335 research outputs found
Easy identification of generalized common and conserved nested intervals
In this paper we explain how to easily compute gene clusters, formalized by
classical or generalized nested common or conserved intervals, between a set of
K genomes represented as K permutations. A b-nested common (resp. conserved)
interval I of size |I| is either an interval of size 1 or a common (resp.
conserved) interval that contains another b-nested common (resp. conserved)
interval of size at least |I|-b. When b=1, this corresponds to the classical
notion of nested interval. We exhibit two simple algorithms to output all
b-nested common or conserved intervals between K permutations in O(Kn+nocc)
time, where nocc is the total number of such intervals. We also explain how to
count all b-nested intervals in O(Kn) time. New properties of the family of
conserved intervals are proposed to do so
Algorithmic Aspects of Switch Cographs
This paper introduces the notion of involution module, the first
generalization of the modular decomposition of 2-structure which has a unique
linear-sized decomposition tree. We derive an O(n^2) decomposition algorithm
and we take advantage of the involution modular decomposition tree to state
several algorithmic results. Cographs are the graphs that are totally
decomposable w.r.t modular decomposition. In a similar way, we introduce the
class of switch cographs, the class of graphs that are totally decomposable
w.r.t involution modular decomposition. This class generalizes the class of
cographs and is exactly the class of (Bull, Gem, Co-Gem, C_5)-free graphs. We
use our new decomposition tool to design three practical algorithms for the
maximum cut, vertex cover and vertex separator problems. The complexity of
these problems was still unknown for this class of graphs. This paper also
improves the complexity of the maximum clique, the maximum independant set, the
chromatic number and the maximum clique cover problems by giving efficient
algorithms, thanks to the decomposition tree. Eventually, we show that this
class of graphs has Clique-Width at most 4 and that a Clique-Width expression
can be computed in linear time
Algorithmic Aspects of a General Modular Decomposition Theory
A new general decomposition theory inspired from modular graph decomposition
is presented. This helps unifying modular decomposition on different
structures, including (but not restricted to) graphs. Moreover, even in the
case of graphs, the terminology ``module'' not only captures the classical
graph modules but also allows to handle 2-connected components, star-cutsets,
and other vertex subsets. The main result is that most of the nice algorithmic
tools developed for modular decomposition of graphs still apply efficiently on
our generalisation of modules. Besides, when an essential axiom is satisfied,
almost all the important properties can be retrieved. For this case, an
algorithm given by Ehrenfeucht, Gabow, McConnell and Sullivan 1994 is
generalised and yields a very efficient solution to the associated
decomposition problem
Laser induced gas breakdown: spectroscopic and chemical studies
Q switched laser induced gas breakdown used in hydrogen atom detection, spectral analysis, and chemical reaction
A Note On Computing Set Overlap Classes
Let be a finite set of elements and a family of subsets of Two sets and of
overlap if and Two sets
are in the same overlap class if there is a series of
sets of in which each overlaps. In this note, we focus
on efficiently identifying all overlap classes in
time. We thus revisit the clever algorithm of Dahlhaus of which we give a clear
presentation and that we simplify to make it practical and implementable in its
real worst case complexity. An useful variant of Dahlhaus's approach is also
explained
Decomposing a Graph into Shortest Paths with Bounded Eccentricity
We introduce the problem of hub-laminar decomposition which generalizes that of computing a shortest path with minimum eccentricity (MESP). Intuitively, it consists in decomposing a graph into several paths that collectively have small eccentricity and meet only near their extremities. The problem is related to computing an isometric cycle with minimum eccentricity (MEIC). It is also linked to DNA reconstitution in the context of metagenomics in biology. We show that a graph having such a decomposition with long enough paths can be decomposed in polynomial time with approximated guaranties on the parameters of the decomposition. Moreover, such a decomposition with few paths allows to compute a compact representation of distances with additive distortion. We also show that having an isometric cycle with small eccentricity is related to the possibility of embedding the graph in a cycle with low distortion
Minimum Eccentricity Shortest Path Problem: an Approximation Algorithm and Relation with the k-Laminarity Problem
The Minimum Eccentricity Shortest Path (MESP) Problem consists in determining a shortest path (a path whose length is the distance between its extremities) of minimum eccentricity in a graph. It was introduced by Dragan and Leitert [9] who described a linear-time algorithm which is an 8-approximation of the problem. In this paper, we study deeper the double-BFS procedure used in that algorithm and extend it to obtain a linear-time 3-approximation algorithm. We moreover study the link between the MESP problem and the notion of laminarity, introduced by Völkel et al [12], corresponding to its restriction to a diameter (i.e. a shortest path of maximum length), and show tight bounds between MESP and laminarity parameters
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