226,817 research outputs found
The scaling of the minimum sum of edge lengths in uniformly random trees
[Abstract] The minimum linear arrangement problem on a network consists of finding the minimum sum of edge lengths that can be achieved when the vertices are arranged linearly. Although there are algorithms to solve this problem on trees in polynomial time, they have remained theoretical and have not been implemented in practical contexts to our knowledge. Here we use one of those algorithms to investigate the growth of this sum as a function of the size of the tree in uniformly random trees. We show that this sum is bounded above by its value in a star tree. We also show that the mean edge length grows logarithmically in optimal linear arrangements, in stark contrast to the linear growth that is expected on optimal arrangements of star trees or on random linear arrangements.Ministerio de Economía, Industria y Competitividad; TIN2013-48031- C4-1-PXunta de Galicia; R2014/034Agència de Gestió d'Ajuts Universitaris i de Recerca; 2014SGR 890Ministerio de Economía, Industria y Competitividad; TIN2014-57226-PMinisterio de Economía, Industria y Competitividad; FFI2014-51978-C2-2-
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Minimum Cell Connection in Line Segment Arrangements
We study the complexity of the following cell connection problems in segment arrangements. Given a set of straight-line segments in the plane and two points a and b in different cells of the induced arrangement:
[(i)] compute the minimum number of segments one needs to remove so that there is a path connecting a to b that does not intersect any of the remaining segments; [(ii)] compute the minimum number of segments one needs to remove so that the arrangement induced by the remaining segments has a single cell.
We show that problems (i) and (ii) are NP-hard and discuss some special, tractable cases. Most notably, we provide a near-linear-time algorithm for a variant of problem (i) where the path connecting a
to b must stay inside a given polygon P with a constant number of holes, the segments are contained in P, and the endpoints of the segments are on the boundary of P. The approach for this latter result uses homotopy of paths to group the segments into clusters with the property that either all segments in a cluster or none participate in an optimal solution
Bounds of the sum of edge lengths in linear arrangements of trees
A fundamental problem in network science is the normalization of the
topological or physical distance between vertices, that requires understanding
the range of variation of the unnormalized distances. Here we investigate the
limits of the variation of the physical distance in linear arrangements of the
vertices of trees. In particular, we investigate various problems on the sum of
edge lengths in trees of a fixed size: the minimum and the maximum value of the
sum for specific trees, the minimum and the maximum in classes of trees (bistar
trees and caterpillar trees) and finally the minimum and the maximum for any
tree. We establish some foundations for research on optimality scores for
spatial networks in one dimension.Comment: Title changed at proof stag
Minimum projective linearizations of trees in linear time
The Minimum Linear Arrangement problem (MLA) consists of finding a mapping
from vertices of a graph to distinct integers that minimizes
. In that setting, vertices are often
assumed to lie on a horizontal line and edges are drawn as semicircles above
said line. For trees, various algorithms are available to solve the problem in
polynomial time in . There exist variants of the MLA in which the
arrangements are constrained. Iordanskii, and later Hochberg and Stallmann
(HS), put forward -time algorithms that solve the problem when
arrangements are constrained to be planar (also known as one-page book
embeddings). We also consider linear arrangements of rooted trees that are
constrained to be projective (planar embeddings where the root is not covered
by any edge). Gildea and Temperley (GT) sketched an algorithm for projective
arrangements which they claimed runs in but did not provide any
justification of its cost. In contrast, Park and Levy claimed that GT's
algorithm runs in where is the maximum degree but
did not provide sufficient detail. Here we correct an error in HS's algorithm
for the planar case, show its relationship with the projective case, and derive
simple algorithms for the projective and planar cases that run undoubtlessly in
-time.Comment: Improved connection with previous Iordanskii's work
Bond disorder, frustration and polymorphism in the spontaneous crystallization of a polymer melt
The isothermal, isobaric spontaneous crystallization of a supercooled polymer melt is investigated by molecular-dynamics simulation of an ensemble of fully-flexible linear chains. Frustration is introduced via two incommensurate length scales set by the bond length and the position of the minimum of the non- bonding potential. Marked polymorphism with considerable bond disorder, distortions of both the local packing and the global monomer arrangements is observed. The analyses in terms of: i) orientational order parameters characterizing the global and the local order and ii) the angular distribution of the next-nearest neighbors of a monomer reach the conclusion that the polymorphs are arranged in distorted Bcc-like lattice
The sum of edge lengths in random linear arrangements
Spatial networks are networks where nodes are located in a space equipped
with a metric. Typically, the space is two-dimensional and until recently and
traditionally, the metric that was usually considered was the Euclidean
distance. In spatial networks, the cost of a link depends on the edge length,
i.e. the distance between the nodes that define the edge. Hypothesizing that
there is pressure to reduce the length of the edges of a network requires a
null model, e.g., a random layout of the vertices of the network. Here we
investigate the properties of the distribution of the sum of edge lengths in
random linear arrangement of vertices, that has many applications in different
fields. A random linear arrangement consists of an ordering of the elements of
the nodes of a network being all possible orderings equally likely. The
distance between two vertices is one plus the number of intermediate vertices
in the ordering. Compact formulae for the 1st and 2nd moments about zero as
well as the variance of the sum of edge lengths are obtained for arbitrary
graphs and trees. We also analyze the evolution of that variance in Erdos-Renyi
graphs and its scaling in uniformly random trees. Various developments and
applications for future research are suggested
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