|
Unrooted Binary Tree
In mathematics and computer science, an unrooted binary tree is an unrooted tree in which each vertex has either one or three neighbors. Definitions A free tree or unrooted tree is a connected undirected graph with no cycles. The vertices with one neighbor are the ''leaves'' of the tree, and the remaining vertices are the ''internal nodes'' of the tree. The degree of a vertex is its number of neighbors; in a tree with more than one node, the leaves are the vertices of degree one. An unrooted binary tree is a free tree in which all internal nodes have degree exactly three. In some applications it may make sense to distinguish subtypes of unrooted binary trees: a planar embedding of the tree may be fixed by specifying a cyclic ordering for the edges at each vertex, making it into a plane tree. In computer science, binary trees are often rooted and ordered when they are used as data structures, but in the applications of unrooted binary trees in hierarchical clustering and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Phylogenetic Tree
A phylogenetic tree or phylogeny is a graphical representation which shows the evolutionary history between a set of species or taxa during a specific time.Felsenstein J. (2004). ''Inferring Phylogenies'' Sinauer Associates: Sunderland, MA. In other words, it is a branching diagram or a tree showing the evolutionary relationships among various biological species or other entities based upon similarities and differences in their physical or genetic characteristics. In evolutionary biology, all life on Earth is theoretically part of a single phylogenetic tree, indicating common ancestry. Phylogenetics is the study of phylogenetic trees. The main challenge is to find a phylogenetic tree representing optimal evolutionary ancestry between a set of species or taxa. Computational phylogenetics (also phylogeny inference) focuses on the algorithms involved in finding optimal phylogenetic tree in the phylogenetic landscape. Phylogenetic trees may be rooted or unrooted. In a ''rooted'' p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Laurence Wolsey
Laurence Alexander Wolsey is a Belgian-English mathematician working in the field of integer programming. His mother Anna Wolsey-Mautner was the daughter of the Viennese Industrialist Konrad David Mautner. He is a former president and research director of the Center for Operations Research and Econometrics (CORE) at Université catholique de Louvain in Belgium. He is professor emeritus of applied mathematics at the engineering school of the same university. Early life and education Wolsey received a MSc in Mathematics from Cambridge in 1966 and a Ph.D. in Mathematics from the Massachusetts Institute of Technology in 1969 under the supervision of Jeremy F. Shapiro. Career Wolsey was visiting researcher at the Manchester Business School in 1969–1971. He was invited by George L. Nemhauser as a Post-Doctoral student to CORE in Belgium in 1971. He met his future wife, Marguerite Loute, sister of CORE colleague Etienne Loute, and settled in Belgium. He was later a visiting professo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Raffaele Pesenti
Raffaele () is an Italian given name and surname, variant of the English Raphael. Notable people with the name include: Given name *Raffaele Amato (born 1965), Italian mobster *Raffaele Cutolo (1941–2021), Italian mobster *Raffaele Ganci (1932–2022), Italian mobster *Raffaele Cantone (born 1963), Italian magistrate *Raffaele Di Gennaro (born 1993), Italian footballer *Raffaele De Rosa (born 1987), Italian motorcycle racer *Raffaele Di Paco (1908–1996), Italian cyclist *Raffaele Fitto (born 1969), Italian politician *Raffaele Guariglia (1889–1970), Italian politician *Raffaele Lombardo (born 1950), Italian politician *Raffaele Palladino (born 1984), Italian footballer *Raffaele Pinto (1945–2020), Italian racing driver *Raffaele Pisu (1925–2019), Italiano actor *Raffaele Riario (1461–1521), Italian cardinal *Raffaele Rossetti (1881–1951), Italian politician *Raffaele Carlo Rossi (1876–1948), Italian cardinal *Raffaele Viviani (1888–1950), Italian artist *Raffaele C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Double Factorial
In mathematics, the double factorial of a number , denoted by , is the product of all the positive integers up to that have the same Parity (mathematics), parity (odd or even) as . That is, n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots. Restated, this says that for even , the double factorial is n!! = \prod_^\frac (2k) = n(n-2)(n-4)\cdots 4\cdot 2 \,, while for odd it is n!! = \prod_^\frac (2k-1) = n(n-2)(n-4)\cdots 3\cdot 1 \,. For example, . The zero double factorial as an empty product. The sequence of double factorials for even = starts as The sequence of double factorials for odd = starts as The term odd factorial is sometimes used for the double factorial of an odd number. The term semifactorial is also used by Donald Knuth, Knuth as a synonym of double factorial. History and usage In a 1902 paper, the physicist Arthur Schuster wrote: states that the double factorial was originally introduced in order to simplify the expression of certain List of integrals of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Graph Enumeration
In combinatorics, an area of mathematics, graph enumeration describes a class of combinatorial enumeration problems in which one must count undirected graph, undirected or directed graphs of certain types, typically as a function of the number of vertices of the graph. These problems may be solved either exactly (as an algebraic enumeration problem) or asymptotic analysis, asymptotically. The pioneers in this area of mathematics were George Pólya, Arthur Cayley and J. Howard Redfield. Labeled vs unlabeled problems In some graphical enumeration problems, the vertices of the graph are considered to be ''labeled'' in such a way as to be distinguishable from each other, while in other problems any permutation of the vertices is considered to form the same graph, so the vertices are considered identical or ''unlabeled''. In general, labeled problems tend to be easier. As with combinatorial enumeration more generally, the Pólya enumeration theorem is an important tool for reducing unlab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Treewidth
In graph theory, the treewidth of an undirected graph is an integer number which specifies, informally, how far the graph is from being a tree. The smallest treewidth is 1; the graphs with treewidth 1 are exactly the trees and the forests A forest is an ecosystem characterized by a dense community of trees. Hundreds of definitions of forest are used throughout the world, incorporating factors such as tree density, tree height, land use, legal standing, and ecological functio .... An example of graphs with treewidth at most 2 are the series–parallel graphs. The maximal graphs with treewidth exactly are called '' -trees'', and the graphs with treewidth at most are called '' partial -trees''. Many other well-studied graph families also have bounded treewidth. Treewidth may be formally defined in several equivalent ways: in terms of the size of the largest vertex set in a tree decomposition of the graph, in terms of the size of the largest clique in a chordal completi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Branch-decomposition
In graph theory, a branch-decomposition of an undirected graph ''G'' is a hierarchical clustering of the edges of ''G'', represented by an unrooted binary tree ''T'' with the edges of ''G'' as its leaves. Removing any edge from ''T'' partitions the edges of ''G'' into two subgraphs, and the width of the decomposition is the maximum number of shared vertices of any pair of subgraphs formed in this way. The branchwidth of ''G'' is the minimum width of any branch-decomposition of ''G''. Branchwidth is closely related to tree-width: for all graphs, both of these numbers are within a constant factor of each other, and both quantities may be characterized by forbidden minors. And as with treewidth, many graph optimization problems may be solved efficiently for graphs of small branchwidth. However, unlike treewidth, the branchwidth of planar graphs may be computed exactly, in polynomial time. Branch-decompositions and branchwidth may also be generalized from graphs to matroids. Defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Quartet Distance
The quartet distance is a way of measuring the distance between two phylogenetic trees. It is defined as the number of subsets of four leaves that are not related by the same topology in both trees. Computing the quartet distance The most straightforward computation of the quartet distance would require O(N^4) time, where N is the number of leaves in the trees. For binary trees, better algorithms have been found to compute the distance in * O(N^2) time * O(N \log^2 N) time and * O(N \log N) time Gerth Stølting Brodal In computer science, the Brodal queue is a heap/priority queue structure with very low worst case time bounds: O(1) for insertion, find-minimum, meld (merge two queues) and decrease-key and O(\mathrm(n)) for delete-minimum and general deletion. ... ''et al.'' found an algorithm that takes O(D N \log N) time to compute the quartet distance between two multifurcating trees when D is the maximum degree of the trees, which iaccessiblein C, perl, and the R packagQ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
