Approximating Local Graph Structure in Almost Random Order Streams

The random order streaming model has been very fruitful for graph streams, allowing for polylogarithmic or even constant space estimators for fundamental graph problems such as matching size estimation, counting the number of connected components and more. However, the assumption that there are no correlations between the order of arrival of edges in the stream is quite strong. In this paper we introduce (hidden) batch random order streams, where edges are grouped in "batches" (which are unknown to the algorithm) that arrive in a random order, as a natural framework for modelling hidden correlations in the arrival order of edges, and present algorithms and lower bounds for this model. On the algorithmic side, we show how known techniques for connected component counting in constant space due to Peng and Sohler [SODA `18] easily translate from random order streams to our model with only a small loss in parameters. Our algorithm obtains an additive $\varepsilon n$ approximation to the number of connected components in the input graph using space $(1/\varepsilon)^{O(1/\varepsilon)}$ by building a representative sample of vertices in the graph that belong to $O(1/\varepsilon)$-size components to estimate the count. On the lower bound side, we show that $(1/\varepsilon)^{\Omega(1/\varepsilon)}$ space is necessary for finding a connected component of size $O(1/\varepsilon)$ even in graphs where most vertices reside in such components -- this makes progress towards an open problem of Peng and Sohler [SODA `18] and constitutes our main technical contribution. The lower bound uses Fourier analytic techniques inspired by the Boolean Hidden Matching problem. Our main innovation here is the first framework for applying such a Fourier analytic approach to a communication game with a polynomial number of players.