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Informational Complexity and the Direct Sum Problem for Simultaneous Message Complexity
- In Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science
, 2001
"... Given m copies of the same problem, does it take m times the amount of resources to solve these m problems? This is the direct sum problem, a fundamental question that has been studied in many computational models. We study this question in the simultaneous message (SM) model of communication introd ..."
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Cited by 102 (10 self)
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Given m copies of the same problem, does it take m times the amount of resources to solve these m problems? This is the direct sum problem, a fundamental question that has been studied in many computational models. We study this question in the simultaneous message (SM) model of communication introduced by Yao [Y79]. The equality problem for n-bit strings is well known to have SM complexity ( p n). We prove that solving m copies of the problem has complexity m p n); the best lower bound provable using previously known techniques is p mn). We also prove similar lower bounds on certain Boolean combinations of multiple copies of the equality function. These results can be generalized to a broader class of functions. We introduce a new notion of informational complexity which is related to SM complexity and has nice direct sum properties. This notion is used as a tool to prove the above results; it appears to be quite powerful and may be of independent interest. 1
On Randomized One-Round Communication Complexity
- Computational Complexity
, 1995
"... We present several results regarding randomized one-round communication complexity. Our results include a connection to the VCdimension, a study of the problem of computing the inner product of two real valued vectors, and a relation between \simultaneous" protocols and one-round protocols. Ke ..."
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Cited by 76 (0 self)
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We present several results regarding randomized one-round communication complexity. Our results include a connection to the VCdimension, a study of the problem of computing the inner product of two real valued vectors, and a relation between \simultaneous" protocols and one-round protocols. Key words. Communication Complexity; One-round and simultaneous protocols; VC-dimension; Subject classications. 68Q25. 1.
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 62 (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 ɛ-approximation 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 ɛ-approximation 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.
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 well-studied authentication problem in cryptography, and t ..."
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Cited by 54 (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 well-studied authentication problem in cryptography, and the required fingerprint size is well understood. We study the problem of sub-linear 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 one-way functions exist, it is possible to construct much better online memory checkers. T he same is also true for sub-linear authentication schemes. We show that the existence of one-way functions is also a necessary condition: even slightly breaking the s × q = Ω(n) lower bound in a computational setting implies the existence of one-way functions. 1
Exponential separation of quantum and classical one-way communication complexity
- SIAM J. Comput
"... Abstract. We give the first exponential separation between quantum and bounded-error randomized one-way 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 48 (4 self)
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Abstract. We give the first exponential separation between quantum and bounded-error randomized one-way 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 one-way communication complexity of HMn is O(log n), yet any randomized one-way protocol with bounded error must use Ω ( √ 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 HMn, we show that the quantum one-way communication complexity remains O(log n) and that the 0-error randomized one-way communication complexity is Ω(n). We prove that any randomized linear one-way 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
On Distributing Symmetric Streaming Computations
"... A common approach for dealing with large data sets is to stream over the input in one pass, and perform computations using sublinear resources. For truly massive data sets, however, even making a single pass over the data is prohibitive. Therefore, streaming computations must be distributed over man ..."
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Cited by 33 (1 self)
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A common approach for dealing with large data sets is to stream over the input in one pass, and perform computations using sublinear resources. For truly massive data sets, however, even making a single pass over the data is prohibitive. Therefore, streaming computations must be distributed over many machines. In practice, obtaining significant speedups using distributed computation has numerous challenges including synchronization, load balancing, overcoming processor failures, and data distribution. Successful systems in practice such as Google’s MapReduce and Apache’s Hadoop address these problems by only allowing a certain class of highly distributable tasks defined by local computations that can be applied in any order to the input. The fundamental question that arises is: How does the class of computational tasks supported by these systems differ from the class for which streaming solutions exist? We introduce a simple algorithmic model for massive, unordered, distributed (mud) computation, as implemented by these systems. We show that in principle, mud algorithms are equivalent in power to symmetric streaming algorithms. More precisely, we show that any symmetric (orderinvariant) function that can be computed by a streaming algorithm can also be computed by a mud algorithm, with comparable space and communication complexity. Our simulation uses Savitch’s theorem and therefore has superpolynomial time complexity. We extend our simulation result to some natural classes of approximate and randomized streaming algorithms. We also give negative results, using communication complexity arguments to prove that extensions to private randomness, promise problems and indeterminate functions are impossible. We also introduce an extension of the mud model to multiple keys and multiple rounds. 1
Private Simultaneous Messages Protocols with Applications
- In Proc. of 5th ISTCS
, 1997
"... We study the Private Simultaneous Messages (PSM) model which is a variant of the model proposed in [15]. In the PSM model there are n players P 1 ; : : : ; Pn , each player P i holding a secret input x i (say, a bit), and all having access to a common random string. Each player sends a single messag ..."
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Cited by 31 (12 self)
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We study the Private Simultaneous Messages (PSM) model which is a variant of the model proposed in [15]. In the PSM model there are n players P 1 ; : : : ; Pn , each player P i holding a secret input x i (say, a bit), and all having access to a common random string. Each player sends a single message to a special player, Carol, depending on its own input and the random string (and independently of all other messages). Based on these messages, Carol should be able to compute f(x 1 ; : : : ; xn ) (for some predetermined function f) but should learn no additional information on the values of x 1 ; : : : ; xn .
Bounded-error quantum state identification and exponential separations in communication complexity
- In Proc. of the 38th Symposium on Theory of Computing (STOC
, 2006
"... We consider the problem of bounded-error quantum state identification: given either state α0 or state α1, we are required to output ‘0’, ‘1 ’ or ‘? ’ (“don’t know”), such that conditioned on outputting ‘0 ’ or ‘1’, our guess is correct with high probability. The goal is to maximize the probability o ..."
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Cited by 29 (16 self)
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We consider the problem of bounded-error quantum state identification: given either state α0 or state α1, we are required to output ‘0’, ‘1 ’ or ‘? ’ (“don’t know”), such that conditioned on outputting ‘0 ’ or ‘1’, our guess is correct with high probability. The goal is to maximize the probability of not outputting ‘?’. We prove a direct product theorem: if we are given two such problems, with optimal probabilities a and b, respectively, and the states in the first problem are pure, then the optimal probability for the joint bounded-error state identification problem is O(ab). Our proof is based on semidefinite programming duality. Using this result, we present two exponential separations in the simultaneous message passing model of communication complexity. First, we describe a relation that can be computed with O(log n) classical bits of communication in the presence of shared randomness, but needs Ω(n 1/3) communication if the parties don’t share randomness, even if communication is quantum. This shows the optimality of Yao’s recent exponential simulation of shared-randomness protocols by quantum protocols without shared randomness. Combined with an earlier separation in the other direction due to Bar-Yossef et al., this shows that the quantum SMP model is incomparable with the classical shared-randomness SMP model. Second, we describe a relation that can be computed with O(log n) classical bits of communication in the presence of shared entanglement, but needs Ω((n / log n) 1/3) communication if the parties share randomness but no entanglement, even if communication is quantum. This is the first example in communication complexity of a situation where entanglement buys you much more than quantum communication.
Quantum communication complexity
- Foundations of Physics
"... Can quantum communication be more efficient than its classical counterpart? Holevo’s theorem rules out the possibility of communicating more than n bits of classical information by the transmission of n quantum bits—unless the two parties are entangled, in which case twice as many classical bits can ..."
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Cited by 19 (6 self)
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Can quantum communication be more efficient than its classical counterpart? Holevo’s theorem rules out the possibility of communicating more than n bits of classical information by the transmission of n quantum bits—unless the two parties are entangled, in which case twice as many classical bits can be communicated but no more. In apparent contradiction, there are distributed computational tasks for which quantum communication cannot be simulated efficiently by classical means. In some cases, the effect of transmitting quantum bits cannot be achieved classically short of transmitting an exponentially larger number of bits. In a similar vein, can entanglement be used to save on classical communication? It is well known that entanglement on its own is useless for the transmission of information. Yet, there are distributed tasks that cannot be accomplished at all in a classical world when communication is not allowed, but that become possible if the noncommunicating parties share prior entanglement. This leads to the question of how expensive it is, in terms of classical communication, to provide an exact simulation of the spooky power of entanglement. KEY WORDS: Bell’s theorem; communication complexity; distributed computation; entanglement simulation; pseudo-telepathy; spooky communication.
Strengths and weaknesses of quantum fingerprinting
- In Proc. of the 21st Conf. on Computational Complexity (CCC
, 2006
"... We study the power of quantum fingerprints in the simultaneous message passing (SMP) setting of communication complexity. Yao recently showed how to simulate, with exponential overhead, classical shared-randomness SMP protocols by means of quantum SMP protocols without shared randomness (Q �-protoco ..."
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Cited by 17 (2 self)
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We study the power of quantum fingerprints in the simultaneous message passing (SMP) setting of communication complexity. Yao recently showed how to simulate, with exponential overhead, classical shared-randomness SMP protocols by means of quantum SMP protocols without shared randomness (Q �-protocols). Our first result is to extend Yao’s simulation to the strongest possible model: every many-round quantum protocol with unlimited shared entanglement can be simulated, with exponential overhead, by Q �-protocols. We apply our technique to obtain an efficient Q �-protocol for a function which cannot be efficiently solved through more restricted simulations. Second, we tightly characterize the power of the quantum fingerprinting technique by making a connection to arrangements of homogeneous halfspaces with maximal margin. These arrangements have been well studied in computational learning theory, and we use some strong results obtained in this area to exhibit weaknesses of quantum fingerprinting. In particular, this implies that for almost all functions, quantum fingerprinting protocols are exponentially worse than classical deterministic SMP protocols.
