5,919 research outputs found
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
Network Farthest-Point Diagrams
Consider the continuum of points along the edges of a network, i.e., an
undirected graph with positive edge weights. We measure distance between these
points in terms of the shortest path distance along the network, known as the
network distance. Within this metric space, we study farthest points.
We introduce network farthest-point diagrams, which capture how the farthest
points---and the distance to them---change as we traverse the network. We
preprocess a network G such that, when given a query point q on G, we can
quickly determine the farthest point(s) from q in G as well as the farthest
distance from q in G. Furthermore, we introduce a data structure supporting
queries for the parts of the network that are farther away from q than some
threshold R > 0, where R is part of the query.
We also introduce the minimum eccentricity feed-link problem defined as
follows. Given a network G with geometric edge weights and a point p that is
not on G, connect p to a point q on G with a straight line segment pq, called a
feed-link, such that the largest network distance from p to any point in the
resulting network is minimized. We solve the minimum eccentricity feed-link
problem using eccentricity diagrams. In addition, we provide a data structure
for the query version, where the network G is fixed and a query consists of the
point p.Comment: A preliminary version of this work was presented at the 24th Canadian
Conference on Computational Geometr
Discriminative Distance-Based Network Indices with Application to Link Prediction
In large networks, using the length of shortest paths as the distance measure
has shortcomings. A well-studied shortcoming is that extending it to
disconnected graphs and directed graphs is controversial. The second
shortcoming is that a huge number of vertices may have exactly the same score.
The third shortcoming is that in many applications, the distance between two
vertices not only depends on the length of shortest paths, but also on the
number of shortest paths. In this paper, first we develop a new distance
measure between vertices of a graph that yields discriminative distance-based
centrality indices. This measure is proportional to the length of shortest
paths and inversely proportional to the number of shortest paths. We present
algorithms for exact computation of the proposed discriminative indices.
Second, we develop randomized algorithms that precisely estimate average
discriminative path length and average discriminative eccentricity and show
that they give -approximations of these indices. Third, we
perform extensive experiments over several real-world networks from different
domains. In our experiments, we first show that compared to the traditional
indices, discriminative indices have usually much more discriminability. Then,
we show that our randomized algorithms can very precisely estimate average
discriminative path length and average discriminative eccentricity, using only
few samples. Then, we show that real-world networks have usually a tiny average
discriminative path length, bounded by a constant (e.g., 2). Fourth, in order
to better motivate the usefulness of our proposed distance measure, we present
a novel link prediction method, that uses discriminative distance to decide
which vertices are more likely to form a link in future, and show its superior
performance compared to the well-known existing measures
Fast Distributed Computation of Distances in Networks
This paper presents a distributed algorithm to simultaneously compute the
diameter, radius and node eccentricity in all nodes of a synchronous network.
Such topological information may be useful as input to configure other
algorithms. Previous approaches have been modular, progressing in sequential
phases using building blocks such as BFS tree construction, thus incurring
longer executions than strictly required. We present an algorithm that, by
timely propagation of available estimations, achieves a faster convergence to
the correct values. We show local criteria for detecting convergence in each
node. The algorithm avoids the creation of BFS trees and simply manipulates
sets of node ids and hop counts. For the worst scenario of variable start
times, each node i with eccentricity ecc(i) can compute: the node eccentricity
in diam(G)+ecc(i)+2 rounds; the diameter in 2*diam(G)+ecc(i)+2 rounds; and the
radius in diam(G)+ecc(i)+2*radius(G) rounds.Comment: 12 page
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