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An Information Statistics Approach to Data Stream and Communication Complexity
, 2003
"... We present a new method for proving strong lower bounds in communication complexity. ..."
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Cited by 153 (8 self)
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We present a new method for proving strong lower bounds in communication complexity.
Optimal space lower bounds for all frequency moments
 In SODA
, 2004
"... Abstract We prove that any onepass streaming algorithm which (ffl, ffi)approximates the kth frequency moment Fk, for any real k 6 = 1 and any ffl = \Omega i 1pm j, must use \Omega \Gamma 1ffl2 \Delta bits of space, where m is the size of the universe. This is optimal in terms of ffl, resolves the ..."
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Cited by 60 (12 self)
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Abstract We prove that any onepass streaming algorithm which (ffl, ffi)approximates the kth frequency moment Fk, for any real k 6 = 1 and any ffl = \Omega i 1pm j, must use \Omega \Gamma 1ffl2 \Delta bits of space, where m is the size of the universe. This is optimal in terms of ffl, resolves the open questions of BarYossef et al in [3, 4], and extends the \Omega \Gamma 1ffl2 \Delta lower bound for F0 in [11] to much smaller ffl by applying novel techniques. Along the way we lower bound the oneway communication complexity of approximating the Hamming distance and the number of bipartite graphs with minimum/maximum degree constraints. 1 Introduction Computing statistics on massive data sets is increasinglyimportant these days. Advances in communication and storage technology enable large bodies of raw datato be generated daily, and consequently, there is a rising demand to process this data efficiently. Sinceit is impractical for an algorithm to store even a small fraction of the data stream, its performance istypically measured by the amount of space it uses. In many scenarios, such as internet routing, once a streamelement is examined it is lost forever unless explicitly saved by the processing algorithm. This, along with thesheer size of the data, makes multiple passes over the data infeasible. In this paper we restrict our attention toonepass streaming algorithms and we investigate their space complexity.Let a =
Distributed streams algorithms for sliding windows
 In Proc. ACM Symp. on Parallel Algorithms and Architectures (SPAA
, 2002
"... Massive data sets often arise as physically distributed, parallel data streams, and it is important to estimate various aggregates and statistics on the union of these streams. This paper presents algorithms for estimating aggregate functions over a “sliding window ” of the N most recent data items ..."
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Cited by 57 (11 self)
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Massive data sets often arise as physically distributed, parallel data streams, and it is important to estimate various aggregates and statistics on the union of these streams. This paper presents algorithms for estimating aggregate functions over a “sliding window ” of the N most recent data items in one or more streams. Our results include: 1. For a single stream, we present the first ɛapproximation scheme for the number of 1’s in a sliding window that is optimal in both worst case time and space. We also present the first ɛapproximation scheme for the sum of integers in [0..R] in a sliding window that is optimal in both worst case time and space (assuming R is at most polynomial in N). Both algorithms are deterministic and use only logarithmic memory words. 2. In contrast, we show that any deterministic algorithm that estimates, to within a small constant relative error, the number of 1’s (or the sum of integers) in a sliding window on the union of distributed streams requires Ω(N) space.
Tight lower bounds for the distinct elements problem
 In FOCS
, 2003
"... We prove strong lower bounds for the space complexity of ¢¤£¦¥¨§� ©approximating the number of distinct elements �� � in a data stream. Let � be the size of the universe from which the stream elements are drawn. We show that any onepass streaming algorithm for ¢¤£¦¥¨§� ©approximating � � must us ..."
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Cited by 44 (9 self)
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We prove strong lower bounds for the space complexity of ¢¤£¦¥¨§� ©approximating the number of distinct elements �� � in a data stream. Let � be the size of the universe from which the stream elements are drawn. We show that any onepass streaming algorithm for ¢¤£¦¥¨§� ©approximating � � must use ����� space £������¦���� � ������ � when, for ���� � any, improving upon the known lower bound of � ��� � � for this range of £. This lower bound is tight up to a factor of ������������ �. Our lower bound is derived from a reduction from the oneway communication complexity of approximating a boolean function in Euclidean space. The reduction makes use of a lowdistortion embedding from an �� � to an � � norm. 1
Lower bounds in communication complexity based on factorization norms
 In Proc. of the 39th Symposium on Theory of Computing (STOC
, 2007
"... We introduce a new method to derive lower bounds on randomized and quantum communication complexity. Our method is based on factorization norms, a notion from Banach Space theory. As we show, our bounds compare favorably with previously known bounds. Aside from the new results that we derive, our me ..."
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Cited by 40 (6 self)
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We introduce a new method to derive lower bounds on randomized and quantum communication complexity. Our method is based on factorization norms, a notion from Banach Space theory. As we show, our bounds compare favorably with previously known bounds. Aside from the new results that we derive, our method yields new and more transparent proofs of some known results as well. Among our new results we extend some known lower bounds to the realm of quantum communication complexity with entanglement. 1
The complexity of online memory checking
 In Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
, 2005
"... We consider the problem of storing a large file on a remote and unreliable server. To verify that the file has not been corrupted, a user could store a small private (randomized) “fingerprint” on his own computer. This is the setting for the wellstudied authentication problem in cryptography, and t ..."
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Cited by 33 (3 self)
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We consider the problem of storing a large file on a remote and unreliable server. To verify that the file has not been corrupted, a user could store a small private (randomized) “fingerprint” on his own computer. This is the setting for the wellstudied authentication problem in cryptography, and the required fingerprint size is well understood. We study the problem of sublinear authentication: suppose the user would like to encode and store the file in a way that allows him to verify that it has not been corrupted, but without reading the entire file. If the user only wants to read q bits of the file, how large does the size s of the private fingerprint need to be? We define this problem formally, and show a tight lower bound on the relationship between s and q when the adversary is not computationally bounded, namely: s × q = Ω(n), where n is the file size. This is an easier case of the online memory checking problem, introduced by Blum et al. in 1991, and hence the same (tight) lower bound applies also to that problem. It was previously shown that when the adversary is computationally bounded, under the assumption that oneway functions exist, it is possible to construct much better online memory checkers. T he same is also true for sublinear authentication schemes. We show that the existence of oneway functions is also a necessary condition: even slightly breaking the s × q = Ω(n) lower bound in a computational setting implies the existence of oneway functions. 1
Exponential separation of quantum and classical oneway communication complexity
 SIAM J. Comput
"... Abstract. We give the first exponential separation between quantum and boundederror randomized oneway communication complexity. Specifically, we define the Hidden Matching Problem HMn: Alice gets as input a string x ∈ {0, 1} n and Bob gets a perfect matching M on the n coordinates. Bob’s goal is t ..."
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Cited by 32 (2 self)
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Abstract. We give the first exponential separation between quantum and boundederror randomized oneway communication complexity. Specifically, we define the Hidden Matching Problem HMn: Alice gets as input a string x ∈ {0, 1} n and Bob gets a perfect matching M on the n coordinates. Bob’s goal is to output a tuple 〈i, j, b 〉 such that the edge (i, j) belongs to the matching M and b = xi ⊕ xj. We prove that the quantum oneway communication complexity of HMn is O(log n), yet any randomized oneway protocol with bounded error must use Ω ( √ n) bits of communication. No asymptotic gap for oneway 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 HMn, we show that the quantum oneway communication complexity remains O(log n) and that the 0error randomized oneway communication complexity is Ω(n). We prove that any randomized linear oneway protocol with bounded error for this problem requires Ω ( 3 √ n log n) bits of communication. Key words. Communication complexity, quantum computation, separation, hidden matching AMS subject classifications. 68P30,68Q15,68Q17,81P68 1. Introduction. The
Public vs. Private coin flips in one round communication games (Extended Abstract)
 In Proc. 28th ACM Symp. on the Theory of Computing
, 1996
"... ) Ilan Newman and Mario Szegedy y Abstract We study 1round two parties communication complexity games, where private random bits are used. We observe that the existence of good protocols for such games is related to a notion of approximating the matrix that represents the function by a certain ..."
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Cited by 32 (0 self)
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) Ilan Newman and Mario Szegedy y Abstract We study 1round two parties communication complexity games, where private random bits are used. We observe that the existence of good protocols for such games is related to a notion of approximating the matrix that represents the function by a certain low rank matrix. This gives rise to a new notion of rank, analogous of positive rank for 1round communication complexity. We prove that the identity matrix is non approximable by low rank matrices in this sense. As a corollary we prove that any randomized protocol for the equality function requires \Omega\Gamma p n) complexity in the 1round, private bits model, answering an open question raised by Yao [8]. A corollary is an answer to the following graph theoretic question: Assume a graph on n vertices has a set of N = N (n) cliques and so that for each two different cliques of this set, the number of edges between them is at most 0.1 of the product of their sizes. How large can N be ? ...
On quantum and probabilistic communication: Las Vegas and oneway protocols
 in Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, 2000
, 2000
"... We investigate the power of quantum communication protocols compared to classical probabilistic protocols. In our first result we describe a total Boolean function that has a quantum Las Vegas protocol communicating at most O(N^{10/11+ epsilon}) qubits for all epsilon > 0, while any classical probab ..."
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Cited by 30 (5 self)
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We investigate the power of quantum communication protocols compared to classical probabilistic protocols. In our first result we describe a total Boolean function that has a quantum Las Vegas protocol communicating at most O(N^{10/11+ epsilon}) qubits for all epsilon > 0, while any classical probabilistic protocol (with bounded error) needs Omega(N/log N) bits. Then we investigate quantum oneway communication complexity. First we show that the VCdimension lower bound on oneway probabilistic communication of [26] holds for quantum protocols, too. Then we prove that for oneway protocols computing total functions quantum Las Vegas communication is asymptotically as efficient as exact quantum communication, which is exactly as efficient as deterministic communication. We describe applications of the lower bounds for oneway communication complexity to quantum finite automata and quantum formulae.
Information Theory Methods in Communication Complexity
 In Proceedings of the 17th Annual IEEE Conference on Computational Complexity
, 2002
"... We use tools and techniques from information theory to study communication complexity problems in the oneway and simultaneous communication models. Our results include: (1) A tight characterization of multiparty oneway communication complexity for product distributions in terms of VCdimension an ..."
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Cited by 28 (7 self)
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We use tools and techniques from information theory to study communication complexity problems in the oneway and simultaneous communication models. Our results include: (1) A tight characterization of multiparty oneway communication complexity for product distributions in terms of VCdimension and shatter coefficients; (2) An equivalence of multiparty oneway and simultaneous communication models for product distributions; (3) A suite of lower bounds for specific functions in the simultaneous communication model, most notably an optimal lower bound for the multiparty set disjointness problem of Alon et al. [AMS99] and for the generalized addressing function problem of Babai et al. [BGKL96] for arbitrary groups. Methodologically, our main contribution is rendering communication complexity problems in the framework of information theory. This allows us access to the powerful calculus of information theory and the use of fundamental principles such as Fano's inequality and the Maximum Likelihood Estimate Principle.