51 research outputs found
Efficient Parameterized Algorithms for Computing All-Pairs Shortest Paths
Computing all-pairs shortest paths is a fundamental and much-studied problem
with many applications. Unfortunately, despite intense study, there are still
no significantly faster algorithms for it than the time
algorithm due to Floyd and Warshall (1962). Somewhat faster algorithms exist
for the vertex-weighted version if fast matrix multiplication may be used.
Yuster (SODA 2009) gave an algorithm running in time ,
but no combinatorial, truly subcubic algorithm is known.
Motivated by the recent framework of efficient parameterized algorithms (or
"FPT in P"), we investigate the influence of the graph parameters clique-width
() and modular-width () on the running times of algorithms for solving
All-Pairs Shortest Paths. We obtain efficient (and combinatorial) parameterized
algorithms on non-negative vertex-weighted graphs of times
, resp. . If fast matrix
multiplication is allowed then the latter can be improved to
using the algorithm of Yuster as a black box.
The algorithm relative to modular-width is adaptive, meaning that the running
time matches the best unparameterized algorithm for parameter value equal
to , and they outperform them already for for any
Efficient parameterized algorithms on structured graphs
In der klassischen KomplexitĂ€tstheorie werden worst-case Laufzeiten von Algorithmen typischerweise einzig abhĂ€ngig von der EingabegröĂe angegeben. In dem Kontext der parametrisierten KomplexitĂ€tstheorie versucht man die Analyse der Laufzeit dahingehend zu verfeinern, dass man zusĂ€tzlich zu der EingabengröĂe noch einen Parameter berĂŒcksichtigt, welcher angibt, wie strukturiert die Eingabe bezĂŒglich einer gewissen Eigenschaft ist. Ein parametrisierter Algorithmus nutzt dann diese beschriebene Struktur aus und erreicht so eine Laufzeit, welche schneller ist als die eines besten unparametrisierten Algorithmus, falls der Parameter klein ist.
Der erste Hauptteil dieser Arbeit fĂŒhrt die Forschung in diese Richtung weiter aus und untersucht den Einfluss von verschieden Parametern auf die Laufzeit von bekannten effizient lösbaren Problemen. Einige vorgestellte Algorithmen sind dabei adaptive Algorithmen, was bedeutet, dass die Laufzeit von diesen Algorithmen mit der Laufzeit des besten unparametrisierten Algorithm fĂŒr den gröĂtmöglichen Parameterwert ĂŒbereinstimmt und damit theoretisch niemals schlechter als die besten unparametrisierten Algorithmen und ĂŒbertreffen diese bereits fĂŒr leicht nichttriviale Parameterwerte.
Motiviert durch den allgemeinen Erfolg und der Vielzahl solcher parametrisierten Algorithmen, welche eine vielzahl verschiedener Strukturen ausnutzen, untersuchen wir im zweiten Hauptteil dieser Arbeit, wie man solche unterschiedliche homogene Strukturen zu mehr heterogenen Strukturen vereinen kann. Ausgehend von algebraischen AusdrĂŒcken, welche benutzt werden können, um von Parametern beschriebene Strukturen zu definieren, charakterisieren wir klar und robust heterogene Strukturen und zeigen exemplarisch, wie sich die Parameter tree-depth und modular-width heterogen verbinden lassen. Wir beschreiben dazu effiziente Algorithmen auf heterogenen Strukturen mit Laufzeiten, welche im Spezialfall mit den homogenen Algorithmen ĂŒbereinstimmen.In classical complexity theory, the worst-case running times of algorithms depend solely on the size of the input. In parameterized complexity the goal is to refine the analysis of the running time of an algorithm by additionally considering a parameter that measures some kind of structure in the input. A parameterized algorithm then utilizes the structure described by the parameter and achieves a running time that is faster than the best general (unparameterized) algorithm for instances of low parameter value.
In the first part of this thesis, we carry forward in this direction and investigate the influence of several parameters on the running times of well-known tractable problems.
Several presented algorithms are adaptive algorithms, meaning that they match the running time of a best unparameterized algorithm for worst-case parameter values. Thus, an adaptive parameterized algorithm is asymptotically never worse than the best unparameterized algorithm, while it outperforms the best general algorithm already for slightly non-trivial parameter values.
As illustrated in the first part of this thesis, for many problems there exist efficient parameterized algorithms regarding multiple parameters, each describing a different kind of structure.
In the second part of this thesis, we explore how to combine such homogeneous structures to more general and heterogeneous structures.
Using algebraic expressions, we define new combined graph classes
of heterogeneous structure in a clean and robust way, and we showcase this for the heterogeneous merge of the parameters tree-depth and modular-width, by presenting parameterized algorithms
on such heterogeneous graph classes and getting running times that match the homogeneous cases throughout
Efficient and Adaptive Parameterized Algorithms on Modular Decompositions
We study the influence of a graph parameter called modular-width on the time complexity for optimally solving well-known polynomial problems such as Maximum Matching, Triangle Counting, and Maximum s-t Vertex-Capacitated Flow. The modular-width of a graph depends on its (unique) modular decomposition tree, and can be computed in linear time O(n+m) for graphs with n vertices and m edges. Modular decompositions are an important tool for graph algorithms, e.g., for linear-time recognition of certain graph classes.
Throughout, we obtain efficient parameterized algorithms of running times O(f(mw)n+m), O(n+f(mw)m)or O(f(mw)+n+m) for low polynomial functions f and graphs of modular-width mw. Our algorithm for Maximum Matching, running in time O(mw^2 log mw n+m), is both faster and simpler than the recent O(mw^4n+m) time algorithm of Coudert et al. (SODA 2018). For several other problems, e.g., Triangle Counting and Maximum b-Matching, we give adaptive algorithms, meaning that their running times match the best unparameterized algorithms for worst-case modular-width of mw=Theta(n) and they outperform them already for mw=o(n), until reaching linear time for mw=O(1)
Structural origins of the cohesive energy in metal-terpyridine oligomer thin-films
FeII-terpyridine based oligomers have attracted considerable interest as key constituents for the realization of highly robust, ultra-thin ordered layers of metal center oligomers (MCOs) for organic electronics applications. By using molecular simulations and nanotribology investigations, we report on the origins of the surprisingly high mechanical and thermal stability in this type of MCO layers, which finds its expression in nanowear resistance values of up to 1.5 ÎŒN for the MCO films, as well as in a thermal stability of two-terminal MCO junctions to temperatures up to âŒ100 °C under electrical load. A theoretical analysis of the fundamental cohesive forces among the constituents within the context of an electrostatic model reveal that the cohesive energy is essentially based on Coulomb interactions among the ionic constituents of the oligomers, leading to an estimated cohesive energy per molar mass of 0.0132 eV mol gâ1 for MCO layers that advantageously compare to the 0.0061 eV mol gâ1 reported for pentacene crystals
Ultrarobust ThinâFilm Devices from SelfâAssembled MetalâTerpyridine Oligomers
Ultrathin molecular layers of Fe(II) -terpyridine oligomers allow the fabrication of large-area crossbar junctions by conventional electrode vapor deposition. The junctions are electrically stable for over 2.5 years and operate over a wide range of temperatures (150-360 K) and voltages (±3 V) due to the high cohesive energy and packing density of the oligomer layer. Electrical measurements reveal ideal Richardson-Shottky emission in surprising agreement with electrochemical, optical, and photoemission data
Measurement of the cosmic ray spectrum above eV using inclined events detected with the Pierre Auger Observatory
A measurement of the cosmic-ray spectrum for energies exceeding
eV is presented, which is based on the analysis of showers
with zenith angles greater than detected with the Pierre Auger
Observatory between 1 January 2004 and 31 December 2013. The measured spectrum
confirms a flux suppression at the highest energies. Above
eV, the "ankle", the flux can be described by a power law with
index followed by
a smooth suppression region. For the energy () at which the
spectral flux has fallen to one-half of its extrapolated value in the absence
of suppression, we find
eV.Comment: Replaced with published version. Added journal reference and DO
Energy Estimation of Cosmic Rays with the Engineering Radio Array of the Pierre Auger Observatory
The Auger Engineering Radio Array (AERA) is part of the Pierre Auger
Observatory and is used to detect the radio emission of cosmic-ray air showers.
These observations are compared to the data of the surface detector stations of
the Observatory, which provide well-calibrated information on the cosmic-ray
energies and arrival directions. The response of the radio stations in the 30
to 80 MHz regime has been thoroughly calibrated to enable the reconstruction of
the incoming electric field. For the latter, the energy deposit per area is
determined from the radio pulses at each observer position and is interpolated
using a two-dimensional function that takes into account signal asymmetries due
to interference between the geomagnetic and charge-excess emission components.
The spatial integral over the signal distribution gives a direct measurement of
the energy transferred from the primary cosmic ray into radio emission in the
AERA frequency range. We measure 15.8 MeV of radiation energy for a 1 EeV air
shower arriving perpendicularly to the geomagnetic field. This radiation energy
-- corrected for geometrical effects -- is used as a cosmic-ray energy
estimator. Performing an absolute energy calibration against the
surface-detector information, we observe that this radio-energy estimator
scales quadratically with the cosmic-ray energy as expected for coherent
emission. We find an energy resolution of the radio reconstruction of 22% for
the data set and 17% for a high-quality subset containing only events with at
least five radio stations with signal.Comment: Replaced with published version. Added journal reference and DO
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