|
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 pat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''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 of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Order
In mathematics, a total 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 total). 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, and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but refers generally to some sort of totally ordered subsets of a given partially ordered set. An extension of a given partial order to a total order is called a linear extension of that partial order. Strict and non-strict total orders A on a set X is a strict partial ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a 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 cycles. The two sets U and V may be thought of as a 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 triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denoting ... [...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''). It is the directed analog of the minimum spanning tree problem. The algorithm was proposed independently first by Yoeng-Jin Chu and Tseng-Hong Liu (1965) and then by Jack Edmonds (1967). Algorithm Description The algorithm takes as input a directed graph D = \langle V, E \rangle where V is the set of nodes and E is the set of directed edges, a distinguished vertex r \in V called the ''root'', and a real-valued weight w(e) for each edge e \in E. It returns a spanning arborescence A rooted at r of minimum weight, where the weight of an arborescence is defined to be the sum of its edge weights, w(A) = \sum_. The algorithm has a recursive description. Let f(D, r, w) denote the function which returns a spanning arborescence rooted at r of minimum weight. We first remove any edge from E whose destina ... [...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 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 Other *Jonathan (apple), a variety of apple * "Jonathan" (song), a 2015 song by French singer and songwrit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for ''Ordnung'', meaning the order of approximation. In computer science, big O notation is used to 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 arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rates: d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Run-time Complexity
In 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 generally expressed ... [...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 problem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kidney Exchange
Kidney paired donation (KPD), or paired exchange, is an approach to living donor kidney transplantation where patients with incompatible donors swap kidneys to receive a compatible kidney. KPD is used in situations where a potential donor is incompatible. Because better donor HLA and age matching are correlated with lower lifetime mortality and longer lasting kidney transplants, many compatible pairs are also participating in swaps to find better matched kidneys. In the United States, the National Kidney Registry organizes the majority of U.S. KPD transplants, including the largest swaps. The first large swap was a 60 participant chain in 2012 that appeared on the front page of the ''New York Times'' and the second, even larger swap, included 70 participants and was completed in 2014. Other KPD programs in the U.S. include the UNOS program, which was launched in 2010 and completed its 100th KPD transplant in 2014, and the Alliance for Paired Donation. According to a 2019 study, kid ... [...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 is a set of edges without common vertices. Finding a matching in a bipartite graph can be treated as a network flow problem. Definitions Given a graph a matching ''M'' in ''G'' is a set of pairwise non-adjacent edges, none of which are 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 intersection with at least one edge in ''M''. The following figure shows examples of maximal matchings (red) in three graphs. : A maximum matching (also known as maximum-cardinality matching) is a matching that contains the largest possible number of edges. ... [...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, Utkuhikhalik, Utkuhikhaliq, Utkuhiksalingmiutitut, Utkuhiksalingmiutut,Briggs, J. L. (1970), Never in anger. Portrait of an Eskimo family. Harvard University Press. Utkuhiksalingmiut Inuktitut, Utku,, Gjoa Haven dialect, is a sub-di ... References {{Reflist Turkish masculine given names Turkish 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] |
