42,175 research outputs found
The Metric Nearness Problem
Metric nearness refers to the problem of optimally restoring metric properties to
distance measurements that happen to be nonmetric due to measurement errors or otherwise. Metric
data can be important in various settings, for example, in clustering, classification, metric-based
indexing, query processing, and graph theoretic approximation algorithms. This paper formulates
and solves the metric nearness problem: Given a set of pairwise dissimilarities, find a “nearest” set
of distances that satisfy the properties of a metric—principally the triangle inequality. For solving
this problem, the paper develops efficient triangle fixing algorithms that are based on an iterative
projection method. An intriguing aspect of the metric nearness problem is that a special case turns
out to be equivalent to the all pairs shortest paths problem. The paper exploits this equivalence and
develops a new algorithm for the latter problem using a primal-dual method. Applications to graph
clustering are provided as an illustration. We include experiments that demonstrate the computational
superiority of triangle fixing over general purpose convex programming software. Finally, we
conclude by suggesting various useful extensions and generalizations to metric nearness
All geographical distances are optimal
International audienceTriangular inequality is one of the four mathematical properties of distance. Its respect derives from the optimal nature of the measurement of distance. This demonstration (L'Hostis 2016, 2017) reveals key aspects of distances and geographical spaces. We develop this argument by investigating the idea of the optimality of distance through a mathematical and geometric discussion, and by dealing with empirical approaches to applied geography.The first part of the paper explores the consequences of considering that the mathematical property of triangle inequality is always respected. In fact, no violations of triangle inequality are observed in geographical spaces. The study of the optimality of distances in empirical approaches confirms the key role of the property of triangle inequality. The general principle of least-effort applies for most movements and spacings. In addition, however, trajectories with multiple detours, like those of shoppers, runners and nomads, are optimal from a certain point of view. This is also the case for excess travel, i.e. a situation of disjunction between an optimum as perceived by a person in movement and an optimum as perceived by an external observer. Any movement, any spacing within and between cities and in geographical space in general, exhibits a form of optimality, and all geographical distances are optimal
Violation of the Finner inequality in the four-output triangle network
Network nonlocality allows one to demonstrate non-classicality in networks
with fixed joint measurements, that is without random measurement settings. The
simplest network in a loop, the triangle, with 4 outputs per party is
especially intriguing. The "elegant distribution" [N. Gisin, Entropy 21, 325
(2019)] still resists analytic proofs, despite its many symmetries. In
particular, this distribution is invariant under any output permutation. The
Finner inequality, which holds for all local and quantum distributions, has
been conjectured to be also valid for all no-signalling distributions with
independent sources (NSI distributions). Here we provide evidence that this
conjecture is false by constructing a 4-output network box that violate the
Finner inequality and prove that it satisfies all NSI inflations up to the
enneagon. As a first step toward the proof of the nonlocality of the elegant
distribution, we prove the nonlocality of the distributions that saturates the
Finner inequality by using geometrical arguments.Comment: 8 pages, 8 figures, Any comments are welcome ([email protected]
Leggett-Garg inequalities and the geometry of the cut polytope
The Bell and Leggett-Garg tests offer operational ways to demonstrate that
non-classical behavior manifests itself in quantum systems, and
experimentalists have implemented these protocols to show that classical
worldviews such as local realism and macrorealism are false, respectively.
Previous theoretical research has exposed important connections between more
general Bell inequalities and polyhedral combinatorics. We show here that
general Leggett-Garg inequalities are closely related to the cut polytope of
the complete graph, a geometric object well-studied in combinatorics. Building
on that connection, we offer a family of Leggett-Garg inequalities that are not
trivial combinations of the most basic Leggett-Garg inequalities. We then show
that violations of macrorealism can occur in surprising ways, by giving an
example of a quantum system that violates the new "pentagon" Leggett-Garg
inequality but does not violate any of the basic "triangle" Leggett-Garg
inequalities.Comment: 5 pages, 1 figur
Joint Measurability, Einstein-Podolsky-Rosen Steering, and Bell Nonlocality
We investigate the relation between the incompatibility of quantum
measurements and quantum nonlocality. We show that any set of measurements that
is not jointly measurable (i.e. incompatible) can be used for demonstrating EPR
steering, a form of quantum nonlocality. This implies that EPR steering and
(non) joint measurability can be viewed as equivalent. Moreover, we discuss the
connection between Bell nonlocality and joint measurability, and give evidence
that both notions are inequivalent. Specifically, we exhibit a set of
incompatible quantum measurements and show that it does not violate a large
class of Bell inequalities. This suggest the existence of incompatible quantum
measurements which are Bell local, similarly to certain entangled states which
admit a local hidden variable model.Comment: 6 pages, 1 figure, 2 tables, title slightly changed, one reference
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Shortcuts through Colocation Facilities
Network overlays, running on top of the existing Internet substrate, are of
perennial value to Internet end-users in the context of, e.g., real-time
applications. Such overlays can employ traffic relays to yield path latencies
lower than the direct paths, a phenomenon known as Triangle Inequality
Violation (TIV). Past studies identify the opportunities of reducing latency
using TIVs. However, they do not investigate the gains of strategically
selecting relays in Colocation Facilities (Colos). In this work, we answer the
following questions: (i) how Colo-hosted relays compare with other relays as
well as with the direct Internet, in terms of latency (RTT) reductions; (ii)
what are the best locations for placing the relays to yield these reductions.
To this end, we conduct a large-scale one-month measurement of inter-domain
paths between RIPE Atlas (RA) nodes as endpoints, located at eyeball networks.
We employ as relays Planetlab nodes, other RA nodes, and machines in Colos. We
examine the RTTs of the overlay paths obtained via the selected relays, as well
as the direct paths. We find that Colo-based relays perform the best and can
achieve latency reductions against direct paths, ranging from a few to 100s of
milliseconds, in 76% of the total cases; 75% (58% of total cases) of these
reductions require only 10 relays in 6 large Colos.Comment: In Proceedings of the ACM Internet Measurement Conference (IMC '17),
London, GB, 201
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