Optimal lower bounds for universal relation, and for samplers and finding duplicates in streams

In the communication problem $\mathbf{UR}$ (universal relation) [KRW95], Alice and Bob respectively receive $x, y \in\{0,1\}^n$ with the promise that $x\neq y$. The last player to receive a message must output an index $i$ such that $x_i\neq y_i$. We prove that the randomized one-way communication complexity of this problem in the public coin model is exactly $\Theta(\min\{n,\log(1/\delta)\log^2(\frac n{\log(1/\delta)})\})$ for failure probability $\delta$. Our lower bound holds even if promised $\mathop{support}(y)\subset \mathop{support}(x)$. As a corollary, we obtain optimal lower bounds for $\ell_p$-sampling in strict turnstile streams for $0\le p < 2$, as well as for the problem of finding duplicates in a stream. Our lower bounds do not need to use large weights, and hold even if promised $x\in\{0,1\}^n$ at all points in the stream. We give two different proofs of our main result. The first proof demonstrates that any algorithm $\mathcal A$ solving sampling problems in turnstile streams in low memory can be used to encode subsets of $[n]$ of certain sizes into a number of bits below the information theoretic minimum. Our encoder makes adaptive queries to $\mathcal A$ throughout its execution, but done carefully so as to not violate correctness. This is accomplished by injecting random noise into the encoder's interactions with $\mathcal A$, which is loosely motivated by techniques in differential privacy. Our second proof is via a novel randomized reduction from Augmented Indexing [MNSW98] which needs to interact with $\mathcal A$ adaptively. To handle the adaptivity we identify certain likely interaction patterns and union bound over them to guarantee correct interaction on all of them. To guarantee correctness, it is important that the interaction hides some of its randomness from $\mathcal A$ in the reduction.Comment: merge of arXiv:1703.08139 and of work of Kapralov, Woodruff, and Yahyazade

The Sketching Complexity of Graph and Hypergraph Counting

Subgraph counting is a fundamental primitive in graph processing, with applications in social network analysis (e.g., estimating the clustering coefficient of a graph), database processing and other areas. The space complexity of subgraph counting has been studied extensively in the literature, but many natural settings are still not well understood. In this paper we revisit the subgraph (and hypergraph) counting problem in the sketching model, where the algorithm's state as it processes a stream of updates to the graph is a linear function of the stream. This model has recently received a lot of attention in the literature, and has become a standard model for solving dynamic graph streaming problems. In this paper we give a tight bound on the sketching complexity of counting the number of occurrences of a small subgraph $H$ in a bounded degree graph $G$ presented as a stream of edge updates. Specifically, we show that the space complexity of the problem is governed by the fractional vertex cover number of the graph $H$. Our subgraph counting algorithm implements a natural vertex sampling approach, with sampling probabilities governed by the vertex cover of $H$. Our main technical contribution lies in a new set of Fourier analytic tools that we develop to analyze multiplayer communication protocols in the simultaneous communication model, allowing us to prove a tight lower bound. We believe that our techniques are likely to find applications in other settings. Besides giving tight bounds for all graphs $H$, both our algorithm and lower bounds extend to the hypergraph setting, albeit with some loss in space complexity

Graph Sketching Against Adaptive Adversaries Applied to the Minimum Degree Algorithm

Motivated by the study of matrix elimination orderings in combinatorial scientific computing, we utilize graph sketching and local sampling to give a data structure that provides access to approximate fill degrees of a matrix undergoing elimination in $O(\text{polylog}(n))$ time per elimination and query. We then study the problem of using this data structure in the minimum degree algorithm, which is a widely-used heuristic for producing elimination orderings for sparse matrices by repeatedly eliminating the vertex with (approximate) minimum fill degree. This leads to a nearly-linear time algorithm for generating approximate greedy minimum degree orderings. Despite extensive studies of algorithms for elimination orderings in combinatorial scientific computing, our result is the first rigorous incorporation of randomized tools in this setting, as well as the first nearly-linear time algorithm for producing elimination orderings with provable approximation guarantees. While our sketching data structure readily works in the oblivious adversary model, by repeatedly querying and greedily updating itself, it enters the adaptive adversarial model where the underlying sketches become prone to failure due to dependency issues with their internal randomness. We show how to use an additional sampling procedure to circumvent this problem and to create an independent access sequence. Our technique for decorrelating the interleaved queries and updates to this randomized data structure may be of independent interest.Comment: 58 pages, 3 figures. This is a substantially revised version of arXiv:1711.08446 with an emphasis on the underlying theoretical problem