114,202 research outputs found
Correlation in Hard Distributions in Communication Complexity
We study the effect that the amount of correlation in a bipartite
distribution has on the communication complexity of a problem under that
distribution. We introduce a new family of complexity measures that
interpolates between the two previously studied extreme cases: the (standard)
randomised communication complexity and the case of distributional complexity
under product distributions.
We give a tight characterisation of the randomised complexity of Disjointness
under distributions with mutual information , showing that it is
for all . This smoothly interpolates
between the lower bounds of Babai, Frankl and Simon for the product
distribution case (), and the bound of Razborov for the randomised case.
The upper bounds improve and generalise what was known for product
distributions, and imply that any tight bound for Disjointness needs
bits of mutual information in the corresponding distribution.
We study the same question in the distributional quantum setting, and show a
lower bound of , and an upper bound, matching up to a
logarithmic factor.
We show that there are total Boolean functions on inputs that have
distributional communication complexity under all distributions of
information up to , while the (interactive) distributional complexity
maximised over all distributions is for .
We show that in the setting of one-way communication under product
distributions, the dependence of communication cost on the allowed error
is multiplicative in -- the previous upper bounds
had the dependence of more than
New bounds on classical and quantum one-way communication complexity
In this paper we provide new bounds on classical and quantum distributional
communication complexity in the two-party, one-way model of communication. In
the classical model, our bound extends the well known upper bound of Kremer,
Nisan and Ron to include non-product distributions. We show that for a boolean
function f:X x Y -> {0,1} and a non-product distribution mu on X x Y and
epsilon in (0,1/2) constant: D_{epsilon}^{1, mu}(f)= O((I(X:Y)+1) vc(f)), where
D_{epsilon}^{1, mu}(f) represents the one-way distributional communication
complexity of f with error at most epsilon under mu; vc(f) represents the
Vapnik-Chervonenkis dimension of f and I(X:Y) represents the mutual
information, under mu, between the random inputs of the two parties. For a
non-boolean function f:X x Y ->[k], we show a similar upper bound on
D_{epsilon}^{1, mu}(f) in terms of k, I(X:Y) and the pseudo-dimension of f' =
f/k. In the quantum one-way model we provide a lower bound on the
distributional communication complexity, under product distributions, of a
function f, in terms the well studied complexity measure of f referred to as
the rectangle bound or the corruption bound of f . We show for a non-boolean
total function f : X x Y -> Z and a product distribution mu on XxY,
Q_{epsilon^3/8}^{1, mu}(f) = Omega(rec_ epsilon^{1, mu}(f)), where
Q_{epsilon^3/8}^{1, mu}(f) represents the quantum one-way distributional
communication complexity of f with error at most epsilon^3/8 under mu and rec_
epsilon^{1, mu}(f) represents the one-way rectangle bound of f with error at
most epsilon under mu . Similarly for a non-boolean partial function f:XxY -> Z
U {*} and a product distribution mu on X x Y, we show, Q_{epsilon^6/(2 x
15^4)}^{1, mu}(f) = Omega(rec_ epsilon^{1, mu}(f)).Comment: ver 1, 19 page
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
Energy Efficiency in Cache Enabled Small Cell Networks With Adaptive User Clustering
Using a network of cache enabled small cells, traffic during peak hours can
be reduced considerably through proactively fetching the content that is most
probable to be requested. In this paper, we aim at exploring the impact of
proactive caching on an important metric for future generation networks,
namely, energy efficiency (EE). We argue that, exploiting the correlation in
user content popularity profiles in addition to the spatial repartitions of
users with comparable request patterns, can result in considerably improving
the achievable energy efficiency of the network. In this paper, the problem of
optimizing EE is decoupled into two related subproblems. The first one
addresses the issue of content popularity modeling. While most existing works
assume similar popularity profiles for all users in the network, we consider an
alternative caching framework in which, users are clustered according to their
content popularity profiles. In order to showcase the utility of the proposed
clustering scheme, we use a statistical model selection criterion, namely
Akaike information criterion (AIC). Using stochastic geometry, we derive a
closed-form expression of the achievable EE and we find the optimal active
small cell density vector that maximizes it. The second subproblem investigates
the impact of exploiting the spatial repartitions of users with comparable
request patterns. After considering a snapshot of the network, we formulate a
combinatorial optimization problem that enables to optimize content placement
such that the used transmission power is minimized. Numerical results show that
the clustering scheme enable to considerably improve the cache hit probability
and consequently the EE compared with an unclustered approach. Simulations also
show that the small base station allocation algorithm results in improving the
energy efficiency and hit probability.Comment: 30 pages, 5 figures, submitted to Transactions on Wireless
Communications (15-Dec-2016
MIMO Detection for High-Order QAM Based on a Gaussian Tree Approximation
This paper proposes a new detection algorithm for MIMO communication systems
employing high order QAM constellations. The factor graph that corresponds to
this problem is very loopy; in fact, it is a complete graph. Hence, a
straightforward application of the Belief Propagation (BP) algorithm yields
very poor results. Our algorithm is based on an optimal tree approximation of
the Gaussian density of the unconstrained linear system. The finite-set
constraint is then applied to obtain a loop-free discrete distribution. It is
shown that even though the approximation is not directly applied to the exact
discrete distribution, applying the BP algorithm to the loop-free factor graph
outperforms current methods in terms of both performance and complexity. The
improved performance of the proposed algorithm is demonstrated on the problem
of MIMO detection
Welfare Maximization with Limited Interaction
We continue the study of welfare maximization in unit-demand (matching)
markets, in a distributed information model where agent's valuations are
unknown to the central planner, and therefore communication is required to
determine an efficient allocation. Dobzinski, Nisan and Oren (STOC'14) showed
that if the market size is , then rounds of interaction (with
logarithmic bandwidth) suffice to obtain an -approximation to the
optimal social welfare. In particular, this implies that such markets converge
to a stable state (constant approximation) in time logarithmic in the market
size.
We obtain the first multi-round lower bound for this setup. We show that even
if the allowable per-round bandwidth of each agent is , the
approximation ratio of any -round (randomized) protocol is no better than
, implying an lower bound on the
rate of convergence of the market to equilibrium.
Our construction and technique may be of interest to round-communication
tradeoffs in the more general setting of combinatorial auctions, for which the
only known lower bound is for simultaneous () protocols [DNO14]
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