11 research outputs found
Approximating the Diameter of Planar Graphs in Near Linear Time
We present a -approximation algorithm running in
time for finding the diameter of an undirected
planar graph with non-negative edge lengths
On The Center Sets and Center Numbers of Some Graph Classes
For a set of vertices and the vertex in a connected graph ,
is called the -eccentricity of in
. The set of vertices with minimum -eccentricity is called the -center
of . Any set of vertices of such that is an -center for some
set of vertices of is called a center set. We identify the center sets
of certain classes of graphs namely, Block graphs, , , wheel
graphs, odd cycles and symmetric even graphs and enumerate them for many of
these graph classes. We also introduce the concept of center number which is
defined as the number of distinct center sets of a graph and determine the
center number of some graph classes
Fast approximation of centrality and distances in hyperbolic graphs
We show that the eccentricities (and thus the centrality indices) of all
vertices of a -hyperbolic graph can be computed in linear
time with an additive one-sided error of at most , i.e., after a
linear time preprocessing, for every vertex of one can compute in
time an estimate of its eccentricity such that
for a small constant . We
prove that every -hyperbolic graph has a shortest path tree,
constructible in linear time, such that for every vertex of ,
. These results are based on an
interesting monotonicity property of the eccentricity function of hyperbolic
graphs: the closer a vertex is to the center of , the smaller its
eccentricity is. We also show that the distance matrix of with an additive
one-sided error of at most can be computed in
time, where is a small constant. Recent empirical studies show that
many real-world graphs (including Internet application networks, web networks,
collaboration networks, social networks, biological networks, and others) have
small hyperbolicity. So, we analyze the performance of our algorithms for
approximating centrality and distance matrix on a number of real-world
networks. Our experimental results show that the obtained estimates are even
better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author
Revisiting Radius, Diameter, and all Eccentricity Computation in Graphs through Certificates
We introduce notions of certificates allowing to bound eccentricities in a
graph. In particular , we revisit radius (minimum eccentricity) and diameter
(maximum eccentricity) computation and explain the efficiency of practical
radius and diameter algorithms by the existence of small certificates for
radius and diameter plus few additional properties. We show how such
computation is related to covering a graph with certain balls or complementary
of balls. We introduce several new algorithmic techniques related to
eccentricity computation and propose algorithms for radius, diameter and all
eccentricities with theoretical guarantees with respect to certain graph
parameters. This is complemented by experimental results on various real-world
graphs showing that these parameters appear to be low in practice. We also
obtain refined results in the case where the input graph has low doubling
dimension, has low hyperbolicity, or is chordal
Beyond Helly graphs: the diameter problem on absolute retracts
Characterizing the graph classes such that, on -vertex -edge graphs in
the class, we can compute the diameter faster than in time is an
important research problem both in theory and in practice. We here make a new
step in this direction, for some metrically defined graph classes.
Specifically, a subgraph of a graph is called a retract of if it is
the image of some idempotent endomorphism of . Two necessary conditions for
being a retract of is to have is an isometric and isochromatic
subgraph of . We say that is an absolute retract of some graph class
if it is a retract of any of which it is an
isochromatic and isometric subgraph. In this paper, we study the complexity of
computing the diameter within the absolute retracts of various hereditary graph
classes. First, we show how to compute the diameter within absolute retracts of
bipartite graphs in randomized time. For the
special case of chordal bipartite graphs, it can be improved to linear time,
and the algorithm even computes all the eccentricities. Then, we generalize
these results to the absolute retracts of -chromatic graphs, for every fixed
. Finally, we study the diameter problem within the absolute retracts
of planar graphs and split graphs, respectively
Revisiting Radius, Diameter, and all Eccentricity Computation in Graphs through Certificates
We introduce notions of certificates allowing to bound eccentricities in a graph. In particular , we revisit radius (minimum eccentricity) and diameter (maximum eccentricity) computation and explain the efficiency of practical radius and diameter algorithms by the existence of small certificates for radius and diameter plus few additional properties. We show how such computation is related to covering a graph with certain balls or complementary of balls. We introduce several new algorithmic techniques related to eccentricity computation and propose algorithms for radius, diameter and all eccentricities with theoretical guarantees with respect to certain graph parameters. This is complemented by experimental results on various real-world graphs showing that these parameters appear to be low in practice. We also obtain refined results in the case where the input graph has low doubling dimension, has low hyperbolicity, or is chordal