Communication complexity

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

Computational complexity questions related to finite monoids and semigroups

Abstract: In this thesis, we address a number of issues pertaining to the computational power of monoids and semigroups as machines and to the computational complexity of problems whose difficulty is parametrized by an underlying semigroup or monoid and find that these two axes of research are deeply intertwined. We first consider the “program over monoid” model of D. Barrington and D. Therien [BT88] and set out to answer two fundamental questions: which monoids are rich enough to recognize arbitrary languages via programs of arbitrary length, and which monoids are so weak that any program over them has an equivalent of polynomial length? We find evidence that the two notions are dual and in particular prove that every monoid in DS has exactly one of these two properties. We also prove that for certain “weak” varieties of monoids, programs can only recognize those languages with a “neutral letter” that can be recognized via morphisms over that variety. We then build an algebraic approach to communication complexity, a field which has been of great importance in the study of small complexity classes. We prove that every monoid has communication complexity O(1), T(log n) or T(n) in this model. We obtain similar classifications for the communication complexity of finite monoids in the probabilistic, simultaneous, probabilistic simultaneous and MOD p-counting variants of this two-party model and thus characterize the communication complexity (in a worst-case partition sense) of every regular language in these five models. Furthermore, we study the same questions in the Chandra-Furst-Lipton multiparty extension of the classical communication model and describe the variety of monoids which have bounded 3-party communication complexity and bounded k-party communication complexity for some k. We also show how these bounds can be used to establish computational limitations of programs over certain classes of monoids. Finally, we consider the computational complexity of testing if an equation or a system of equations over some fixed finite monoid (or semigroup) has a solution. (Abstract shortened by UMI.)

Many-to-many personalized communication with bounded traffic

Abstract: This paper presents solutions for the problem of many-to-many personalized communication, with bounded incoming and outgoing traffic, on a distributed memory parallel machine. We present a two-stage algorithm that decomposes the many-to-many communication with possibly high variance in message size into two communications with low message size variance. The algorithm is deterministic and takes time 2t/spl mu/(+lower order terms) when t/spl ges/0(p/sup 2/+p/spl tau///spl mu/) Here t is the maximum outgoing or incoming traffic at any processor, /spl tau/ is the startup overhead and /spl mu/ is the inverse of the data transfer rate. Optimality is achieved when the traffic is large, a condition that is usually satisfied in practice on coarse-grained architectures. The algorithm was implemented on the Connection Machine CM-5. The implementation used the low latency communication primitives (active messages) available on the CM-5, but the algorithm as such is architecture-independent. An alternate single-stage algorithm using distributed random scheduling for the CM-5 was implemented and the performance of the two algorithms were compared. >

Distributed Stochastic Variance Reduced Gradient Methods and A Lower Bound for Communication Complexity

Abstract: We study distributed optimization algorithms for minimizing the average of convex functions. The applications include empirical risk minimization problems in statistical machine learning where the datasets are large and have to be stored on different machines. We design a distributed stochastic variance reduced gradient algorithm that, under certain conditions on the condition number, simultaneously achieves the optimal parallel runtime, amount of communication and rounds of communication among all distributed first-order methods up to constant factors. Our method and its accelerated extension also outperform existing distributed algorithms in terms of the rounds of communication as long as the condition number is not too large compared to the size of data in each machine. We also prove a lower bound for the number of rounds of communication for a broad class of distributed first-order methods including the proposed algorithms in this paper. We show that our accelerated distributed stochastic variance reduced gradient algorithm achieves this lower bound so that it uses the fewest rounds of communication among all distributed first-order algorithms.

Exponential Separation of Quantum and Classical One-Way Communication Complexity

Abstract: We give the first exponential separation between quantum and bounded-error randomized one-way communication complexity. Specifically, we define the Hidden Matching Problem HM$_n$: Alice gets as input a string ${\bf x}\in\{0, 1\}^n$, and Bob gets a perfect matching $M$ on the $n$ coordinates. Bob's goal is to output a tuple $\langle i,j,b \rangle$ such that the edge $(i,j)$ belongs to the matching $M$ and $b=x_i\oplus x_j$. We prove that the quantum one-way communication complexity of HM$_n$ is $O(\log n)$, yet any randomized one-way protocol with bounded error must use $\Omega({\sqrt{n}})$ bits of communication. No asymptotic gap for one-way communication was previously known. Our bounds also hold in the model of Simultaneous Messages (SM), and hence we provide the first exponential separation between quantum SM and randomized SM with public coins. For a Boolean decision version of HM$_n$, we show that the quantum one-way communication complexity remains $O(\log n)$ and that the 0-error randomized one-way communication complexity is $\Omega(n)$. We prove that any randomized linear one-way protocol with bounded error for this problem requires $\Omega(\sqrt[3]{n \log n})$ bits of communication.

Improved Efficient Arguments (Preliminary Version)

Abstract: We consider complexity of perfect zero-knowledge arguments [4]. Let T denote the time needed to (deterministically) check a proof and let L denote an appropriate security parameter. We introduce new techniques for implementing very efficient zero-knowledge arguments. The resulting argument has the following features: ? The arguer can, if provided with the proof that can be deterministically checked in O(T) time, run in time O(TLO(1)). The best previous bound was O(T1+?LO(1)). ? The protocol can be simulated in time O(LO(1)). The best previous bound was O(T1+?LO(1)). ? A communication complexity of O(LlgL), where L is the security parameter against the prover. The best previous known bound was O(LlgT).This can be based on fairly general algebraic assumptions, such as the hardness of discrete logarithms.Aside from the quantitative improvements, our results become qualitatively different when considering arguers that can run for some super-polynomial but bounded amount of time. In this scenario, we give the first arguments zero-knowledge arguments and the first "constructive" arguments in which the complexity of arguing a proof is tightly bounded by the complexity of verifying the proof.We obtain our results by a hybrid construction that combines the best features of different PCPs. This allows us to obtain better bounds than the previous technique, which only used a single PCP. In our proof of soundness we exploit the error correction capabilities as well as the soundness of the known PCPs.