On the shortest spanning subtree of a graph and the traveling salesman problem
01 Feb 1956-Vol. 7, Iss: 1, pp 48-50
TL;DR: Kurosh and Levitzki as discussed by the authors, on the radical of a general ring and three problems concerning nil rings, Bull Amer Math Soc vol 49 (1943) pp 913-919 10 -, On the structure of algebraic algebras and related rings.
Abstract: 7 A Kurosh, Ringtheoretische Probleme die mit dem Burnsideschen Problem uber periodische Gruppen in Zussammenhang stehen, Bull Acad Sei URSS, Ser Math vol 5 (1941) pp 233-240 8 J Levitzki, On the radical of a general ring, Bull Amer Math Soc vol 49 (1943) pp 462^66 9 -, On three problems concerning nil rings, Bull Amer Math Soc vol 49 (1943) pp 913-919 10 -, On the structure of algebraic algebras and related rings, Trans Amer Math Soc vol 74 (1953) pp 384-409
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TL;DR: A tree is a graph with one and only one path between every two nodes, where at least one path exists between any two nodes and the length of each branch is given.
Abstract: We consider n points (nodes), some or all pairs of which are connected by a branch; the length of each branch is given. We restrict ourselves to the case where at least one path exists between any two nodes. We now consider two problems. Problem 1. Constrnct the tree of minimum total length between the n nodes. (A tree is a graph with one and only one path between every two nodes.) In the course of the construction that we present here, the branches are subdivided into three sets: I. the branches definitely assignec~ to the tree under construction (they will form a subtree) ; II. the branches from which the next branch to be added to set I, will be selected ; III. the remaining branches (rejected or not yet considered). The nodes are subdivided into two sets: A. the nodes connected by the branches of set I, B. the remaining nodes (one and only one branch of set II will lead to each of these nodes), We start the construction by choosing an arbitrary node as the only member of set A, and by placing all branches that end in this node in set II. To start with, set I is empty. From then onwards we perform the following two steps repeatedly. Step 1. The shortest branch of set II is removed from this set and added to
22,704 citations
TL;DR: Arlequin ver 3.0 as discussed by the authors is a software package integrating several basic and advanced methods for population genetics data analysis, like the computation of standard genetic diversity indices, the estimation of allele and haplotype frequencies, tests of departure from linkage equilibrium, departure from selective neutrality and demographic equilibrium, estimation or parameters from past population expansions, and thorough analyses of population subdivision under the AMOVA framework.
Abstract: Arlequin ver 3.0 is a software package integrating several basic and advanced methods for population genetics data analysis, like the computation of standard genetic diversity indices, the estimation of allele and haplotype frequencies, tests of departure from linkage equilibrium, departure from selective neutrality and demographic equilibrium, estimation or parameters from past population expansions, and thorough analyses of population subdivision under the AMOVA framework. Arlequin 3 introduces a completely new graphical interface written in C++, a more robust semantic analysis of input files, and two new methods: a Bayesian estimation of gametic phase from multi-locus genotypes, and an estimation of the parameters of an instantaneous spatial expansion from DNA sequence polymorphism. Arlequin can handle several data types like DNA sequences, microsatellite data, or standard multi-locus genotypes. A Windows version of the software is freely available on http://cmpg.unibe.ch/software/arlequin3.
14,271 citations
TL;DR: A method for constructing networks from recombination-free population data that combines features of Kruskal's algorithm for finding minimum spanning trees by favoring short connections, and Farris's maximum-parsimony (MP) heuristic algorithm, which sequentially adds new vertices called "median vectors", except that the MJ method does not resolve ties.
Abstract: Reconstructing phylogenies from intraspecific data (such as human mitochondrial DNA variation) is often a challenging task because of large sample sizes and small genetic distances between individuals. The resulting multitude of plausible trees is best expressed by a network which displays alternative potential evolutionary paths in the form of cycles. We present a method ("median joining" [MJ]) for constructing networks from recombination-free population data that combines features of Kruskal's algorithm for finding minimum spanning trees by favoring short connections, and Farris's maximum-parsimony (MP) heuristic algorithm, which sequentially adds new vertices called "median vectors", except that our MJ method does not resolve ties. The MJ method is hence closely related to the earlier approach of Foulds, Hendy, and Penny for estimating MP trees but can be adjusted to the level of homoplasy by setting a parameter epsilon. Unlike our earlier reduced median (RM) network method, MJ is applicable to multistate characters (e.g., amino acid sequences). An additional feature is the speed of the implemented algorithm: a sample of 800 worldwide mtDNA hypervariable segment I sequences requires less than 3 h on a Pentium 120 PC. The MJ method is demonstrated on a Tibetan mitochondrial DNA RFLP data set.
9,937 citations
Book•
01 Jan 1974
TL;DR: This text introduces the basic data structures and programming techniques often used in efficient algorithms, and covers use of lists, push-down stacks, queues, trees, and graphs.
Abstract: From the Publisher:
With this text, you gain an understanding of the fundamental concepts of algorithms, the very heart of computer science. It introduces the basic data structures and programming techniques often used in efficient algorithms. Covers use of lists, push-down stacks, queues, trees, and graphs. Later chapters go into sorting, searching and graphing algorithms, the string-matching algorithms, and the Schonhage-Strassen integer-multiplication algorithm. Provides numerous graded exercises at the end of each chapter.
0201000296B04062001
9,262 citations
Book•
01 Jan 1976
TL;DR: In this paper, the authors present Graph Theory with Applications: Graph theory with applications, a collection of applications of graph theory in the field of Operational Research and Management. Journal of the Operational research Society: Vol. 28, Volume 28, issue 1, pp. 237-238.
Abstract: (1977). Graph Theory with Applications. Journal of the Operational Research Society: Vol. 28, Volume 28, issue 1, pp. 237-238.
7,497 citations