We study the distributed simulation problem where @math players aim to generate sequences of random coin flips where some subsets of the players share an independent common coin which can be tossed multiple times, and there is a publicly seen blackboard through which the players communicate with each other. We provide a tight representation of the optimal communication rates via linear programming, and more importantly, propose explicit algorithms for the optimal distributed simulation for a wide class of hypergraphs. In particular, the optimal communication rate in complete hypergraphs is still achievable in sparser hypergraphs containing a path-connected cycle-free cluster of topologically connected components. Some key steps in analyzing the upper bounds rely on two different definitions of connectivity in hypergraphs, which may be of independent interest.
We also review some literature on the communication complexity. First introduced in @cite_22 , the blackboard communication protocol serves as an elegant mathematical framework for the study of communication complexity. A series of research are devoted to the lower bounds in communication complexity, where the is the prominent tool for all the deterministic @cite_14 @cite_32 , nondeterministic @cite_25 and randomized communication complexities @cite_42 @cite_6 @cite_43 . We refer to @cite_26 for a survey of these methods. Another closely-related problem is distributed inference under communication constraints @cite_12 , where distributed simulation of private public randomness is useful for distributed learning and property testing @cite_1 @cite_44 . To establish lower bounds on the communication complexity in distributed inference, the copy-paste property of the blackboard communication model typically plays an important role @cite_31 @cite_9 . However, our technique to establish the lower bound is different, where only the sequential nature of the blackboard communication protocol is used in the proof of Theorem (cf. Lemma ), which may be of independent interest.
@cite_14 @cite_26 @cite_22 @cite_9 @cite_42 @cite_1 @cite_32 @cite_6 @cite_44 @cite_43 @cite_31 @cite_25 @cite_12
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In this paper we prove lower bounds on randomized multiparty communication complexity, mainly in the message-passing model, where messages are sent player-to-player. Some of our results apply to the blackboard model, where each message is written on a blackboard for all players to see. We introduce a new technique for proving such bounds, called symmetrization, which is natural, intuitive, and often easy to use. For example, for the problem where each of @math players gets a bit-vector of length @math , and the goal is to compute the coordinatewise XOR of these vectors, we prove a tight lower bound of @math in the blackboard model. For the same problem with AND instead of XOR, we prove a lower bound of roughly @math in the message-passing model (assuming @math ) and @math in the blackboard model. We also prove lower bounds for bitwise majority, for a graph-connectivity problem, and for other problems; the technique seems applicable to a wide range of other problems as well. A... This paper provides the first general technique for proving information lower bounds on two-party unbounded-rounds communication problems. We show that the discrepancy lower bound, which applies to randomized communication complexity, also applies to information complexity. More precisely, if the discrepancy of a two-party function @math with respect to a distribution @math is @math , then any two party randomized protocol computing @math must reveal at least @math bits of information to the participants. As a corollary, we obtain that any two-party protocol for computing a random function on @math must reveal @math bits of information to the participants. In addition, we prove that the discrepancy of the Greater-Than function is @math , which provides an alternative proof to the recent proof of Viola Viola11 of the @math lower bound on the communication complexity of this well-studied function and, combined with our main result, proves the tight @math lower bound on its information complexity. The proof of our main result develops a new simulation procedure that may be of an independent interest. In a very recent breakthrough work of kerenidis2012lower , this simulation procedure was the main building block for proving that almost all known lower bound techniques for communication complexity (and not just discrepancy) apply to information complexity. We study a model of communication complexity that encompasses many well-studied problems, including classical and quantum communication complexity, the complexity of simulating distributions arising from bipartite measurements of shared quantum states, and XOR games. In this model, Alice gets an input x, Bob gets an input y, and their goal is to each produce an output a, b distributed according to some pre-specified joint distribution p(a, b|x, y). Our results apply to any non-signaling distribution, that is, those where Alice's marginal distribution does not depend on Bob's input, and vice versa. By taking a geometric view of the non-signaling distributions, we introduce a simple new technique based on affine combinations of lower-complexity distributions, and we give the first general technique to apply to all these settings, with elementary proofs and very intuitive interpretations. Specifically, we introduce two complexity measures, one which gives lower bounds on classical communication, and one for quantum communication. These measures can be expressed as convex optimization problems. We show that the dual formulations have a striking interpretation, since they coincide with maximum violations of Bell and Tsirelson inequalities. The dual expressions are closely related to the winning probability of XOR games. Despite their apparent simplicity, these lower bounds subsume many known communication complexity lower bound methods, most notably the recent lower bounds of Linial and Shraibman for the special case of Boolean functions. We show that as in the case of Boolean functions, the gap between the quantum and classical lower bounds is at most linear in the size of the support of the distribution, and does not depend on the size of the inputs. This translates into a bound on the gap between maximal Bell and Tsirelson inequality violations, which was previously known only for the case of distributions with Boolean outcomes and uniform marginals. It also allows us to show that for some distributions, information theoretic methods are necessary to prove strong lower bounds. Finally, we give an exponential upper bound on quantum and classical communication complexity in the simultaneous messages model, for any non-signaling distribution. One consequence of this is a simple proof that any quantum distribution can be approximated with a constant number of bits of communication. Communication is a bottleneck in many distributed computations. In VLSI, communication constraints dictate lower bounds on the performance of chips. The two-processor information transfer model measures the communication requirements to compute functions. We study the unbounded error probabilistic version of this model. Because of its weak notion of correct output, we believe that this model measures the “intrinsic” communication complexity of functions. We present exact characterizations of the unbounded error communication complexity in terms of arrangements of hyperplanes and approximations of matrices. These characterizations establish the connection with certain classical problems in combinatorial geometry which are concerned with the configurations of points in d-dimensional real space. With the help of these characterizations, we obtain some upper and lower bounds on communication complexity. The upper bounds which we obtained for the functions—equality and verification of Hamming distance—are considerably better than their counterparts in the deterministic, the nondeterministic, and the bounded error probabilistic models. We also exhibit a function which has log n complexity. We present a counting argument to show that most functions have linear complexity. Further, we apply the logarithmic lower bound on communication complexity to obtain an Ω(n log n) bound on the time of 1-tape unbounded error probabilistic Turing machines. We believe that this is the first nontrivial lower bound obtained for such machines. Recently, Forster [7] proved a new lower bound on probabilistic communication complexity in terms of the operator norm of the communication matrix. In this paper, we want to exploit the various relations between communication complexity of distributed Boolean functions, geometric questions related to half space representations of these functions, and the computational complexity of these functions in various restricted models of computation. In order to widen the range of applicability of Forster's bound, we start with the derivation of a generalized lower bound. We present a concrete family of distributed Boolean functions where the generalized bound leads to a linear lower bound on the probabilistic communication complexity (and thus to an exponential lower bound on the number of Euclidean dimensions needed for a successful half space representation), whereas the old bound fails. We move on to a geometric characterization of the well known communication complexity class C-PP in terms of half space representations achieving a large margin. Our characterization hints to a close connection between the bounded error model of probabilistic communication complexity and the area of large margin classification. In the final section of the paper, we describe how our techniques can be used to prove exponential lower bounds on the size of depth-2 threshold circuits (with still some technical restrictions). Similar results can be obtained for read-k-times randomized ordered binary decision diagram and related models. We show that almost all known lower bound methods for communication complexity are also lower bounds for the information complexity. In particular, we define a relaxed version of the partition bound of Jain and Klauck and prove that it lower bounds the information complexity of any function. Our relaxed partition bound subsumes all norm based methods (e.g. the ?2 method) and rectangle-based methods (e.g. the rectangle corruption bound, the smooth rectangle bound, and the discrepancy bound), except the partition bound. Our result uses a new connection between rectangles and zero-communication protocols where the players can either output a value or abort. We prove the following compression lemma: given a protocol for a function f with information complexity I, one can construct a zero-communication protocol that has non-abort probability at least 2^ -O(I) and that computes f correctly with high probability conditioned on not aborting. Then, we show how such a zero-communication protocol relates to the relaxed partition bound. We use our main theorem to resolve three of the open questions raised by Braver man. First, we show that the information complexity of the Vector in Subspace Problem is O(n^ 1 3 ), which, in turn, implies that there exists an exponential separation between quantum communication complexity and classical information complexity. Moreover, we provide an O(n) lower bound on the information complexity of the Gap Hamming Distance Problem. In much of the theoretical literature on wireless algorithms, issues of message dissemination are considered together with issues of contention management. This combination leads to complicated algorithms and analysis, and makes it difficult to extend the work to harder communication problems. In this paper, we present results of a current project aimed at simplifying such algorithms and analysis by decomposing the treatment into two levels, using abstract "MAC layer" specifications to encapsulate the contention management. We use two different abstract MAC layers: the basic one of [14, 15] and a new probabilistic layer. We first present a typical randomized contention-manageent algorithm for a standard graph-based radio network model We show that it implements both abstract MAC layers. We combine this algorithm with greedy algorithms for single-message and multi-message global broadcast and analyze the combination, using both abstract MAC layers as intermediate layers. Using the basic MAC layer, we prove a bound of O(D log(n ∈) log Δ) for the time to deliver a single message everywhere with probability 1 -- ∈, where D is the network diameter, n is the number of nodes, and Δ is the maximum node degree. Using the probabilistic layer, we prove a bound of O((D + log(n ∈)) log Δ), which matches the best previously-known bound for single-message broadcast over the physical network model. For multi-message broadcast, we obtain bounds of O((D + kΔ) log(n ∈) log Δ) using the basic layer and O((D + kΔ log(n ∈)) log Δ) using the probabilistic layer, for the time to deliver a message everywhere in the presence of at most k concurrent messages. We present a new method for proving strong lower bounds in communication complexity. This method is based on the notion of the conditional information complexity, of a function which is the minimum amount of information about the inputs that has to be revealed by a communication protocol for the function. While conditional information complexity is a lower bound on communication complexity, we show that it also admits a direct sum theorem. Direct sum decomposition reduces our task to that of proving conditional information complexity lower bounds for simple problems (such as the AND of two bits). For the latter, we develop novel techniques based on Hellinger distance and its generalizations.Our paradigm leads to two main results: (1) An improved lower bound for the multi-party set-disjointness problem in the general communication complexity model, and a nearly optimal lower bound in the one-way communication model. As a consequence, we show that for any real k > 2, approximating the kth frequency moment in the data stream model requires essentially Ω(n1-2 k) space; this resolves a conjecture of (J. Comput. System Sci. 58(1) (1999) 137).(2) A lower bound for the Lp approximation problem in the general communication model: this solves an open problem of Saks and Sun (in: Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC), 2002, pp. 360-369). As a consequence, we show that for p > 2, approximating the Lp norm to within a factor of ne in the data stream model with constant number of passes requires Ω(n1-4e-2 p) space. Numerous massive data sets, ranging from flows of Internet traffic to logs of supermarket transactions, have emerged during the past few years. Their overwhelming size and the typically restricted access to them call for new computational models. This thesis studies three such models: sampling computations, data stream computations, and sketch computations. While most of the previous work focused on designing algorithms in the new models, this thesis revolves around the limitations of the models. We develop a suite of lower bound techniques that characterize the complexity of functions in these models, indicating which problems can be solved efficiently in them. We derive specific bounds for a multitude of practical problems, arising from applications in database, networking, and information retrieval, such as frequency statistics, selection functions, statistical moments, and distance estimation. We present general, powerful, and easy to use lower bound techniques for the sampling model. The techniques apply to all functions and address both oblivious and adaptive sampling. They frequently produce optimal bounds for a wide range of functions. They are stated in terms of new combinatorial and statistical properties of functions, which are easy to calculate. We obtain lower bounds for the data stream and sketch models through one-way and simultaneous communication complexity. We develop lower bounds for the latter via a new information-theoretic view of communication complexity. A highlight of this work is an optimal simultaneous communication complexity lower bound for the important multi-party set-disjointness problem. Finally, we present a powerful method for proving lower bounds for general communication complexity. The method is based on a direct sum property of a new measure of complexity for communication complexity protocols and on a novel statistical view of communication complexity. We use the technique to obtain improved communication complexity and data stream lower bounds for several problems, including multi-party set-disjointness, frequency moments, and Lp distance estimation. These results solve open problems of Alon, Matias, and Szegedy and of Saks and Sun. The time-complexity of deterministic and randomized protocols for achieving broadcast (distributing a message from a source to all other nodes) in arbitrary multi-hop radio networks is investigated. In many such networks, communication takes place in synchronous time-slots. A processor receives a message at a certain time-slot if exactly one of its neighbors transmits at that time-slot. We assume no collision-detection mechanism; i.e., it is not always possible to distinguish the case where no neighbor transmits from the case where several neighbors transmit simultaneously. We present a randomized protocol that achieves broadcast in time which is optimal up to a logarithmic factor. In particular, with probability 1 --E, the protocol achieves broadcast within O((D + log n s) ‘log n) time-slots, where n is the number of processors in the network and D its diameter. On the other hand, we prove a linear lower bound on the deterministic time-complexity of broadcast in this model. Namely, we show that any deterministic broadcast protocol requires 8(n) time-slots, even if the network has diameter 3, and n is known to all processors. These two results demonstrate an exponential gap in complexity between randomization and determinism. Consider two parties who wish to communicate in order to execute some interactive protocol π. However, the communication channel between them is noisy: An adversary sees everything that is transmitted over the channel and can change a constant fraction of the bits arbitrarily, thus interrupting the execution of π (which was designed for an error-free channel). If π only contains a single long message, then a good error correcting code would overcome the noise with only a constant overhead in communication. However, this solution is not applicable to interactive protocols consisting of many short messages. Schulman [1992, 1993] introduced the notion of interactive coding: A simulator that, given any protocol π, is able to simulate it (i.e., produce its intended transcript) even in the presence of constant rate adversarial channel errors, and with only constant (multiplicative) communication overhead. However, the running time of Schulman's simulator, and of all simulators that followed, has been exponential (or subexponential) in the communication complexity of π (which we denote by N). In this work, we present three efficient simulators, all of which are randomized and have a certain failure probability (over the choice of coins). The first runs in time poly(N), has failure probability roughly 2-N, and is resilient to 1 32-fraction of adversarial error. The second runs in time O(N log N), has failure probability roughly 2-N, and is resilient to some constant fraction of adversarial error. The third runs in time O(N), has failure probability 1 poly(N), and is resilient to some constant fraction of adversarial error. (Computational complexity is measured in the RAM model.) The first two simulators can be made deterministic if they are a priori given a random string (which may be known to the adversary ahead of time). In particular, the simulators can be made to be nonuniform and deterministic (with equivalent performance). Three decades of research in communication complexity have led to the invention of a number of techniques to lower bound randomized communication complexity. The majority of these techniques involve properties of large submatrices (rectangles) of the truth-table matrix defining a communication problem. The only technique that does not quite fit is information complexity, which has been investigated over the last decade. Here, we connect information complexity to one of the most powerful "rectangular" techniques: the recently-introduced smooth corruption (or "smooth rectangle") bound. We show that the former subsumes the latter under rectangular input distributions. We conjecture that this subsumption holds more generally, under arbitrary distributions, which would resolve the long-standing direct sum question for randomized communication. As an application, we obtain an optimal @math lower bound on the information complexity---under the uniform distribution ---of the so-called orthogonality problem (ORT), which is in turn closely related to the much-studied Gap-Hamming-Distance (GHD). The proof of this bound is along the lines of recent communication lower bounds for GHD, but we encounter a surprising amount of additional technical detail. We consider a distributed task allocation problem in which m players must divide a set of n tasks between them. Each player i receives as input a set Xi of tasks such that the union of all input sets covers the task set. The goal is for each player to output a subset Yi ⊆ Xi, such that the outputs (Y1,...,Ym) form a partition of the set of tasks. The problem can be viewed as a distributed one-shot variant of the well-known k-server problem, and we also show that it is closely related to the problem of finding a rooted spanning tree in directed broadcast networks. We study the communication complexity and round complexity of the task allocation problem. We begin with the classical two-player communication model, and show that the randomized communication complexity of task allocation is Ω(n), even when the set of tasks is known to the players in advance. For the multi-player setting with m = O(n) we give two upper bounds in the shared-blackboard model of communication. We show that the problem can be solved in O(log n) rounds and O(n log n) total bits for arbitrary inputs; moreover, if for any set X of tasks, there are at least α|X| players that have at least one task from X in their inputs, then O((1 α + log m)log n) rounds suffice even if each player can only write O(log n) bits on the blackboard in each round. Finally, we extend our results to the case where the players communicate over an arbitrary directed communication graph instead of a shared blackboard. As an application of these results, we also consider the related problem of constructing a directed spanning tree in strongly-connected directed networks and we show lower and upper bounds for that problem.