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

Half-Duplex Communication Complexity

Suppose Alice and Bob are communicating in order to compute some function f, but instead of a classical communication channel they have a pair of walkie-talkie devices. They can use some classical communication protocol for f where in each round one player sends a bit and the other one receives it. The question is whether talking via walkie-talkie gives them more power? Using walkie-talkies instead of a classical communication channel allows players two extra possibilities: to speak simultaneously (but in this case they do not hear each other) and to listen at the same time (but in this case they do not transfer any bits). The motivation for this kind of a communication model comes from the study of the KRW conjecture. We show that for some definitions this non-classical communication model is, in fact, more powerful than the classical one as it allows to compute some functions in a smaller number of rounds. We also prove lower bounds for these models using both combinatorial and information theoretic methods

Super-Cubic Lower Bound for Generalized Karchmer-Wigderson Games

In this paper, we prove a super-cubic lower bound on the size of a communication protocol for generalized Karchmer-Wigderson game for an explicit function f: {0,1}? ? {0,1}^{log n}. Lower bounds for original Karchmer-Wigderson games correspond to De Morgan formula lower bounds, thus the best known size lower bound is cubic. The generalized Karchmer-Wigderson games are similar to the original ones, so we hope that our approach can provide an insight for proving better lower bounds on the original Karchmer-Wigderson games, and hence for proving new lower bounds on De Morgan formula size. To achieve super-cubic lower bound we adapt several techniques used in formula complexity to communication protocols, prove communication complexity lower bound for a composition of several functions with a multiplexer relation, and use a technique from [Ivan Mihajlin and Alexander Smal, 2021] to extract the "hardest" function from it. As a result, in this setting we are able to show that there is a relatively small set of functions such that at least one of them does not have a small protocol. The resulting lower bound of ??(n^3.156) is significantly better than the bound obtained from the counting argument

Toward the KRW Composition Conjecture: Cubic Formula Lower Bounds via Communication Complexity

One of the major challenges of the research in circuit complexity is proving super-polynomial lower bounds for de-Morgan formulas. Karchmer, Raz, and Wigderson suggested to approach this problem by proving that formula complexity behaves "as expected" with respect to the composition of functions f * g. They showed that this conjecture, if proved, would imply super-polynomial formula lower bounds. The first step toward proving the KRW conjecture was made by Edmonds et al., who proved an analogue of the conjecture for the composition of "universal relations". In this work, we extend the argument of Edmonds et al. further to f * g where f is an arbitrary function and g is the parity function. While this special case of the KRW conjecture was already proved implicitly in Hastad\u27s work on random restrictions, our proof seems more likely to be generalizable to other cases of the conjecture. In particular, our proof uses an entirely different approach, based on communication complexity technique of Karchmer and Wigderson. In addition, our proof gives a new structural result, which roughly says that the naive way for computing f * g is the only optimal way. Along the way, we obtain a new proof of the state-of-the-art formula lower bound of n^{3-o(1)} due to Hastad