226,817 research outputs found

    The scaling of the minimum sum of edge lengths in uniformly random trees

    Get PDF
    [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-

    Bounds of the sum of edge lengths in linear arrangements of trees

    Full text link
    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

    Get PDF
    The Minimum Linear Arrangement problem (MLA) consists of finding a mapping π\pi from vertices of a graph to distinct integers that minimizes {u,v}Eπ(u)π(v)\sum_{\{u,v\}\in E}|\pi(u) - \pi(v)|. 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 n=Vn=|V|. There exist variants of the MLA in which the arrangements are constrained. Iordanskii, and later Hochberg and Stallmann (HS), put forward O(n)O(n)-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 O(n)O(n) but did not provide any justification of its cost. In contrast, Park and Levy claimed that GT's algorithm runs in O(nlogdmax)O(n \log d_{max}) where dmaxd_{max} 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 O(n)O(n)-time.Comment: Improved connection with previous Iordanskii's work

    Bond disorder, frustration and polymorphism in the spontaneous crystallization of a polymer melt

    Get PDF
    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

    Get PDF
    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
    corecore