A correction on Shiloach's algorithm for minimum linear arrangement of trees

More than 30 years ago, Shiloach published an algorithm to solve the minimum linear arrangement problem for undirected trees. Here we fix a small error in the original version of the algorithm and discuss its effect on subsequent literature. We also improve some aspects of the notation.Comment: A new introductory paragraph has been added; error solutions and notation improvements are discussed with more dept

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-

Minimum projective linearizations of trees in linear time

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

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

The optimality of syntactic dependency distances

It is often stated that human languages, as other biological systems, are shaped by cost-cutting pressures but, to what extent? Attempts to quantify the degree of optimality of languages by means of an optimality score have been scarce and focused mostly on English. Here we recast the problem of the optimality of the word order of a sentence as an optimization problem on a spatial network where the vertices are words, arcs indicate syntactic dependencies and the space is defined by the linear order of the words in the sentence. We introduce a new score to quantify the cognitive pressure to reduce the distance between linked words in a sentence. The analysis of sentences from 93 languages representing 19 linguistic families reveals that half of languages are optimized to a 70% or more. The score indicates that distances are not significantly reduced in a few languages and confirms two theoretical predictions, i.e. that longer sentences are more optimized and that distances are more likely to be longer than expected by chance in short sentences. We present a new hierarchical ranking of languages by their degree of optimization. The statistical advantages of the new score call for a reevaluation of the evolution of dependency distance over time in languages as well as the relationship between dependency distance and linguistic competence. Finally, the principles behind the design of the score can be extended to develop more powerful normalizations of topological distances or physical distances in more dimensions

Dynamics and thermodynamics of linear quantum open systems

We analyze the behavior of a network of quantum oscillators coupled with a number of external environments. We show that the dynamics is such that the quantum state of the network always obeys a local master equation with a simple analytical solution. We use this to study the emergence of thermodynamical laws in the stationary regime, achieved for sufficiently long times if the environments are dissipative. We show that the validity of the second law implies the impossibility of building a quantum refrigerator without moving parts (therefore, a quantum absorption refrigerators requires non-linearity as an crucial ingredient, as recently proposed by Kosloff and others cite{Kosloff1,Kosloff2}). We also show that the third law imposes strong constraints on the low frequency behavior of the environmental spectral densities.Comment: 4 pages of main text, 6 pages of supplementary material, 1 figure; substantially modified, detailed derivations presented in the supplementary materia

Comparando procesos nivales sobre diferentes cordilleras ibéricas

13 páginas.- Presentación elaborada para las Jornadas IPErinas 2016, celebradas en Zaragoza, el 15 de diciembre de 2016Peer reviewe

Population History and Gene Divergence in Native Mexicans Inferred from 76 Human Exomes.

Native American genetic variation remains underrepresented in most catalogs of human genome sequencing data. Previous genotyping efforts have revealed that Mexico's Indigenous population is highly differentiated and substructured, thus potentially harboring higher proportions of private genetic variants of functional and biomedical relevance. Here we have targeted the coding fraction of the genome and characterized its full site frequency spectrum by sequencing 76 exomes from five Indigenous populations across Mexico. Using diffusion approximations, we modeled the demographic history of Indigenous populations from Mexico with northern and southern ethnic groups splitting 7.2 KYA and subsequently diverging locally 6.5 and 5.7 KYA, respectively. Selection scans for positive selection revealed BCL2L13 and KBTBD8 genes as potential candidates for adaptive evolution in Rarámuris and Triquis, respectively. BCL2L13 is highly expressed in skeletal muscle and could be related to physical endurance, a well-known phenotype of the northern Mexico Rarámuri. The KBTBD8 gene has been associated with idiopathic short stature and we found it to be highly differentiated in Triqui, a southern Indigenous group from Oaxaca whose height is extremely low compared to other Native populations

Study of the genetic diversity of almond seedling populations in Morocco: application of chemometric approach

Almond (Prunus amygdalus Batsch) in Morocco is still propagated by farmers mostly from seed, generating a large genetic diversity. Evaluation of the almond diversity in Morocco from the point of view of kernel quality, oil and protein contents, and major fatty acid composition were determined. Principal component analysis (PCA) was used to compare the kernel components among 46 genotypes selected from different production regions, as well as five introduced cultivars. Oil and protein contents were highly variable between genotypes, ranging respectively from 48.29 to 65.19% and from 14.07 to 36.48% of the total kernel dry weight. Fatty acid composition of the oil fraction ranged between 5.6-8.34% for palmitic acid, 0.37-0.87% for palmitoleic acid, 1.3-3.3% for stearic acid, 57.95-81.97% for oleic acid, and 9.69-29.98% for linoleic acid. Clustering of genotypes from similar regions suggested the existence of parental relationship among these genotypes and, as a consequence, a common ancestral origin. The Moroccan genotypes did not cluster separately from the introduced varieties. Two genotypes had very high oil contents whereas four genotypes had exceptionally high protein contents (>30%). One genotype had a very high oleic content (82%). The large variability observed for oil and fatty acid composition and the presence of genotypes with higher oil and fatty acid contents than the foreign cultivars represent a very promising base to obtain new Moroccan almond cultivars with oil of higher quality.Peer ReviewedPrunus amygdalusOil contentFatty acidsGenetic resourcesBreedingPublishe