10,390 research outputs found
Quantifying multipartite nonlocality
The nonlocal correlations of multipartite entangled states can be reproduced
by a classical model if sufficiently many parties join together or if
sufficiently many parties broadcast their measurement inputs. The maximal
number m of groups and the minimal number k of broadcasting parties that allow
for the reproduction of a given set of correlations quantify their multipartite
nonlocal content. We show how upper-bounds on m and lower-bounds on k can be
computed from the violation of the Mermin-Svetlichny inequalities. While
n-partite GHZ states violate these inequalities maximally, we find that W
states violate them only by a very small amount.Comment: 4 pages, 3 figures. v2: minor change
On the Computational Power of Radio Channels
Radio networks can be a challenging platform for which to develop distributed algorithms, because the network nodes must contend for a shared channel. In some cases, though, the shared medium is an advantage rather than a disadvantage: for example, many radio network algorithms cleverly use the shared channel to approximate the degree of a node, or estimate the contention. In this paper we ask how far the inherent power of a shared radio channel goes, and whether it can efficiently compute "classicaly hard" functions such as Majority, Approximate Sum, and Parity.
Using techniques from circuit complexity, we show that in many cases, the answer is "no". We show that simple radio channels, such as the beeping model or the channel with collision-detection, can be approximated by a low-degree polynomial, which makes them subject to known lower bounds on functions such as Parity and Majority; we obtain round lower bounds of the form Omega(n^{delta}) on these functions, for delta in (0,1). Next, we use the technique of random restrictions, used to prove AC^0 lower bounds, to prove a tight lower bound of Omega(1/epsilon^2) on computing a (1 +/- epsilon)-approximation to the sum of the nodes\u27 inputs. Our techniques are general, and apply to many types of radio channels studied in the literature
Beyond Geometry : Towards Fully Realistic Wireless Models
Signal-strength models of wireless communications capture the gradual fading
of signals and the additivity of interference. As such, they are closer to
reality than other models. However, nearly all theoretic work in the SINR model
depends on the assumption of smooth geometric decay, one that is true in free
space but is far off in actual environments. The challenge is to model
realistic environments, including walls, obstacles, reflections and anisotropic
antennas, without making the models algorithmically impractical or analytically
intractable.
We present a simple solution that allows the modeling of arbitrary static
situations by moving from geometry to arbitrary decay spaces. The complexity of
a setting is captured by a metricity parameter Z that indicates how far the
decay space is from satisfying the triangular inequality. All results that hold
in the SINR model in general metrics carry over to decay spaces, with the
resulting time complexity and approximation depending on Z in the same way that
the original results depends on the path loss term alpha. For distributed
algorithms, that to date have appeared to necessarily depend on the planarity,
we indicate how they can be adapted to arbitrary decay spaces.
Finally, we explore the dependence on Z in the approximability of core
problems. In particular, we observe that the capacity maximization problem has
exponential upper and lower bounds in terms of Z in general decay spaces. In
Euclidean metrics and related growth-bounded decay spaces, the performance
depends on the exact metricity definition, with a polynomial upper bound in
terms of Z, but an exponential lower bound in terms of a variant parameter phi.
On the plane, the upper bound result actually yields the first approximation of
a capacity-type SINR problem that is subexponential in alpha
Exploiting spontaneous transmissions for broadcasting and leader election in radio networks
We study two fundamental communication primitives: broadcasting and leader election in the classical model of multi-hop radio networks with unknown topology and without collision detection mechanisms. It has been known for almost 20 years that in undirected networks with n nodes and diameter D, randomized broadcasting requires Ω(D log n/D + log2 n) rounds, assuming that uninformed nodes are not allowed to communicate (until they are informed). Only very recently, Haeupler and Wajc (PODC'2016) showed that this bound can be improved for the model with spontaneous transmissions, providing an O(D log n log log n/log D + logO(1) n)-time broadcasting algorithm. In this article, we give a new and faster algorithm that completes broadcasting in O(D log n/log D + logO(1) n) time, succeeding with high probability. This yields the first optimal O(D)-time broadcasting algorithm whenever n is polynomial in D.
Furthermore, our approach can be applied to design a new leader election algorithm that matches the performance of our broadcasting algorithm. Previously, all fast randomized leader election algorithms have used broadcasting as a subroutine and their complexity has been asymptotically strictly larger than the complexity of broadcasting. In particular, the fastest previously known randomized leader election algorithm of Ghaffari and Haeupler (SODA'2013) requires O(D log n/D min {log log n, log n/D} + logO(1) n)-time, succeeding with high probability. Our new algorithm again requires O(D log n/log D + logO(1) n) time, also succeeding with high probability
Distributed Approximation Algorithms for Weighted Shortest Paths
A distributed network is modeled by a graph having nodes (processors) and
diameter . We study the time complexity of approximating {\em weighted}
(undirected) shortest paths on distributed networks with a {\em
bandwidth restriction} on edges (the standard synchronous \congest model). The
question whether approximation algorithms help speed up the shortest paths
(more precisely distance computation) was raised since at least 2004 by Elkin
(SIGACT News 2004). The unweighted case of this problem is well-understood
while its weighted counterpart is fundamental problem in the area of
distributed approximation algorithms and remains widely open. We present new
algorithms for computing both single-source shortest paths (\sssp) and
all-pairs shortest paths (\apsp) in the weighted case.
Our main result is an algorithm for \sssp. Previous results are the classic
-time Bellman-Ford algorithm and an -time
-approximation algorithm, for any integer
, which follows from the result of Lenzen and Patt-Shamir (STOC 2013).
(Note that Lenzen and Patt-Shamir in fact solve a harder problem, and we use
to hide the O(\poly\log n) term.) We present an -time -approximation algorithm for \sssp. This
algorithm is {\em sublinear-time} as long as is sublinear, thus yielding a
sublinear-time algorithm with almost optimal solution. When is small, our
running time matches the lower bound of by Das Sarma
et al. (SICOMP 2012), which holds even when , up to a
\poly\log n factor.Comment: Full version of STOC 201
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