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François Le Gall - One of the best experts on this subject based on the ideXlab platform.
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Quantum Communication Complexity of Distribution Testing.
arXiv: Computational Complexity, 2020Co-Authors: Aleksandrs Belovs, François Le Gall, Arturo Castellanos, Guillaume Malod, Alexander A. SherstovAbstract:The classical Communication Complexity of testing closeness of discrete distributions has recently been studied by Andoni, Malkin and Nosatzki (ICALP'19). In this problem, two players each receive $t$ samples from one distribution over $[n]$, and the goal is to decide whether their two distributions are equal, or are $\epsilon$-far apart in the $l_1$-distance. In the present paper we show that the quantum Communication Complexity of this problem is $\tilde{O}(n/(t\epsilon^2))$ qubits when the distributions have low $l_2$-norm, which gives a quadratic improvement over the classical Communication Complexity obtained by Andoni, Malkin and Nosatzki. We also obtain a matching lower bound by using the pattern matrix method. Let us stress that the samples received by each of the parties are classical, and it is only Communication between them that is quantum. Our results thus give one setting where quantum protocols overcome classical protocols for a testing problem with purely classical samples.
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Quantum Communication Complexity of Distributed Set Joins
arXiv: Quantum Physics, 2016Co-Authors: Stacey Jeffery, François Le GallAbstract:Computing set joins of two inputs is a common task in database theory. Recently, Van Gucht, Williams, Woodruff and Zhang [PODS 2015] considered the Complexity of such problems in the natural model of (classical) two-party Communication Complexity and obtained tight bounds for the Complexity of several important distributed set joins. In this paper we initiate the study of the *quantum* Communication Complexity of distributed set joins. We design a quantum protocol for distributed Boolean matrix multiplication, which corresponds to computing the composition join of two databases, showing that the product of two $n\times n$ Boolean matrices, each owned by one of two respective parties, can be computed with $\widetilde{O}(\sqrt{n}\ell^{3/4})$ qubits of Communication, where $\ell$ denotes the number of non-zero entries of the product. Since Van Gucht et al. showed that the classical Communication Complexity of this problem is $\widetilde{\Theta}(n\sqrt{\ell})$, our quantum algorithm outperforms classical protocols whenever the output matrix is sparse. We also show a quantum lower bound and a matching classical upper bound on the Communication Complexity of distributed matrix multiplication over $\mathbb{F}_2$. Besides their applications to database theory, the Communication Complexity of set joins is interesting due to its connections to direct product theorems in Communication Complexity. In this work we also introduce a notion of *all-pairs* product theorem, and relate this notion to standard direct product theorems in Communication Complexity.
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MFCS - Quantum Communication Complexity of Distributed Set Joins
2016Co-Authors: Stacey Jeffery, François Le GallAbstract:Computing set joins of two inputs is a common task in database theory. Recently, Van Gucht, Williams, Woodruff and Zhang [PODS 2015] considered the Complexity of such problems in the natural model of (classical) two-party Communication Complexity and obtained tight bounds for the Complexity of several important distributed set joins. In this paper we initiate the study of the quantum Communication Complexity of distributed set joins. We design a quantum protocol for distributed Boolean matrix multiplication, which corresponds to computing the composition join of two databases, showing that the product of two n times n Boolean matrices, each owned by one of two respective parties, can be computed with widetilde-O(sqrt{n} ell^{3/4}) qubits of Communication, where ell denotes the number of non-zero entries of the product. Since Van Gucht et al. showed that the classical Communication Complexity of this problem is widetilde-Theta(n sqrt{ell}), our quantum algorithm outperforms classical protocols whenever the output matrix is sparse. We also show a quantum lower bound and a matching classical upper bound on the Communication Complexity of distributed matrix multiplication over F_2. Besides their applications to database theory, the Communication Complexity of set joins is interesting due to its connections to direct product theorems in Communication Complexity. In this work we also introduce a notion of all-pairs product theorem, and relate this notion to standard direct product theorems in Communication Complexity.
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Quantum weakly nondeterministic Communication Complexity
Theoretical Computer Science, 2013Co-Authors: François Le GallAbstract:In this paper we study a weak version of quantum nondeterministic Communication Complexity, in which a classical proof has to be checked with probability one by a quantum protocol. We prove that, in the framework of Communication Complexity, even this weak version of quantum nondeterminism is strictly stronger than classical nondeterminism. More precisely, we show a separation, for a total function, of quantum weakly nondeterministic and classical nondeterministic Communication Complexity. This separation is quadratic and shows that classical proofs can be checked more efficiently by quantum protocols than by classical ones.
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Quantum weakly nondeterministic Communication Complexity
Lecture Notes in Computer Science, 2006Co-Authors: François Le GallAbstract:In this paper we study a weak version of quantum nondeterministic Communication Complexity, corresponding to the most natural generalization of classical nondeterminism, in which a classical proof has to be checked with probability one by a quantum protocol. We prove that, in the framework of Communication Complexity, even the weak version of quantum nondeterminism is strictly stronger than classical nondeterminism. More precisely, we show the first separation, for a total function, of quantum weakly nondeterministic and classical nondeterministic Communication Complexity. This separation is quadratic and shows that classical proofs can be checked more efficiently by quantum protocols than by classical ones.
Madhu Sudan - One of the best experts on this subject based on the ideXlab platform.
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Communication Complexity of permutation invariant functions
Symposium on Discrete Algorithms, 2016Co-Authors: Badih Ghazi, Pritish Kamath, Madhu SudanAbstract:Motivated by the quest for a broader understanding of upper bounds in Communication Complexity, at least for simple functions, we introduce the class of "permutation-invariant" functions. A partial function f: {0, 1}n × {0, 1}n → {0, 1, ?} is permutation-invariant if for every bijection π: {1, . . ., n} → {1, . . ., n} and every x, y ∈ {0, 1}n, it is the case that f(x, y) = f(xπ, yπ). Most of the commonly studied functions in Communication Complexity are permutation-invariant. For such functions, we present a simple Complexity measure (computable in time polynomial in n given an implicit description of f) that describes their Communication Complexity up to polynomial factors and up to an additive error that is logarithmic in the input size. This gives a coarse taxonomy of the Communication Complexity of simple functions. Our work highlights the role of the well-known lower bounds of functions such as S et -D isjointness and I ndexing , while complementing them with the relatively lesser-known upper bounds for G ap -I nner -P roduct (from the sketching literature) and S parse -G ap -I nner -P roduct (from the recent work of Canonne et al. [ITCS 2015]). We also present consequences to the study of Communication Complexity with imperfectly shared randomness where we show that for total permutation-invariant functions, imperfectly shared randomness results in only a polynomial blow-up in Communication Complexity after an additive O(log log n) overhead.
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SODA - Communication Complexity of permutation-invariant functions
Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, 2015Co-Authors: Badih Ghazi, Pritish Kamath, Madhu SudanAbstract:Motivated by the quest for a broader understanding of upper bounds in Communication Complexity, at least for simple functions, we introduce the class of "permutation-invariant" functions. A partial function f: {0, 1}n × {0, 1}n → {0, 1, ?} is permutation-invariant if for every bijection π: {1, . . ., n} → {1, . . ., n} and every x, y ∈ {0, 1}n, it is the case that f(x, y) = f(xπ, yπ). Most of the commonly studied functions in Communication Complexity are permutation-invariant. For such functions, we present a simple Complexity measure (computable in time polynomial in n given an implicit description of f) that describes their Communication Complexity up to polynomial factors and up to an additive error that is logarithmic in the input size. This gives a coarse taxonomy of the Communication Complexity of simple functions. Our work highlights the role of the well-known lower bounds of functions such as S et -D isjointness and I ndexing , while complementing them with the relatively lesser-known upper bounds for G ap -I nner -P roduct (from the sketching literature) and S parse -G ap -I nner -P roduct (from the recent work of Canonne et al. [ITCS 2015]). We also present consequences to the study of Communication Complexity with imperfectly shared randomness where we show that for total permutation-invariant functions, imperfectly shared randomness results in only a polynomial blow-up in Communication Complexity after an additive O(log log n) overhead.
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Communication Complexity of Permutation-Invariant Functions
arXiv: Computational Complexity, 2015Co-Authors: Badih Ghazi, Pritish Kamath, Madhu SudanAbstract:Motivated by the quest for a broader understanding of Communication Complexity of simple functions, we introduce the class of "permutation-invariant" functions. A partial function $f:\{0,1\}^n \times \{0,1\}^n\to \{0,1,?\}$ is permutation-invariant if for every bijection $\pi:\{1,\ldots,n\} \to \{1,\ldots,n\}$ and every $\mathbf{x}, \mathbf{y} \in \{0,1\}^n$, it is the case that $f(\mathbf{x}, \mathbf{y}) = f(\mathbf{x}^{\pi}, \mathbf{y}^{\pi})$. Most of the commonly studied functions in Communication Complexity are permutation-invariant. For such functions, we present a simple Complexity measure (computable in time polynomial in $n$ given an implicit description of $f$) that describes their Communication Complexity up to polynomial factors and up to an additive error that is logarithmic in the input size. This gives a coarse taxonomy of the Communication Complexity of simple functions. Our work highlights the role of the well-known lower bounds of functions such as 'Set-Disjointness' and 'Indexing', while complementing them with the relatively lesser-known upper bounds for 'Gap-Inner-Product' (from the sketching literature) and 'Sparse-Gap-Inner-Product' (from the recent work of Canonne et al. [ITCS 2015]). We also present consequences to the study of Communication Complexity with imperfectly shared randomness where we show that for total permutation-invariant functions, imperfectly shared randomness results in only a polynomial blow-up in Communication Complexity after an additive $O(\log \log n)$ overhead.
Badih Ghazi - One of the best experts on this subject based on the ideXlab platform.
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Communication Complexity of permutation invariant functions
Symposium on Discrete Algorithms, 2016Co-Authors: Badih Ghazi, Pritish Kamath, Madhu SudanAbstract:Motivated by the quest for a broader understanding of upper bounds in Communication Complexity, at least for simple functions, we introduce the class of "permutation-invariant" functions. A partial function f: {0, 1}n × {0, 1}n → {0, 1, ?} is permutation-invariant if for every bijection π: {1, . . ., n} → {1, . . ., n} and every x, y ∈ {0, 1}n, it is the case that f(x, y) = f(xπ, yπ). Most of the commonly studied functions in Communication Complexity are permutation-invariant. For such functions, we present a simple Complexity measure (computable in time polynomial in n given an implicit description of f) that describes their Communication Complexity up to polynomial factors and up to an additive error that is logarithmic in the input size. This gives a coarse taxonomy of the Communication Complexity of simple functions. Our work highlights the role of the well-known lower bounds of functions such as S et -D isjointness and I ndexing , while complementing them with the relatively lesser-known upper bounds for G ap -I nner -P roduct (from the sketching literature) and S parse -G ap -I nner -P roduct (from the recent work of Canonne et al. [ITCS 2015]). We also present consequences to the study of Communication Complexity with imperfectly shared randomness where we show that for total permutation-invariant functions, imperfectly shared randomness results in only a polynomial blow-up in Communication Complexity after an additive O(log log n) overhead.
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SODA - Communication Complexity of permutation-invariant functions
Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, 2015Co-Authors: Badih Ghazi, Pritish Kamath, Madhu SudanAbstract:Motivated by the quest for a broader understanding of upper bounds in Communication Complexity, at least for simple functions, we introduce the class of "permutation-invariant" functions. A partial function f: {0, 1}n × {0, 1}n → {0, 1, ?} is permutation-invariant if for every bijection π: {1, . . ., n} → {1, . . ., n} and every x, y ∈ {0, 1}n, it is the case that f(x, y) = f(xπ, yπ). Most of the commonly studied functions in Communication Complexity are permutation-invariant. For such functions, we present a simple Complexity measure (computable in time polynomial in n given an implicit description of f) that describes their Communication Complexity up to polynomial factors and up to an additive error that is logarithmic in the input size. This gives a coarse taxonomy of the Communication Complexity of simple functions. Our work highlights the role of the well-known lower bounds of functions such as S et -D isjointness and I ndexing , while complementing them with the relatively lesser-known upper bounds for G ap -I nner -P roduct (from the sketching literature) and S parse -G ap -I nner -P roduct (from the recent work of Canonne et al. [ITCS 2015]). We also present consequences to the study of Communication Complexity with imperfectly shared randomness where we show that for total permutation-invariant functions, imperfectly shared randomness results in only a polynomial blow-up in Communication Complexity after an additive O(log log n) overhead.
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Communication Complexity of Permutation-Invariant Functions
arXiv: Computational Complexity, 2015Co-Authors: Badih Ghazi, Pritish Kamath, Madhu SudanAbstract:Motivated by the quest for a broader understanding of Communication Complexity of simple functions, we introduce the class of "permutation-invariant" functions. A partial function $f:\{0,1\}^n \times \{0,1\}^n\to \{0,1,?\}$ is permutation-invariant if for every bijection $\pi:\{1,\ldots,n\} \to \{1,\ldots,n\}$ and every $\mathbf{x}, \mathbf{y} \in \{0,1\}^n$, it is the case that $f(\mathbf{x}, \mathbf{y}) = f(\mathbf{x}^{\pi}, \mathbf{y}^{\pi})$. Most of the commonly studied functions in Communication Complexity are permutation-invariant. For such functions, we present a simple Complexity measure (computable in time polynomial in $n$ given an implicit description of $f$) that describes their Communication Complexity up to polynomial factors and up to an additive error that is logarithmic in the input size. This gives a coarse taxonomy of the Communication Complexity of simple functions. Our work highlights the role of the well-known lower bounds of functions such as 'Set-Disjointness' and 'Indexing', while complementing them with the relatively lesser-known upper bounds for 'Gap-Inner-Product' (from the sketching literature) and 'Sparse-Gap-Inner-Product' (from the recent work of Canonne et al. [ITCS 2015]). We also present consequences to the study of Communication Complexity with imperfectly shared randomness where we show that for total permutation-invariant functions, imperfectly shared randomness results in only a polynomial blow-up in Communication Complexity after an additive $O(\log \log n)$ overhead.
Harry Buhrman - One of the best experts on this subject based on the ideXlab platform.
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Quantum Entanglement and Communication Complexity
BRICS Report Series, 2016Co-Authors: Harry Buhrman, Richard Cleve, Wim Van DamAbstract:We consider a variation of the multi-party Communication Complexity scenario where the parties are supplied with an extra resource: particlesin an entangled quantum state. We show that, although a priorquantum entanglement cannot be used to simulate a Communication channel, it can reduce the Communication Complexity of functions insome cases. Specifically, we show that, for a particular function among three parties (each of which possesses part of the function's input), a prior quantum entanglement enables them to learn the value of thefunction with only three bits of Communication occurring among the parties, whereas, without quantum entanglement, four bits of Communication are necessary. We also show that, for a particular two-party probabilistic Communication Complexity problem, quantum entanglementresults in less Communication than is required with only classicalrandom correlations (instead of quantum entanglement). These results are a noteworthy contrast to the well-known fact that quantum entanglement cannot be used to actually simulate Communication amongremote parties.
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Nonlocality and Communication Complexity
Reviews of Modern Physics, 2010Co-Authors: Harry Buhrman, Richard Cleve, Serge Massar, Ronald De WolfAbstract:Quantum information processing is the emerging field that defines and realizes computing devices that make use of quantum mechanical principles, like the superposition principle, entanglement, and interference. Until recently the common notion of computing was based on classical mechanics, and did not take into account all the possibilities that physically-realizable computing devices offer in principle. The field gained momentum after Peter Shor developed an efficient algorithm for factoring numbers, demonstrating the potential computing powers that quantum computing devices can unleash. In this review we study the information counterpart of computing. It was realized early on by Holevo, that quantum bits, the quantum mechanical counterpart of classical bits, cannot be used for efficient transformation of information, in the sense that arbitrary k-bit messages can not be compressed into messages of k − 1 qubits. The abstract form of the distributed computing setting is called Communication Complexity. It studies the amount of information, in terms of bits or in our case qubits, that two spatially separated computing devices need to exchange in order to perform some computational task. Surprisingly, quantum mechanics can be used to obtain dramatic advantages for such tasks. We review the area of quantum Communication Complexity, and show how it connects the foundational physics questions regarding non-locality with those of Communication Complexity studied in theoretical computer science. The first examples exhibiting the advantage of the use of qubits in distributed information-processing tasks were based on non-locality tests. However, by now the field has produced strong and interesting quantum protocols and algorithms of its own that demonstrate that entanglement, although it cannot be used to replace Communication, can be used to reduce the Communication exponentially. In turn, these new advances yield a new outlook on the foundations of physics, and could even yield new proposals for experiments that test the foundations of physics.
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Individual Communication Complexity
Journal of Computer and System Sciences, 2007Co-Authors: Harry Buhrman, Hartmut Klauck, Nikolai K. Vereshchagin, Paul M. B. VitányiAbstract:We initiate the theory of Communication Complexity of individual inputs held by the agents. This contrasts with the usual Communication Complexity model, where one counts the amount of Communication for the worst-case or the average-case inputs. The individual Communication Complexity gives more information (the worst-case and the average-case can be derived from it but not vice versa) and may in some cases be of more interest. It is given in terms of the Kolmogorov complexities of the individual inputs. There are different measures of Communication Complexity depending on whether the protocol is guaranteed to be correct for all inputs or not, and whether there's one-way or two-way Communication. Bounds are provided for the Communication of specific functions and connections between the different Communication measures are shown. Some counter-intuitive results: for deterministic protocols that need to communicate Bob's input to Alice they need to communicate all of Bob's input (rather than the information difference with Alice's input), and there are so-called ''non-communicable'' inputs.
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Individual Communication Complexity
Lecture Notes in Computer Science, 2004Co-Authors: Harry Buhrman, Hartmut Klauck, Nikolai K. Vereshchagin, Paul M. B. VitányiAbstract:We initiate the theory of Communication Complexity of individual inputs held by the agents, rather than worst-case or average-case. We consider total, partial, and partially correct protocols, one-way versus two-way, with (not in this version) and without help bits.
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Individual Communication Complexity
arXiv: Computational Complexity, 2003Co-Authors: Harry Buhrman, Hartmut Klauck, Nikolai K. Vereshchagin, Paul M. B. VitányiAbstract:We initiate the theory of Communication Complexity of individual inputs held by the agents, rather than worst-case or average-case. We consider total, partial, and partially correct protocols, one-way versus two-way, with and without help bits. The results are expressed in trems of Kolmogorov Complexity.
Pritish Kamath - One of the best experts on this subject based on the ideXlab platform.
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Communication Complexity of permutation invariant functions
Symposium on Discrete Algorithms, 2016Co-Authors: Badih Ghazi, Pritish Kamath, Madhu SudanAbstract:Motivated by the quest for a broader understanding of upper bounds in Communication Complexity, at least for simple functions, we introduce the class of "permutation-invariant" functions. A partial function f: {0, 1}n × {0, 1}n → {0, 1, ?} is permutation-invariant if for every bijection π: {1, . . ., n} → {1, . . ., n} and every x, y ∈ {0, 1}n, it is the case that f(x, y) = f(xπ, yπ). Most of the commonly studied functions in Communication Complexity are permutation-invariant. For such functions, we present a simple Complexity measure (computable in time polynomial in n given an implicit description of f) that describes their Communication Complexity up to polynomial factors and up to an additive error that is logarithmic in the input size. This gives a coarse taxonomy of the Communication Complexity of simple functions. Our work highlights the role of the well-known lower bounds of functions such as S et -D isjointness and I ndexing , while complementing them with the relatively lesser-known upper bounds for G ap -I nner -P roduct (from the sketching literature) and S parse -G ap -I nner -P roduct (from the recent work of Canonne et al. [ITCS 2015]). We also present consequences to the study of Communication Complexity with imperfectly shared randomness where we show that for total permutation-invariant functions, imperfectly shared randomness results in only a polynomial blow-up in Communication Complexity after an additive O(log log n) overhead.
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SODA - Communication Complexity of permutation-invariant functions
Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, 2015Co-Authors: Badih Ghazi, Pritish Kamath, Madhu SudanAbstract:Motivated by the quest for a broader understanding of upper bounds in Communication Complexity, at least for simple functions, we introduce the class of "permutation-invariant" functions. A partial function f: {0, 1}n × {0, 1}n → {0, 1, ?} is permutation-invariant if for every bijection π: {1, . . ., n} → {1, . . ., n} and every x, y ∈ {0, 1}n, it is the case that f(x, y) = f(xπ, yπ). Most of the commonly studied functions in Communication Complexity are permutation-invariant. For such functions, we present a simple Complexity measure (computable in time polynomial in n given an implicit description of f) that describes their Communication Complexity up to polynomial factors and up to an additive error that is logarithmic in the input size. This gives a coarse taxonomy of the Communication Complexity of simple functions. Our work highlights the role of the well-known lower bounds of functions such as S et -D isjointness and I ndexing , while complementing them with the relatively lesser-known upper bounds for G ap -I nner -P roduct (from the sketching literature) and S parse -G ap -I nner -P roduct (from the recent work of Canonne et al. [ITCS 2015]). We also present consequences to the study of Communication Complexity with imperfectly shared randomness where we show that for total permutation-invariant functions, imperfectly shared randomness results in only a polynomial blow-up in Communication Complexity after an additive O(log log n) overhead.
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Communication Complexity of Permutation-Invariant Functions
arXiv: Computational Complexity, 2015Co-Authors: Badih Ghazi, Pritish Kamath, Madhu SudanAbstract:Motivated by the quest for a broader understanding of Communication Complexity of simple functions, we introduce the class of "permutation-invariant" functions. A partial function $f:\{0,1\}^n \times \{0,1\}^n\to \{0,1,?\}$ is permutation-invariant if for every bijection $\pi:\{1,\ldots,n\} \to \{1,\ldots,n\}$ and every $\mathbf{x}, \mathbf{y} \in \{0,1\}^n$, it is the case that $f(\mathbf{x}, \mathbf{y}) = f(\mathbf{x}^{\pi}, \mathbf{y}^{\pi})$. Most of the commonly studied functions in Communication Complexity are permutation-invariant. For such functions, we present a simple Complexity measure (computable in time polynomial in $n$ given an implicit description of $f$) that describes their Communication Complexity up to polynomial factors and up to an additive error that is logarithmic in the input size. This gives a coarse taxonomy of the Communication Complexity of simple functions. Our work highlights the role of the well-known lower bounds of functions such as 'Set-Disjointness' and 'Indexing', while complementing them with the relatively lesser-known upper bounds for 'Gap-Inner-Product' (from the sketching literature) and 'Sparse-Gap-Inner-Product' (from the recent work of Canonne et al. [ITCS 2015]). We also present consequences to the study of Communication Complexity with imperfectly shared randomness where we show that for total permutation-invariant functions, imperfectly shared randomness results in only a polynomial blow-up in Communication Complexity after an additive $O(\log \log n)$ overhead.
