Open AccessJournal Article
Communication Lower Bounds Using Dual Polynomials.
TLDR
In this paper, the authors provide a unified guide to all the key proofs of Sherstov's degree/discrepancy theorem, which translates lower bounds on the threshold degree of a Boolean function into upper bounds on discrepancy of a related function.Abstract:
Representations of Boolean functions by real polynomials play an important role in complexity theory. Typically, one is interested in the least degree of a polynomial p(x_1,...,x_n) that approximates or sign-represents a given Boolean function f(x_1,...,x_n). This article surveys a new and growing body of work in communication complexity that centers around the dual objects, i.e., polynomials that certify the difficulty of approximating or sign-representing a given function. We provide a unified guide to the following results, complete with all the key proofs:
(1) Sherstov's Degree/Discrepancy Theorem, which translates lower bounds on the threshold degree of a Boolean function into upper bounds on the discrepancy of a related function;
(2) Two different methods for proving lower bounds on bounded-error communication based on the approximate degree: Sherstov's pattern matrix method and Shi and Zhu's block composition method;
(3) Extension of the pattern matrix method to the multiparty model, obtained by Lee and Shraibman and by Chattopadhyay and Ada, and the resulting improved lower bounds for DISJOINTNESS;
(4) David and Pitassi's separation of NP and BPP in multiparty communication complexity for k=(1-eps)log n players.read more
Citations
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Journal Article
Rectangles Are Nonnegative Juntas.
TL;DR: In this paper, the communication lower bound for composed functions of the form $f\circ g^n, where f is any boolean function on n inputs and g is a sufficiently hard two-party gadget, was established.
Book ChapterDOI
On Computation and Communication with Small Bias
TL;DR: In this article, it was shown that PPcc is strictly included in UPPcc, which is the same class of complexity classes as UPPCC with weakly restricted bias and unrestricted bias.
Journal Article
Discrepancy and the power of bottom fan-in in depth-three circuits.
TL;DR: It is proved that depth three circuits consisting of a MAJORITY gate at the output, gates computing arbitrary symmetric function at the second layer and arbitrary gates of bounded fan-in at the base layer cannot simulate the circuit class AC0 in sub-exponential size.
Journal ArticleDOI
Improved Separations between Nondeterministic and Randomized Multiparty Communication
TL;DR: An explicit function f that can be computed by a nondeterministic number-on-forehead protocol communicating O(logn) bits, but that requires n bits of communication for randomized number- on-fore head protocols is exhibited.
Book ChapterDOI
Improved Separations between Nondeterministic and Randomized Multiparty Communication
TL;DR: An explicit function f: {0, 1}ni¾?{0,1} that can be computed by a nondeterministic number-on-forehead protocol communicating O(logn) bits, but that requires ni½?(1)bits of communication for randomized number- on-fore head protocols with k= i¾?,·lognplayers, is exhibited.
References
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Book
Matrix Analysis
Roger A. Horn,Charles R. Johnson +1 more
TL;DR: In this article, the authors present results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrate their importance in a variety of applications, such as linear algebra and matrix theory.
Book
Communication Complexity
Eyal Kushilevitz,Noam Nisan +1 more
TL;DR: This chapter surveys the theory of two-party communication complexity and presents results regarding the following models of computation: • Finite automata • Turing machines • Decision trees • Ordered binary decision diagrams • VLSI chips • Networks of threshold gates.
Journal ArticleDOI
Quantum Communication
Nicolas Gisin,Rob Thew +1 more
TL;DR: The current state of research and future directions in quantum key distribution and quantum networks are reviewed in this paper, with a special emphasis on quantum key distributions and quantum key sharing in quantum networks.
Journal ArticleDOI
The probabilistic communication complexity of set intersection
TL;DR: It is shown that, for inputs of length n, the probabilistic (bounded error) communication complexity of set intersection is $\Theta ( n )$.