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Blossom Algorithm
In graph theory, the blossom algorithm is an algorithm for constructing maximum matchings on graphs. The algorithm was developed by Jack Edmonds in 1961, and published in 1965. Given a general graph , the algorithm finds a matching such that each vertex in is incident with at most one edge in and is maximized. The matching is constructed by iteratively improving an initial empty matching along augmenting paths in the graph. Unlike bipartite matching, the key new idea is that an odd-length cycle in the graph (blossom) is contracted to a single vertex, with the search continuing iteratively in the contracted graph. The algorithm runs in time , where is the number of edges of the graph and is its number of vertices. A better running time of O( , E, \sqrt ) for the same task can be achieved with the much more complex algorithm of Micali and Vazirani. A major reason that the blossom algorithm is important is that it gave the first proof that a maximum-size matching could be ... [...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] |
Blossom (graph Theory)
In graph theory, a mathematical discipline, a factor-critical graph (or hypomatchable graph.) is a graph with vertices in which every subgraph of vertices has a perfect matching. (A perfect matching in a graph is a subset of its edges with the property that each of its vertices is the endpoint of exactly one of the edges in the subset.) A matching that covers all but one vertex of a graph is called a near-perfect matching. So equivalently, a factor-critical graph is a graph in which there are near-perfect matchings that avoid every possible vertex. Examples Any odd-length cycle graph is factor-critical, as is any complete graph with an odd number of vertices. More generally, every Hamiltonian graph with an odd number of vertices is factor-critical. The friendship graphs (graphs formed by connecting a collection of triangles at a single common vertex) provide examples of graphs that are factor-critical but not Hamiltonian. If a graph is factor-critical, then so is the Myciels ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Weight Matching
In computer science and graph theory, the maximum weight matching problem is the problem of finding, in a weighted graph, a matching in which the sum of weights is maximized. A special case of it is the assignment problem, in which the input is restricted to be a bipartite graph, and the matching constrained to be have cardinality that of the smaller of the two partitions. Another special case is the problem of finding a maximum cardinality matching on an unweighted graph: this corresponds to the case where all edge weights are the same. Algorithms There is a O(V^E) time algorithm to find a maximum matching or a maximum weight matching in a graph that is not bipartite; it is due to Jack Edmonds, is called the ''paths, trees, and flowers'' method or simply Edmonds' algorithm, and uses bidirected edges. A generalization of the same technique can also be used to find maximum independent sets in claw-free graphs. More elaborate algorithms exist and are reviewed by Duan and Pettie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path Lifting
A path is a route for physical travel – see Trail. Path or PATH may also refer to: Physical paths of different types * Bicycle path * Bridle path, used by people on horseback * Course (navigation), the intended path of a vehicle * Desire path, created by human or animal foot traffic * Footpath, intended for use only by pedestrians * Shared-use path, intended for multiple modes such as walking, bicycling, in-line skating or others * Sidewalk, a paved path along the side of a road * Hoggin, a buff-coloured gravel & clay pathway often seen in gardens of Stately Homes, Parks etc. * Trail, an unpaved lane or road Mathematics, physics, and computing * Path (computing), in file systems, the human-readable address of a resource ** PATH (variable), in computing, a way to specify a list of directories containing executable programs * Path (graph theory), a sequence of edges of a graph that form a trail ** st-connectivity problem, sometimes known as the "path problem" * Path (to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path Detection
A path is a route for physical travel – see Trail. Path or PATH may also refer to: Physical paths of different types * Bicycle path * Bridle path, used by people on horseback * Course (navigation), the intended path of a vehicle * Desire path, created by human or animal foot traffic * Footpath, intended for use only by pedestrians * Shared-use path, intended for multiple modes such as walking, bicycling, in-line skating or others * Sidewalk, a paved path along the side of a road * Hoggin, a buff-coloured gravel & clay pathway often seen in gardens of Stately Homes, Parks etc. * Trail, an unpaved lane or road Mathematics, physics, and computing * Path (computing), in file systems, the human-readable address of a resource ** PATH (variable), in computing, a way to specify a list of directories containing executable programs * Path (graph theory), a sequence of edges of a graph that form a trail ** st-connectivity problem, sometimes known as the "path problem" * Path (to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blossom Contraction
In botany, blossoms are the flowers of stone fruit trees (genus ''Prunus'') and of some other plants with a similar appearance that flower profusely for a period of time in spring. Colloquially, flowers of orange are referred to as such as well. Peach blossoms (including nectarine), most cherry blossoms, and some almond blossoms are usually pink. Plum blossoms, apple blossoms, orange blossoms, some cherry blossoms, and most almond blossoms are white. Blossoms provide pollen to pollinators such as bees, and initiate cross-pollination necessary for the trees to reproduce by producing fruit. Herbal use The ancient Phoenicians used almond blossoms with honey and urine as a tonic, and sprinkled them into stews and gruels to give muscular strength. Crushed petals were also used as a poultice on skin spots and mixed with banana oil, for dry skin and sunburn. In herbalism the crab apple was used as treatment for boils, abscesses, splinters, wounds, coughs, colds and a host of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forest Expansion
A forest is an area of land dominated by 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 function. The United Nations' Food and Agriculture Organization (FAO) defines a forest as, "Land spanning more than 0.5 hectares with trees higher than 5 meters and a canopy cover of more than 10 percent, or trees able to reach these thresholds ''in situ''. It does not include land that is predominantly under agricultural or urban use." Using this definition, '' Global Forest Resources Assessment 2020'' (FRA 2020) found that forests covered , or approximately 31 percent of the world's land area in 2020. Forests are the predominant terrestrial ecosystem of Earth, and are found around the globe. More than half of the world's forests are found in only five countries (Brazil, Canada, China, Russia, and the United States). The largest share of forests (45 percent) are in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forest (graph Theory)
In graph theory, a tree is an undirected graph in which any two vertices are connected by ''exactly one'' path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by ''at most one'' path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. A polytreeSee . (or directed tree or oriented treeSee .See . or singly connected networkSee .) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees. A rooted tree may be directed, called a directed rooted tree, either making all its edges point away from the root—in which case it is called an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edmonds Lifting End Point
Edmonds may refer to: * Edmonds (surname), a surname (including a list of people with the surname) * Edmonds, Washington, a city in Washington, US ** Edmonds station (Washington), a passenger train station in Washington, US * Edmonds station (SkyTrain), a SkyTrain station in Burnaby, British Columbia, Canada See also * Burnaby-Edmonds, an electoral district in British Columbia, Canada * Edmond (other) * Edmunds (other) Edmunds may refer to: People * Edmunds (given name) * Edmunds (surname) Places * Edmunds Center, an arena in Deland, Florida * Edmunds County, South Dakota Companies * Edmunds (company), provider of automotive information See also * Edmonds ... {{disambiguation cs:Edmonds ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edmonds Lifting Path
Edmonds may refer to: * Edmonds (surname), a surname (including a list of people with the surname) * Edmonds, Washington, a city in Washington, US ** Edmonds station (Washington), a passenger train station in Washington, US * Edmonds station (SkyTrain), a SkyTrain station in Burnaby, British Columbia, Canada See also * Burnaby-Edmonds, an electoral district in British Columbia, Canada * Edmond (other) * Edmunds (other) Edmunds may refer to: People * Edmunds (given name) * Edmunds (surname) Places * Edmunds Center, an arena in Deland, Florida * Edmunds County, South Dakota Companies * Edmunds (company), provider of automotive information See also * Edmonds ... {{disambiguation cs:Edmonds ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
