimate within $(1\pm\epsilon)T_k(H)$ with probability at least $1-\delta$. We also give a simpler $1$-pass algorithm that achieves $O(\epsilon^{-2} \log\delta^{-1} \log n\cdot (m/T) ( \Delta_E + \Delta_V^{1-1/k} ))$ space, where $\Delta_E$ (respectively, $\Delta_V$) denotes the maximum number of $k$-simplices that share a hyperedge (respectively, a vertex). We complement these algorithmic results with space lower bounds of the form $\Omega(\epsilon^{-2})$, $\Omega(m^{1+1/k}/T)$, $\Omega(m/T^{1-1/k})$ and $\Omega(m\Delta_V^{1/k}/T)$ for multi-pass algorithms and $\Omega(m\Delta_E/T)$ for $1$-pass algorithms, which show that some of the dependencies on parameters in our upper bounds are nearly tight. Our techniques extend and generalize several different ideas previously developed for triangle counting in graphs, using appropriate innovations to handle the more complicated combinatorics of hypergraphs.