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The White-Box Adversarial Data Stream Model
Authors:
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Award ID(s):
Publication Date:
NSF-PAR ID:
10341769
Journal Name:
Proceedings of the 41st ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems (PODS 2022)
Page Range or eLocation-ID:
15-27
5. We revisit the much-studied problem of space-efficiently estimating the number of triangles in a graph stream, and extensions of this problem to counting fixed-sized cliques and cycles, obtaining a number of new upper and lower bounds. For the important special case of counting triangles, we give a $4$-pass, $(1\pm\varepsilon)$-approximate, randomized algorithm that needs at most $\widetilde{O}(\varepsilon^{-2}\cdot m^{3/2}/T)$ space, where $m$ is the number of edges and $T$ is a promised lower bound on the number of triangles. This matches the space bound of a very recent algorithm (McGregor et al., PODS 2016), with an arguably simpler and more general technique. We give an improved multi-pass lower bound of $\Omega(\min\{m^{3/2}/T, m/\sqrt{T}\})$, applicable at essentially all densities $\Omega(n) \le m \le O(n^2)$. We also prove other multi-pass lower bounds in terms of various structural parameters of the input graph. Together, our results resolve a couple of open questions raised in recent work (Braverman et al., ICALP 2013). Our presentation emphasizes more general frameworks, for both upper and lower bounds. We give a sampling algorithm for counting arbitrary subgraphs and then improve it via combinatorial means in the special cases of counting odd cliques and odd cycles. Our results show that these problemsmore »