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Priority Matching
In graph theory, a priority matching (also called: maximum priority matching) is a matching that maximizes the number of high-priority vertices that participate in the matching. Formally, we are given a graph , and a partition of the vertex-set into some subsets, , called ''priority classes''. A priority matching is a matching that, among all possible matchings, saturates the largest number of vertices from ; subject to this, it saturates the largest number of vertices from ; subject to this, it saturates the largest number of vertices from ; and so on. Priority matchings were introduced by Alvin Roth, Tayfun Sonmez and Utku Unver in the context of kidney exchange. In this problem, the vertices are patient-donor pairs, and each edge represents a mutual medical compatibility. For example, an edge between pair 1 and pair 2 indicates that donor 1 is compatible with patient 2 and donor 2 is compatible with patient 1. The priority classes correspond to medical priority among pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') which are connected by ''Glossary of graph theory terms#edge, edges'' (also called ''arcs'', ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a Set (mathematics), set of vertices (also called nodes or points); * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Order
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a ( strongly connected, formerly called totality). Requirements 1. to 3. just make up the definition of a partial order. Reflexivity (1.) already follows from strong connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders. Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, toset and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but generally refers to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theory), independent sets U and V, that is, every edge (graph theory), edge connects a Vertex (graph theory), vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycle (graph theory), cycles. The two sets U and V may be thought of as a graph coloring, coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a Gallery of named graphs, triangle: after one node is colored blue and another red, the third vertex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edmonds' Algorithm
In graph theory, Edmonds' algorithm or Chu–Liu/Edmonds' algorithm is an algorithm for finding a spanning arborescence of minimum weight (sometimes called an ''optimum branching''). The algorithm is applicable to finding a minimum spanning forest with given roots. However, when searching for the minimum spanning forest among all k-component spanning forests, a multiplier arises in the complexity of the algorithm C_V^k, corresponding to the choice of a subset of vertices designated as roots. This makes it unsuitable for such a task. Even when constructing a minimum spanning tree, regardless of the root, the algorithm must be used V times, sequentially assigning each vertex as the root. An efficient algorithm for finding minimum spanning forests that solves the root assignment problem is presented in (https://link.springer.com/article/10.1007/s10958-023-06666-w). It builds a sequence of minimal k-component spanning forests for all k up to the minimum spanning tree. The Chu-Liu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jonathan S
Jonathan may refer to: *Jonathan (name), a masculine given name Media * ''Jonathan'' (1970 film), a German film directed by Hans W. Geißendörfer * ''Jonathan'' (2016 film), a German film directed by Piotr J. Lewandowski * ''Jonathan'' (2018 film), an American film directed by Bill Oliver * ''Jonathan'' (Buffy comic), a 2001 comic book based on the ''Buffy the Vampire Slayer'' television series *Jonathan (TV show), a Welsh-language television show hosted by ex-rugby player Jonathan Davies People and biblical figures Bible * Jonathan (1 Samuel), son of King Saul of Israel and friend of David, in the Books of Samuel * Jonathan (Judges), in the Book of Judges * Jonathan (son of Abiathar), in 2 Samuel and 1 Kings Judaism *Jonathan Apphus, fifth son of Mattathias and leader of the Hasmonean dynasty of Judea from 161 to 143 BCE * Rabbi Jonathan, 2nd century * Jonathan (High Priest), a High Priest of Israel in the 1st century Footballers * Jonathan (footballer, born 1991) * Jonat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a member of a #Related asymptotic notations, family of notations invented by German mathematicians Paul Gustav Heinrich Bachmann, Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for '':wikt:Ordnung#German, Ordnung'', meaning the order of approximation. In computer science, big O notation is used to Computational complexity theory, classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetic function, arithmetical function and a better understood approximation; one well-known exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Run-time Complexity
In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is general ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Cardinality Matching
Maximum cardinality matching is a fundamental problem in graph theory. We are given a graph , and the goal is to find a matching containing as many edges as possible; that is, a maximum cardinality subset of the edges such that each vertex is adjacent to at most one edge of the subset. As each edge will cover exactly two vertices, this problem is equivalent to the task of finding a matching that covers as many vertices as possible. An important special case of the maximum cardinality matching problem is when is a bipartite graph, whose vertices are partitioned between left vertices in and right vertices in , and edges in always connect a left vertex to a right vertex. In this case, the problem can be efficiently solved with simpler algorithms than in the general case. Algorithms for bipartite graphs Flow-based algorithm The simplest way to compute a maximum cardinality matching is to follow the Ford–Fulkerson algorithm. This algorithm solves the more general probl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kidney Exchange
In humans, the kidneys are two reddish-brown bean-shaped blood-filtering organs that are a multilobar, multipapillary form of mammalian kidneys, usually without signs of external lobulation. They are located on the left and right in the retroperitoneal space, and in adult humans are about in length. They receive blood from the paired renal arteries; blood exits into the paired renal veins. Each kidney is attached to a ureter, a tube that carries excreted urine to the bladder. The kidney participates in the control of the volume of various body fluids, fluid osmolality, acid-base balance, various electrolyte concentrations, and removal of toxins. Filtration occurs in the glomerulus: one-fifth of the blood volume that enters the kidneys is filtered. Examples of substances reabsorbed are solute-free water, sodium, bicarbonate, glucose, and amino acids. Examples of substances secreted are hydrogen, ammonium, potassium and uric acid. The nephron is the structural and functional uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matching (graph Theory)
In the mathematical discipline of graph theory, a matching or independent edge set in an undirected Graph (discrete mathematics), graph is a set of Edge (graph theory), edges without common vertex (graph theory), vertices. In other words, a subset of the edges is a matching if each vertex appears in at most one edge of that matching. Finding a matching in a bipartite graph can be treated as a Flow network, network flow problem. Definitions Given a Graph (discrete mathematics), graph a matching ''M'' in ''G'' is a set of pairwise non-adjacent edges, none of which are loop (graph theory), loops; that is, no two edges share common vertices. A vertex is matched (or saturated) if it is an endpoint of one of the edges in the matching. Otherwise the vertex is unmatched (or unsaturated). A maximal matching is a matching ''M'' of a graph ''G'' that is not a subset of any other matching. A matching ''M'' of a graph ''G'' is maximal if every edge in ''G'' has a non-empty intersectio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Utku Unver
Utku is a Turkish given name for females and (mostly) males; it means 'victory'. People named Utku include: Given name (male) * Utku Dalmaz (born 1985), Turkish electronic dance music producer, jazz guitarist and web developer * Utku Sen (born 1998), German footballer * Utku Ünal (born 1983), Turkish musician * Utku Yuvakuran (born 1997), Turkish football goalkeeper See also *Utkuhiksalik Utkuhiksalik, also known as Utkuhikhalik, Utkuhikhaliq, Utkuhiksalingmiutitut, Utkuhiksalingmiutut,Briggs, J. L. (1970), Never in anger. Portrait of an Eskimo family. Harvard University Press. Utkuhiksalingmiut Inuktitut, Utku,, or the Gjoa Haven ... References {{given name Turkish masculine given names Masculine given names Turkish feminine given names Feminine given names ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tayfun Sönmez
Tayfun Sönmez is a Turkish-American professor of economics at Boston College. He is a Fellow of the Econometric Society and the 2008 winner of the Social Choice and Welfare Prize, which honors scholars under the age of 40 for excellent accomplishment in the area of social choice theory and welfare economics. Sönmez has made significant contributions in the areas of microeconomic theory, mechanism/market design, and game theory. His work has been featured by the U.S. National Science Foundation for its practical relevance. Work Sönmez's early work studied the mathematical properties of allocation systems without transfers. His PhD dissertation established the relationship between the core and strategy-proof mechanisms. He was awarded University of Rochester's Conibear Prize as a graduate student for the best third-year paper. Following his first job at the University of Michigan, Sönmez identified a connection between the housing market model of Lloyd Shapley and Herbert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
