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Lower Bounds for NumberinHand Multiparty Communication Complexity, Made Easy ∗
"... In this paper we prove lower bounds on randomized multiparty communication complexity, both in the blackboard model (where each message is written on a blackboard for all players to see) and (mainly) in the messagepassing model, where messages are sent playertoplayer. We introduce a new technique ..."
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In this paper we prove lower bounds on randomized multiparty communication complexity, both in the blackboard model (where each message is written on a blackboard for all players to see) and (mainly) in the messagepassing model, where messages are sent playertoplayer. 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 k players gets a bitvector of length n, and the goal is to compute the coordinatewise XOR of these vectors, we prove a tight lower bounds of Ω(nk) in the blackboard model. For the same problem with AND instead of XOR, we prove a lower bounds of roughly Ω(nk) in the messagepassing model (assuming k ≤ n/3200) and Ω(n log k) in the blackboard model. We also prove lower bounds for bitwise majority, for a graphconnectivity problem, and for other problems; the technique seems applicable to a wide range of other problems as well. The obtained communication lower bounds imply new lower bounds in the functional monitoring model [11] (also called the distributed streaming model). All of our lower bounds allow randomized communication protocols with twosided error. We also use the symmetrization technique to prove several directsumlike results for multiparty communication. 1
When Distributed Computation is Communication Expensive
"... We consider a number of fundamental statistical and graph problems in the messagepassing model, where we have k machines (sites), each holding a piece of data, and the machines want to jointly solve a problem defined on the union of the k data sets. The communication is pointtopoint, and the goal ..."
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We consider a number of fundamental statistical and graph problems in the messagepassing model, where we have k machines (sites), each holding a piece of data, and the machines want to jointly solve a problem defined on the union of the k data sets. The communication is pointtopoint, and the goal is to minimize the total communication among the k machines. This model captures all pointtopoint distributed computational models with respect to minimizing communication costs. Our analysis shows that exact computation of many statistical and graph problems in this distributed setting requires a prohibitively large amount of communication, and often one cannot improve upon the communication of the simple protocol in which all machines send their data to a centralized server. Thus, in order to obtain protocols that are communicationefficient, one has to allow approximation, or investigate the distribution or layout of the data sets. 1
An optimal lower bound for distinct elements in the message passing model
 In Proc. ACMSIAM Symposium on Discrete Algorithms
, 2014
"... In the messagepassing model of communication, there are k players each with their own private input, who try to compute or approximate a function of their inputs by sending messages to one another over private channels. We consider the setting in which each player holds a subset Si of elements of a ..."
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In the messagepassing model of communication, there are k players each with their own private input, who try to compute or approximate a function of their inputs by sending messages to one another over private channels. We consider the setting in which each player holds a subset Si of elements of a universe of size n, and their goal is to output a (1 + )approximation to the total number of distinct elements in the union of the sets Si with constant probability, which can be amplified by independent repetition. This problem has applications in data mining, sensor networks, and network monitoring. We resolve the communication complexity of this problem up to a constant factor, for all settings of n, k and , by showing a lower bound of Ω(k · min(n, 1/2) + k log n) bits. This improves upon previous results, which either had nontrivial restrictions on the relationships between the values of n, k, and , or were suboptimal by logarithmic factors, or both. 1
Approximating matching size from random streams
, 2014
"... We present a streaming algorithm that makes one pass over the edges of an unweighted graph presented in random order, and produces a polylogarithmic approximation to the size of the maximum matching in the graph, while using only polylogarithmic space. Prior to this work the only approximations know ..."
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We present a streaming algorithm that makes one pass over the edges of an unweighted graph presented in random order, and produces a polylogarithmic approximation to the size of the maximum matching in the graph, while using only polylogarithmic space. Prior to this work the only approximations known were a folklore Õ(pn) approximation with polylogarithmic space in an n vertex graph and a constant approximation with ⌦(n) space. Our work thus gives the first algorithm where both the space and approximation factors are smaller than any polynomial in n. Our algorithm is obtained by e↵ecting a streaming implementation of a simple “local ” algorithm that we design for this problem. The local algorithm produces a O(k · n1/k) approximation to the size of a maximum matching by exploring the radius k neighborhoods of vertices, for any parameter k. We show, somewhat surprisingly, that our local algorithm can be implemented in the streaming setting even for k = ⌦(log n / log log n). Our analysis exposes some of the problems that arise in such conversions of local algorithms into streaming ones, and gives techniques to overcome such problems.
Streaming Lower Bounds for Approximating MAXCUT
, 2014
"... We consider the problem of estimating the value of max cut in a graph in the streaming model of computation. At one extreme, there is a trivial 2approximation for this problem that uses only O(log n) space, namely, count the number of edges and output half of this value as the estimate for max cut ..."
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We consider the problem of estimating the value of max cut in a graph in the streaming model of computation. At one extreme, there is a trivial 2approximation for this problem that uses only O(log n) space, namely, count the number of edges and output half of this value as the estimate for max cut value. On the other extreme, if one allows Õ(n) space, then a nearoptimal solution to the max cut value can be obtained by storing an Õ(n)size sparsifier that essentially preserves the max cut. An intriguing question is if polylogarithmic space suffices to obtain a nontrivial approximation to the maxcut value (that is, beating the factor 2). It was recently shown that the problem of estimating the size of a maximum matching in a graph admits a nontrivial approximation in polylogarithmic space. Our main result is that any streaming algorithm that breaks the 2approximation barrier requires Ω̃( n) space even if the edges of the input graph are presented in random order. Our result is obtained by exhibiting a distribution over graphs which are either bipartite or 12far from being bipartite, and establishing that Ω̃( n) space is necessary to differentiate between these two cases. Thus as a direct corollary we obtain that
Communication Complexity of Approximate Matching in Distributed Graphs
"... In this paper we consider the communication complexity of approximation algorithms for maximum matching in a graph in the messagepassing model of distributed computation. The input graph consists of n vertices and edges partitioned over a set of k sites. The output is an αapproximate maximum matc ..."
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In this paper we consider the communication complexity of approximation algorithms for maximum matching in a graph in the messagepassing model of distributed computation. The input graph consists of n vertices and edges partitioned over a set of k sites. The output is an αapproximate maximum matching in the input graph which has to be reported by one of the sites. We show a lower bound on the communication complexity of Ω(α2kn) and show that it is tight up to polylogarithmic factors. This lower bound also applies to other combinatorial problems on graphs in the messagepassing computation model, including maxflow and graph sparsification.
Welfare Maximization with Limited Interaction
, 2015
"... We continue the study of welfare maximization in unitdemand (matching) markets, in a distributed information model where agent’s valuations are unknown to the central planner, and therefore communication is required to determine an efficient allocation. Dobzinski, Nisan and Oren (STOC’14) showed t ..."
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We continue the study of welfare maximization in unitdemand (matching) markets, in a distributed information model where agent’s valuations are unknown to the central planner, and therefore communication is required to determine an efficient allocation. Dobzinski, Nisan and Oren (STOC’14) showed that if the market size is n, then r rounds of interaction (with logarithmic bandwidth) suffice to obtain an n1/(r+1)approximation to the optimal social welfare. In particular, this implies that such markets converge to a stable state (constant approximation) in time logarithmic in the market size. We obtain the first multiround lower bound for this setup. We show that even if the allowable perround bandwidth of each agent is nε(r), the approximation ratio of any rround (randomized) protocol is no better than Ω(n1/5 r+1), implying an Ω(log logn) lower bound on the rate of convergence of the market to equilibrium. Our construction and technique may be of interest to roundcommunication tradeoffs in the more general setting of combinatorial auctions, for which the only known lower bound is for simultaneous (r = 1) protocols [DNO14].