64 research outputs found
Separating NOF communication complexity classes RP and NP
We provide a non-explicit separation of the number-on-forehead communication
complexity classes RP and NP when the number of players is up to \delta log(n)
for any \delta<1. Recent lower bounds on Set-Disjointness [LS08,CA08] provide
an explicit separation between these classes when the number of players is only
up to o(loglog(n))
Tight Bounds for Set Disjointness in the Message Passing Model
In a multiparty message-passing model of communication, there are
players. Each player has a private input, and they communicate by sending
messages to one another over private channels. While this model has been used
extensively in distributed computing and in multiparty computation, lower
bounds on communication complexity in this model and related models have been
somewhat scarce. In recent work \cite{phillips12,woodruff12,woodruff13}, strong
lower bounds of the form were obtained for several
functions in the message-passing model; however, a lower bound on the classical
Set Disjointness problem remained elusive.
In this paper, we prove tight lower bounds of the form
for the Set Disjointness problem in the message passing model. Our bounds are
obtained by developing information complexity tools in the message-passing
model, and then proving an information complexity lower bound for Set
Disjointness. As a corollary, we show a tight lower bound for the task
allocation problem \cite{DruckerKuhnOshman} via a reduction from Set
Disjointness
A Lower Bound for Sampling Disjoint Sets
Suppose Alice and Bob each start with private randomness and no other input, and they wish to engage in a protocol in which Alice ends up with a set x subseteq[n] and Bob ends up with a set y subseteq[n], such that (x,y) is uniformly distributed over all pairs of disjoint sets. We prove that for some constant beta0 of the uniform distribution over all pairs of disjoint sets of size sqrt{n}
A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing and LDCs
The Bonami-Beckner hypercontractive inequality is a powerful tool in Fourier
analysis of real-valued functions on the Boolean cube. In this paper we present
a version of this inequality for matrix-valued functions on the Boolean cube.
Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also
present a number of applications. First, we analyze maps that encode
classical bits into qubits, in such a way that each set of bits can be
recovered with some probability by an appropriate measurement on the quantum
encoding; we show that if , then the success probability is
exponentially small in . This result may be viewed as a direct product
version of Nayak's quantum random access code bound. It in turn implies strong
direct product theorems for the one-way quantum communication complexity of
Disjointness and other problems. Second, we prove that error-correcting codes
that are locally decodable with 2 queries require length exponential in the
length of the encoded string. This gives what is arguably the first
``non-quantum'' proof of a result originally derived by Kerenidis and de Wolf
using quantum information theory, and answers a question by Trevisan.Comment: This is the full version of a paper that will appear in the
proceedings of the IEEE FOCS 08 conferenc
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