24,419 research outputs found
Improved Algorithms for Parity and Streett objectives
The computation of the winning set for parity objectives and for Streett
objectives in graphs as well as in game graphs are central problems in
computer-aided verification, with application to the verification of closed
systems with strong fairness conditions, the verification of open systems,
checking interface compatibility, well-formedness of specifications, and the
synthesis of reactive systems. We show how to compute the winning set on
vertices for (1) parity-3 (aka one-pair Streett) objectives in game graphs in
time and for (2) k-pair Streett objectives in graphs in time
. For both problems this gives faster algorithms for dense
graphs and represents the first improvement in asymptotic running time in 15
years
Improved Parallel Algorithms for Spanners and Hopsets
We use exponential start time clustering to design faster and more
work-efficient parallel graph algorithms involving distances. Previous
algorithms usually rely on graph decomposition routines with strict
restrictions on the diameters of the decomposed pieces. We weaken these bounds
in favor of stronger local probabilistic guarantees. This allows more direct
analyses of the overall process, giving: * Linear work parallel algorithms that
construct spanners with stretch and size in unweighted
graphs, and size in weighted graphs. * Hopsets that lead
to the first parallel algorithm for approximating shortest paths in undirected
graphs with work
Quantifying loopy network architectures
Biology presents many examples of planar distribution and structural networks
having dense sets of closed loops. An archetype of this form of network
organization is the vasculature of dicotyledonous leaves, which showcases a
hierarchically-nested architecture containing closed loops at many different
levels. Although a number of methods have been proposed to measure aspects of
the structure of such networks, a robust metric to quantify their hierarchical
organization is still lacking. We present an algorithmic framework, the
hierarchical loop decomposition, that allows mapping loopy networks to binary
trees, preserving in the connectivity of the trees the architecture of the
original graph. We apply this framework to investigate computer generated
graphs, such as artificial models and optimal distribution networks, as well as
natural graphs extracted from digitized images of dicotyledonous leaves and
vasculature of rat cerebral neocortex. We calculate various metrics based on
the Asymmetry, the cumulative size distribution and the Strahler bifurcation
ratios of the corresponding trees and discuss the relationship of these
quantities to the architectural organization of the original graphs. This
algorithmic framework decouples the geometric information (exact location of
edges and nodes) from the metric topology (connectivity and edge weight) and it
ultimately allows us to perform a quantitative statistical comparison between
predictions of theoretical models and naturally occurring loopy graphs.Comment: 17 pages, 8 figures. During preparation of this manuscript the
authors became aware of the work of Mileyko at al., concurrently submitted
for publicatio
On functional module detection in metabolic networks
Functional modules of metabolic networks are essential for understanding the metabolism of an organism as a whole. With the vast amount of experimental data and the construction of complex and large-scale, often genome-wide, models, the computer-aided identification of functional modules becomes more and more important. Since steady states play a key role in biology, many methods have been developed in that context, for example, elementary flux modes, extreme pathways, transition invariants and place invariants. Metabolic networks can be studied also from the point of view of graph theory, and algorithms for graph decomposition have been applied for the identification of functional modules. A prominent and currently intensively discussed field of methods in graph theory addresses the Q-modularity. In this paper, we recall known concepts of module detection based on the steady-state assumption, focusing on transition-invariants (elementary modes) and their computation as minimal solutions of systems of Diophantine equations. We present the Fourier-Motzkin algorithm in detail. Afterwards, we introduce the Q-modularity as an example for a useful non-steady-state method and its application to metabolic networks. To illustrate and discuss the concepts of invariants and Q-modularity, we apply a part of the central carbon metabolism in potato tubers (Solanum tuberosum) as running example. The intention of the paper is to give a compact presentation of known steady-state concepts from a graph-theoretical viewpoint in the context of network decomposition and reduction and to introduce the application of Q-modularity to metabolic Petri net models
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