412 research outputs found
Optimal decremental connectivity in planar graphs
We show an algorithm for dynamic maintenance of connectivity information in
an undirected planar graph subject to edge deletions. Our algorithm may answer
connectivity queries of the form `Are vertices and connected with a
path?' in constant time. The queries can be intermixed with any sequence of
edge deletions, and the algorithm handles all updates in time. This
results improves over previously known time algorithm
Min-Cost Flow in Unit-Capacity Planar Graphs
In this paper we give an O~((nm)^(2/3) log C) time algorithm for computing min-cost flow (or min-cost circulation) in unit capacity planar multigraphs where edge costs are integers bounded by C. For planar multigraphs, this improves upon the best known algorithms for general graphs: the O~(m^(10/7) log C) time algorithm of Cohen et al. [SODA 2017], the O(m^(3/2) log(nC)) time algorithm of Gabow and Tarjan [SIAM J. Comput. 1989] and the O~(sqrt(n) m log C) time algorithm of Lee and Sidford [FOCS 2014]. In particular, our result constitutes the first known fully combinatorial algorithm that breaks the Omega(m^(3/2)) time barrier for min-cost flow problem in planar graphs.
To obtain our result we first give a very simple successive shortest paths based scaling algorithm for unit-capacity min-cost flow problem that does not explicitly operate on dual variables. This algorithm also runs in O~(m^(3/2) log C) time for general graphs, and, to the best of our knowledge, it has not been described before. We subsequently show how to implement this algorithm faster on planar graphs using well-established tools: r-divisions and efficient algorithms for computing (shortest) paths in so-called dense distance graphs
Algorithmic Applications of Baur-Strassen's Theorem: Shortest Cycles, Diameter and Matchings
Consider a directed or an undirected graph with integral edge weights from
the set [-W, W], that does not contain negative weight cycles. In this paper,
we introduce a general framework for solving problems on such graphs using
matrix multiplication. The framework is based on the usage of Baur-Strassen's
theorem and of Strojohann's determinant algorithm. It allows us to give new and
simple solutions to the following problems:
* Finding Shortest Cycles -- We give a simple \tilde{O}(Wn^{\omega}) time
algorithm for finding shortest cycles in undirected and directed graphs. For
directed graphs (and undirected graphs with non-negative weights) this matches
the time bounds obtained in 2011 by Roditty and Vassilevska-Williams. On the
other hand, no algorithm working in \tilde{O}(Wn^{\omega}) time was previously
known for undirected graphs with negative weights. Furthermore our algorithm
for a given directed or undirected graph detects whether it contains a negative
weight cycle within the same running time.
* Computing Diameter and Radius -- We give a simple \tilde{O}(Wn^{\omega})
time algorithm for computing a diameter and radius of an undirected or directed
graphs. To the best of our knowledge no algorithm with this running time was
known for undirected graphs with negative weights.
* Finding Minimum Weight Perfect Matchings -- We present an
\tilde{O}(Wn^{\omega}) time algorithm for finding minimum weight perfect
matchings in undirected graphs. This resolves an open problem posted by
Sankowski in 2006, who presented such an algorithm but only in the case of
bipartite graphs.
In order to solve minimum weight perfect matching problem we develop a novel
combinatorial interpretation of the dual solution which sheds new light on this
problem. Such a combinatorial interpretation was not know previously, and is of
independent interest.Comment: To appear in FOCS 201
Spin-dependent tunneling in modulated structures of (Ga,Mn)As
A model of coherent tunneling, which combines multi-orbital tight-binding
approximation with Landauer-B\"uttiker formalism, is developed and applied to
all-semiconductor heterostructures containing (Ga,Mn)As ferromagnetic layers. A
comparison of theoretical predictions and experimental results on
spin-dependent Zener tunneling, tunneling magnetoresistance (TMR), and
anisotropic magnetoresistance (TAMR) is presented. The dependence of spin
current on carrier density, magnetization orientation, strain, voltage bias,
and spacer thickness is examined theoretically in order to optimize device
design and performance.Comment: 9 pages, 13 figures, submitted to PR
Global Journey to Post-Pandemic Normalcy and Revival
After a year of COVID-19, countries, societies, and individuals are longing for normalcy and beginning to consider what life will be like post-pandemic. Efforts and experiences of countries in the European Union, Asia, Asia-Pacific, Australia, Africa, Latin America, the Caribbean, and the United States are examined as they face challenges to end the pandemic and prepare for the post-pandemic reality. What will be the post-pandemic new normalcy ? What changes caused by the pandemic are permanent in societies and the world? What are the necessary reforms that have to take place as part of normalcy? Reflections on the impacts of vaccinations, herd immunity, societal improvements and reorganizations, trends, and actions in the post-COVID-19 world
Algorithmic Complexity of Power Law Networks
It was experimentally observed that the majority of real-world networks
follow power law degree distribution. The aim of this paper is to study the
algorithmic complexity of such "typical" networks. The contribution of this
work is twofold.
First, we define a deterministic condition for checking whether a graph has a
power law degree distribution and experimentally validate it on real-world
networks. This definition allows us to derive interesting properties of power
law networks. We observe that for exponents of the degree distribution in the
range such networks exhibit double power law phenomenon that was
observed for several real-world networks. Our observation indicates that this
phenomenon could be explained by just pure graph theoretical properties.
The second aim of our work is to give a novel theoretical explanation why
many algorithms run faster on real-world data than what is predicted by
algorithmic worst-case analysis. We show how to exploit the power law degree
distribution to design faster algorithms for a number of classical P-time
problems including transitive closure, maximum matching, determinant, PageRank
and matrix inverse. Moreover, we deal with the problems of counting triangles
and finding maximum clique. Previously, it has been only shown that these
problems can be solved very efficiently on power law graphs when these graphs
are random, e.g., drawn at random from some distribution. However, it is
unclear how to relate such a theoretical analysis to real-world graphs, which
are fixed. Instead of that, we show that the randomness assumption can be
replaced with a simple condition on the degrees of adjacent vertices, which can
be used to obtain similar results. As a result, in some range of power law
exponents, we are able to solve the maximum clique problem in polynomial time,
although in general power law networks the problem is NP-complete
Interlayer Exchange Coupling in (Ga,Mn)As-based Superlattices
The interlayer coupling between (Ga,Mn)As ferromagnetic layers in
all-semiconductor superlattices is studied theoretically within a tight-binding
model, which takes into account the crystal, band and magnetic structure of the
constituent superlattice components. It is shown that the mechanism originally
introduced to describe the spin correlations in antiferromagnetic EuTe/PbTe
superlattices, explains the experimental results observed in ferromagnetic
semiconductor structures, i.e., both the antiferromagnetic coupling between
ferromagnetic layers in IV-VI (EuS/PbS and EuS/YbSe) superlattices as well as
the ferromagnetic interlayer coupling in III-V ((Ga,Mn)As/GaAs) multilayer
structures. The model allows also to predict (Ga,Mn)As-based structures, in
which an antiferromagnetic interlayer coupling could be expected.Comment: 4 pages, 3 figure
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