Computing Lengths of Shortest Non-Crossing Paths in Planar Graphs

Given a plane undirected graph $G$ with non-negative edge weights and a set of $k$ terminal pairs on the external face, it is shown in Takahashi et al., (Algorithmica, 16, 1996, pp. 339-357) that the lengths of $k$ non-crossing shortest paths joining the $k$ terminal pairs (if they exist) can be computed in $O(n \log n)$ worst-case time, where $n$ is the number of vertices of $G$. This technique only applies when the union $U$ of the computed shortest paths is a forest. We show that given a plane undirected weighted graph $U$ and a set of $k$ terminal pairs on the external face, it is always possible to compute the lengths of $k$ non-crossing shortest paths joining the $k$ terminal pairs in linear worst-case time, provided that the graph $U$ is the union of $k$ shortest paths, possibly containing cycles. Moreover, each shortest path $\pi$ can be listed in $O(\ell+\ell\log\lceil{\frac{k}{\ell}}\rceil)$, where $\ell$ is the number of edges in $\pi$. As a consequence, the problem of computing multi-terminal distances in a plane undirected weighted graph can always be solved in $O(n \log k)$ worst-case time in the general case.Comment: 17 pages, 11 figure

How vulnerable is an undirected planar graph with respect to max flow

We study the problem of computing the vitality of edges and vertices with respect to the st-max flow in undirected planar graphs, where the vitality of an edge/vertex is the st-max flow decrease when the edge/vertex is removed from the graph. This allows us to establish the vulnerability of the graph with respect to the st-max flow. We give efficient algorithms to compute an additive guaranteed approximation of the vitality of edges and vertices in planar undirected graphs. We show that in the general case high vitality values are well approximated in time close to the time currently required to compute st-max flow O(n log log n). We also give improved, and sometimes optimal, results in the case of integer capacities. All our algorithms work in O(n) space

Non-crossing shortest paths lengths in planar graphs in linear time

Given a plane graph it is known how to compute the union of non-crossing shortest paths. These algorithms do not allow neither to list each single shortest path nor to compute length of shortest paths. Given the union of non-crossing shortest paths, we introduce the concept of shortcuts that allows us to establish whether a path is a shortest path by checking local properties on faces of the graph. By using shortcuts we can compute the length of each shortest path, given their union, in total linear time, and we can list each shortest path p in O(max{l,l log log kl}), where l is the number of edges in p and k the number of shortest paths

Network homophily via multi-dimensional extensions of Cantelli's inequality

Homophily is the principle whereby "similarity breeds connections". We give a quantitative formulation of this principle within networks. We say that a network is homophillic with respect to a given labeled partition of its vertices, when the classes of the partition induce subgraphs that are significantly denser than what we expect under a random labeled partition into classes maintaining the same cardinalities (type). This is the recently introduced \emph{random coloring model} for network homophily. In this perspective, the vector whose entries are the sizes of the subgraphs induced by the corresponding classes, is viewed as the observed outcome of the random vector described by picking labeled partitions at random among partitions with the same type.\,Consequently, the input network is homophillic at the significance level $\alpha$ whenever the one-sided tail probability of observing an outcome at least as extreme as the observed one, is smaller than $\alpha$. Clearly, $\alpha$ can also be thought of as a quantifier of homophily in the scale $[0,1]$. Since, as we show, even approximating this tail probability is an NP-hard problem, we resort multidimensional extensions of classical Cantelli's inequality to bound $\alpha$ from above. This upper bound is the homophily index we propose. It requires the knowledge of the covariance matrix of the random vector, which was not previously known within the random coloring model. In this paper we close this gap by computing the covariance matrix of subgraph sizes under the random coloring model. Interestingly, the matrix depends on the input partition only through its type and on the network only through its degrees. Furthermore all the covariances have the same sign and this sign is a graph invariant. Plugging this structure into Cantelli's bound yields a meaningful, easy to compute index for measuring network homophily

Max-flow vitality in undirected unweighted planar graphs

We show a fast algorithm for determining the set of relevant edges in a planar undirected unweighted graph with respect to the maximum flow. This is a special case of the \emph{max flow vitality} problem, that has been efficiently solved for general undirected graphs and $st$-planar graphs. The \emph{vitality} of an edge of a graph with respect to the maximum flow between two fixed vertices $s$ and $t$ is defined as the reduction of the maximum flow caused by the removal of that edge. In this paper we show that the set of edges having vitality greater than zero in a planar undirected unweighted graph with $n$ vertices, can be found in $O(n \log n)$ worst-case time and $O(n)$ space.Comment: 9 pages, 4 figure

Evaluating homophily in networks via HONTO (HOmophily Network TOol): a case study of chromosomal interactions in human PPI networks

It has been observed in different kinds of networks, such as social or biological ones, a typical behavior inspired by the general principle 'similarity breeds connections'. These networks are defined as homophilic as nodes belonging to the same class preferentially interact with each other. In this work, we present HONTO (HOmophily Network TOol), a user-friendly open-source Python3 package designed to evaluate and analyze homophily in complex networks. The tool takes in input from the network along with a partition of its nodes into classes and yields a matrix whose entries are the homophily/heterophily z-score values. To complement the analysis, the tool also provides z-score values of nodes that do not interact with any other node of the same class. Homophily/heterophily z-scores values are presented as a heatmap allowing a visual at-a-glance interpretation of results. AVAILABILITY AND IMPLEMENTATION: Tool's source code is available at https://github.com/cumbof/honto under the MIT license, installable as a package from PyPI (pip install honto) and conda-forge (conda install -c conda-forge honto), and has a wrapper for the Galaxy platform available on the official Galaxy ToolShed (Blankenberg et al., 2014) at https://toolshed.g2.bx.psu.edu/view/fabio/honto

IMPLEMENTASI BUDAYA LITERASI DALAM PEMBENTUKAN KARAKTER INTEGRITAS SISWA DI SMA NEGERI 02 BATU

This study aims to describes the implementation of literacy culture in reporting the integrity character of students at SMA Negeri 02 Batu; mistakes and the solution of the problem in implementing literacy culture in order to characterize the integrity of students at SMA Negeri 02 Batu. The research method used qualitative research and descriptive approaches. This research consists of procedures, implementation and data analysis. The data technique are uses observation, interview and documentation. The data analysis technique consist of four stages of activity, namely data collection, data reduction, data presentation, and drawing conclusions. The data validity technique used source triangulation. The results showed that the implementation of Cultural Literacy was carried out through various stages, namely the habituation, development and learning stages. Constraints originating from GLS administrators and teachers are those who lack literacy training, students are low reading interest and different character of students, family is an understanding of the understanding of the character of education at home. Damage to GLS administrators and teachers, the solution is to provide training to improve competence, to students the solution is to motivate and approach each student, trust that comes from parents, the solution provides an understanding of character education at home

Evaluation of Commercial Probiotic Products

Although there is a vast number of probiotic products commercially available due to their acceptability and increasing usage, their quality control has continuously been a major concern. This study aimed to assess some commercially available probiotics on the UK market for content in relation to their label claim. Seven products were used for the study. The bacteria content were isolated, identified and enumerated on selective media. The results revealed that all products evaluated contained viable probiotic bacteria but only three out of the seven products (43%) contained the claimed culture concentration or more. None of the multispecies product contained all the labelled probiotic bacteria. Misidentification of some species occurred. The results concurred with previous studies and showed that quality issues with commercial probiotics remain. Since probiotic activity is linked with probiotic concentration and is strain specific, the need exist for a global comprehensive legislation to control the quality of probiotics whose market is gaining huge momentum

Clostridium botulinum spores and toxin in mascarpone cheese and other milk products

A total of 1,017 mascarpone cheese samples, collected at retail, were analyzed for Clostridium botulinum spores and toxin, aerobic mesophilic spore counts, as well as pH, a(w) (water activity), and Eh (oxidation-reduction potential). In addition 260 samples from other dairy products were also analyzed for spores and botulinum toxin. Experiments were carried out on naturally and artificially contaminated mascarpone to investigate the influence of different temperature conditions on toxin production by C. botulinum. Three hundred and thirty-one samples (32.5%) of mascarpone were positive for botulinal spores, and 7 (0.8%) of the 878 samples produced at the plant involved in an outbreak of foodborne botulism also contained toxin type A. The chemical-physical parameters (pH, a(w), Eh) of all samples were compatible with C. botulinum growth and toxinogenesis. Of the other milk products, 2.7% were positive for C. botulinum spores. Growth and toxin formation occurred in naturally and experimentally contaminated mascarpone samples after 3 and 4 days of incubation at 28 degrees C, respectively

Onopordum platylepis (Murb.) Murb. as a novel source of thistle rennet: First application to the manufacture of traditional Italian raw ewe's milk cheese

In this study, for the very first time, aqueous extracts obtained from flowers of spontaneously grown or cultivated Onopordum platylepis (Murb.) Murb. thistles were used as coagulating agents for the pilot-scale manufacture of Caciofiore, a traditional Italian raw ewe’s milk cheese. Cheese prototypes were compared to control cheeses curdled with a commercial thistle rennet obtained from flowers of Cynara cardunculus L. After 45 days of ripening under controlled conditions, both the experimental and control cheese prototypes were analyzed for: cheese yield, physico-chemical (pH, titratable acidity, aw, proximate composition), morpho-textural (color and texture), and microbiological parameters (viable cell counts and species composition assessed by Illumina sequencing), as well as volatile profile by SPME-GC–MS. Slight variations in titratable acidity, color, and texture were observed among samples. Based on the results overall collected, neither the yield nor the proximate composition was apparently affected by the type of thistle coagulant. However, the experimental cheese prototypes curdled with extracts from flowers of both spontaneous or cultivated thistles showed 10 % higher values of water-soluble nitrogen compared to the control prototypes. On the other hand, these latter showed slightly higher loads of presumptive lactococci, thermophilic cocci, coliforms, and eumycetes, but lower counts of Escherichia coli. No statistically significant differences were revealed by the metataxonomic analysis of the bacterial and fungal biota. Though most volatile organic compounds (VOCs) were consistent among the prototypes, significant variability was observed in the abundance of some key aroma compounds, such as butanoic, hexanoic, and octanoic acids, ethanol, propan-2-ol, isobutyl acetate, 2-methyl butanoic acid, and 3-methyl butanal. However, further investigations are required to attribute these differences to either the type of coagulant or the metabolic activity of the microorganisms occurring in the analyzed cheese samples. The results overall collected support the potential exploitation of O. platylepis as a novel source of thistle coagulant to produce ewe’s milk cheeses