2,013 research outputs found
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
On the communication complexity of sparse set disjointness and exists-equal problems
In this paper we study the two player randomized communication complexity of
the sparse set disjointness and the exists-equal problems and give matching
lower and upper bounds (up to constant factors) for any number of rounds for
both of these problems. In the sparse set disjointness problem, each player
receives a k-subset of [m] and the goal is to determine whether the sets
intersect. For this problem, we give a protocol that communicates a total of
O(k\log^{(r)}k) bits over r rounds and errs with very small probability. Here
we can take r=\log^{*}k to obtain a O(k) total communication \log^{*}k-round
protocol with exponentially small error probability, improving on the O(k)-bits
O(\log k)-round constant error probability protocol of Hastad and Wigderson
from 1997.
In the exist-equal problem, the players receive vectors x,y\in [t]^n and the
goal is to determine whether there exists a coordinate i such that x_i=y_i.
Namely, the exists-equal problem is the OR of n equality problems. Observe that
exists-equal is an instance of sparse set disjointness with k=n, hence the
protocol above applies here as well, giving an O(n\log^{(r)}n) upper bound. Our
main technical contribution in this paper is a matching lower bound: we show
that when t=\Omega(n), any r-round randomized protocol for the exists-equal
problem with error probability at most 1/3 should have a message of size
\Omega(n\log^{(r)}n). Our lower bound holds even for super-constant r <=
\log^*n, showing that any O(n) bits exists-equal protocol should have \log^*n -
O(1) rounds
Some Communication Complexity Results and their Applications
Communication Complexity represents one of the premier techniques for proving lower bounds in theoretical computer science. Lower bounds on communication problems can be leveraged to prove lower bounds in several different areas. In this work, we study three different communication complexity problems. The lower bounds for these problems have applications in circuit complexity, wireless sensor networks, and streaming algorithms. First, we study the multiparty pointer jumping problem. We present the first nontrivial upper bound for this problem. We also provide a suite of strong lower bounds under several restricted classes of protocols. Next, we initiate the study of several non-monotone functions in the distributed functional monitoring setting and provide several lower bounds. In particular, we give a generic adversarial technique and show that when deletions are allowed, no nontrivial protocol is possible. Finally, we study the Gap-Hamming-Distance problem and give tight lower bounds for protocols that use a constant number of messages. As a result, we take a well-known lower bound for one-pass streaming algorithms for a host of problems and extend it so it applies to streaming algorithms that use a constant number of passes
Distributed PCP Theorems for Hardness of Approximation in P
We present a new distributed model of probabilistically checkable proofs
(PCP). A satisfying assignment to a CNF formula is
shared between two parties, where Alice knows , Bob knows
, and both parties know . The goal is to have
Alice and Bob jointly write a PCP that satisfies , while
exchanging little or no information. Unfortunately, this model as-is does not
allow for nontrivial query complexity. Instead, we focus on a non-deterministic
variant, where the players are helped by Merlin, a third party who knows all of
.
Using our framework, we obtain, for the first time, PCP-like reductions from
the Strong Exponential Time Hypothesis (SETH) to approximation problems in P.
In particular, under SETH we show that there are no truly-subquadratic
approximation algorithms for Bichromatic Maximum Inner Product over
{0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate
Regular Expression Matching, and Diameter in Product Metric. All our
inapproximability factors are nearly-tight. In particular, for the first two
problems we obtain nearly-polynomial factors of ; only
-factor lower bounds (under SETH) were known before
One-Sided Error Communication Complexity of Gap Hamming Distance
Assume that Alice has a binary string x and Bob a binary string y, both strings are of length n. Their goal is to output 0, if x and y are at least L-close in Hamming distance, and output 1, if x and y are at least U-far in Hamming distance, where L < U are some integer parameters known to both parties. If the Hamming distance between x and y lies in the interval (L, U), they are allowed to output anything. This problem is called the Gap Hamming Distance. In this paper we study public-coin one-sided error communication complexity of this problem. The error with probability at most 1/2 is allowed only for pairs at Hamming distance at least U. In this paper we determine this complexity up to factors logarithmic in L. The protocol we construct for the upper bound is simultaneous
Communication Complexity of Permutation-Invariant Functions
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 is permutation-invariant if for every bijection
and every , it is the case that . 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 given
an implicit description of ) 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 overhead
On the power of non-local boxes
A non-local box is a virtual device that has the following property: given
that Alice inputs a bit at her end of the device and that Bob does likewise, it
produces two bits, one at Alice's end and one at Bob's end, such that the XOR
of the outputs is equal to the AND of the inputs. This box, inspired from the
CHSH inequality, was first proposed by Popescu and Rohrlich to examine the
question: given that a maximally entangled pair of qubits is non-local, why is
it not maximally non-local? We believe that understanding the power of this box
will yield insight into the non-locality of quantum mechanics. It was shown
recently by Cerf, Gisin, Massar and Popescu, that this imaginary device is able
to simulate correlations from any measurement on a singlet state. Here, we show
that the non-local box can in fact do much more: through the simulation of the
magic square pseudo-telepathy game and the Mermin-GHZ pseudo-telepathy game, we
show that the non-local box can simulate quantum correlations that no entangled
pair of qubits can in a bipartite scenario and even in a multi-party scenario.
Finally we show that a single non-local box cannot simulate all quantum
correlations and propose a generalization for a multi-party non-local box. In
particular, we show quantum correlations whose simulation requires an
exponential amount of non-local boxes, in the number of maximally entangled
qubit pairs.Comment: 14 pages, 1 figur
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