Communication complexity

About: Communication complexity is a research topic. Over the lifetime, 3870 publications have been published within this topic receiving 105832 citations.

One-qubit fingerprinting schemes

Abstract: Fingerprinting is a technique in communication complexity in which two parties (Alice and Bob) with large data sets send short messages to a third party (a referee), who attempts to compute some function of the larger data sets. For the equality function, the referee attempts to determine whether Alice's data and Bob's data are the same. In this paper, we consider the extreme scenario of performing fingerprinting where Alice and Bob both send either one bit (classically) or one qubit (in the quantum regime) messages to the referee for the equality problem. Restrictive bounds are demonstrated for the error probability of one-bit fingerprinting schemes, and show that it is easy to construct one-qubit fingerprinting schemes which can outperform any one-bit fingerprinting scheme. The author hopes that this analysis will provide results useful for performing physical experiments, which may help to advance implementations for more general quantum communication protocols.

Toward Better Formula Lower Bounds: An Information Complexity Approach to the KRW Composition Conjecture.

Abstract: One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (ie, P n NC1) This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the first milestones toward proving super-polynomial circuit lower bounds Karchmer, Raz, and Wigderson [21] suggested to approach this problem by proving the following conjecture: given two boolean functions f and g, the depth complexity of the composed function g o f is roughly the sum of the depth complexities of f and g They showed that the validity of this conjecture would imply that P n NC1 As a starting point for studying the composition of functions, they introduced a relation called "the universal relation", and suggested to study the composition of universal relations This suggestion proved fruitful, and an analogue of the KRW conjecture for the universal relation was proved by Edmonds et al [12] An alternative proof was given later by Hastad and Wigderson [18] However, studying the composition of functions seems more difficult, and the KRW conjecture is still wide open In this work, we make a natural step in this direction, which lies between what is known and the original conjecture: we show that an analogue of the conjecture holds for the composition of a function with a universal relation We also suggest a candidate for the next step and provide initial results toward it Our main technical contribution is developing an approach based on the notion of information complexity for analyzing KW relations -- communication problems that are closely related to questions on circuit depth and formula complexity Recently, information complexity has proved to be a powerful tool, and underlined some major progress on several long-standing open problems in communication complexity In this work, we develop general tools for analyzing the information complexity of KW relations, which may be of independent interest

Communication Efficient, Sample Optimal, Linear Time Locally Private Discrete Distribution Estimation.

Abstract: We consider discrete distribution estimation over $k$ elements under $\varepsilon$-local differential privacy from $n$ samples The samples are distributed across users who send privatized versions of their sample to the server All previously known sample optimal algorithms require linear (in $k$) communication complexity in the high privacy regime $(\varepsilon<1)$, and have a running time that grows as $n\cdot k$, which can be prohibitive for large domain size $k$ We study the task simultaneously under four resource constraints, privacy, sample complexity, computational complexity, and communication complexity We propose \emph{Hadamard Response (HR)}, a local non-interactive privatization mechanism with order optimal sample complexity (for all privacy regimes), a communication complexity of $\log k+2$ bits, and runs in nearly linear time Our encoding and decoding mechanisms are based on Hadamard matrices, and are simple to implement The gain in sample complexity comes from the large Hamming distance between rows of Hadamard matrices, and the gain in time complexity is achieved by using the Fast Walsh-Hadamard transform We compare our approach with Randomized Response (RR), RAPPOR, and subset-selection mechanisms (SS), theoretically, and experimentally For $k=10000$, our algorithm runs about 100x faster than SS, and RAPPOR

Nondeterministic Quantum Query and Quantum Communication Complexities

Abstract: We study nondeterministic quantum algorithms for Boolean functions f. Such algorithms have positive acceptance probability on input x iff f(x)=1. In the setting of query complexity, we show that the nondeterministic quantum complexity of a Boolean function is equal to its ``nondeterministic polynomial'' degree. We also prove a quantum-vs-classical gap of 1 vs n for nondeterministic query complexity for a total function. In the setting of communication complexity, we show that the nondeterministic quantum complexity of a two-party function is equal to the logarithm of the rank of a nondeterministic version of the communication matrix. This implies that the quantum communication complexities of the equality and disjointness functions are n+1 if we do not allow any error probability. We also exhibit a total function in which the nondeterministic quantum communication complexity is exponentially smaller than its classical counterpart.

Detecting Correlations with Little Memory and Communication

Abstract: We study the problem of identifying correlations in multivariate data, under information constraints: Either on the amount of memory that can be used by the algorithm, or the amount of communication when the data is distributed across several machines. We prove a tight trade-off between the memory/communication complexity and the sample complexity, implying (for example) that to detect pairwise correlations with optimal sample complexity, the number of required memory/communication bits is at least quadratic in the dimension. Our results substantially improve those of Shamir [2014], which studied a similar question in a much more restricted setting. To the best of our knowledge, these are the first provable sample/memory/communication trade-offs for a practical estimation problem, using standard distributions, and in the natural regime where the memory/communication budget is larger than the size of a single data point. To derive our theorems, we prove a new information-theoretic result, which may be relevant for studying other information-constrained learning problems.