8,985 research outputs found
Partition bound is quadratically tight for product distributions
Let be a 2-party
function. For every product distribution on ,
we show that
where is the distributional communication
complexity of with error at most under the distribution
and is the {\em partition bound} of , as defined by
Jain and Klauck [{\em Proc. 25th CCC}, 2010]. We also prove a similar bound in
terms of , the {\em information complexity} of ,
namely, The latter bound was recently and
independently established by Kol [{\em Proc. 48th STOC}, 2016] using a
different technique.
We show a similar result for query complexity under product distributions.
Let be a function. For every bit-wise
product distribution on , we show that
where
is the distributional query complexity of
with error at most under the distribution and
is the {\em query partition bound} of the function
.
Partition bounds were introduced (in both communication complexity and query
complexity models) to provide LP-based lower bounds for randomized
communication complexity and randomized query complexity. Our results
demonstrate that these lower bounds are polynomially tight for {\em product}
distributions.Comment: The previous version of the paper erroneously stated the main result
in terms of relaxed partition number instead of partition numbe
An Optimal Lower Bound on the Communication Complexity of Gap-Hamming-Distance
We prove an optimal lower bound on the randomized communication
complexity of the much-studied Gap-Hamming-Distance problem. As a consequence,
we obtain essentially optimal multi-pass space lower bounds in the data stream
model for a number of fundamental problems, including the estimation of
frequency moments.
The Gap-Hamming-Distance problem is a communication problem, wherein Alice
and Bob receive -bit strings and , respectively. They are promised
that the Hamming distance between and is either at least
or at most , and their goal is to decide which of these is the
case. Since the formal presentation of the problem by Indyk and Woodruff (FOCS,
2003), it had been conjectured that the naive protocol, which uses bits of
communication, is asymptotically optimal. The conjecture was shown to be true
in several special cases, e.g., when the communication is deterministic, or
when the number of rounds of communication is limited.
The proof of our aforementioned result, which settles this conjecture fully,
is based on a new geometric statement regarding correlations in Gaussian space,
related to a result of C. Borell (1985). To prove this geometric statement, we
show that random projections of not-too-small sets in Gaussian space are close
to a mixture of translated normal variables
Classical and quantum partition bound and detector inefficiency
We study randomized and quantum efficiency lower bounds in communication
complexity. These arise from the study of zero-communication protocols in which
players are allowed to abort. Our scenario is inspired by the physics setup of
Bell experiments, where two players share a predefined entangled state but are
not allowed to communicate. Each is given a measurement as input, which they
perform on their share of the system. The outcomes of the measurements should
follow a distribution predicted by quantum mechanics; however, in practice, the
detectors may fail to produce an output in some of the runs. The efficiency of
the experiment is the probability that the experiment succeeds (neither of the
detectors fails).
When the players share a quantum state, this gives rise to a new bound on
quantum communication complexity (eff*) that subsumes the factorization norm.
When players share randomness instead of a quantum state, the efficiency bound
(eff), coincides with the partition bound of Jain and Klauck. This is one of
the strongest lower bounds known for randomized communication complexity, which
subsumes all the known combinatorial and algebraic methods including the
rectangle (corruption) bound, the factorization norm, and discrepancy.
The lower bound is formulated as a convex optimization problem. In practice,
the dual form is more feasible to use, and we show that it amounts to
constructing an explicit Bell inequality (for eff) or Tsirelson inequality (for
eff*). We give an example of a quantum distribution where the violation can be
exponentially bigger than the previously studied class of normalized Bell
inequalities.
For one-way communication, we show that the quantum one-way partition bound
is tight for classical communication with shared entanglement up to arbitrarily
small error.Comment: 21 pages, extended versio
The Partition Bound for Classical Communication Complexity and Query Complexity
We describe new lower bounds for randomized communication complexity and
query complexity which we call the partition bounds. They are expressed as the
optimum value of linear programs. For communication complexity we show that the
partition bound is stronger than both the rectangle/corruption bound and the
\gamma_2/generalized discrepancy bounds. In the model of query complexity we
show that the partition bound is stronger than the approximate polynomial
degree and classical adversary bounds. We also exhibit an example where the
partition bound is quadratically larger than polynomial degree and classical
adversary bounds.Comment: 28 pages, ver. 2, added conten
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}
Simulation Theorems via Pseudorandom Properties
We generalize the deterministic simulation theorem of Raz and McKenzie
[RM99], to any gadget which satisfies certain hitting property. We prove that
inner-product and gap-Hamming satisfy this property, and as a corollary we
obtain deterministic simulation theorem for these gadgets, where the gadget's
input-size is logarithmic in the input-size of the outer function. This answers
an open question posed by G\"{o}\"{o}s, Pitassi and Watson [GPW15]. Our result
also implies the previous results for the Indexing gadget, with better
parameters than was previously known. A preliminary version of the results
obtained in this work appeared in [CKL+17]
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