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Karger's Algorithm
In computer science and graph theory, Karger's algorithm is a randomized algorithm to compute a minimum cut of a connected graph. It was invented by David Karger and first published in 1993. The idea of the algorithm is based on the concept of contraction of an edge (u, v) in an undirected graph G = (V, E). Informally speaking, the contraction of an edge merges the nodes u and v into one, reducing the total number of nodes of the graph by one. All other edges connecting either u or v are "reattached" to the merged node, effectively producing a multigraph. Karger's basic algorithm iteratively contracts randomly chosen edges until only two nodes remain; those nodes represent a cut in the original graph. By iterating this basic algorithm a sufficient number of times, a minimum cut can be found with high probability. The global minimum cut problem A ''cut'' (S,T) in an undirected graph G = (V, E) is a partition of the vertices V into two non-empty, disjoint sets S\cup T= V. The ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Min Cut Example
Min or MIN may refer to: Places * Fujian, also called Mǐn, a province of China ** Min Kingdom (909–945), a state in Fujian * Min County, a county of Dingxi, Gansu province, China * Min River (Fujian) * Min River (Sichuan) * Mineola (Amtrak station), station code MIN People Personal names * Min (Korean surname), a Korean surname * Min (surname) (闵/閔), a Chinese surname * Min (Korean given name), a Korean given name and name element Individuals with the name * Min (Vietnamese singer) (born 1988) * Pechaya Wattanamontree (born 1989), Thai actress * Min (South Korean singer) (born 1991), South Korean singer, songwriter and actress Lee Min-young * Min (treasurer), ancient Egyptian official * Min, Marquis of Jin (died 678 BC), Chinese monarch * Empress Myeongseong (1851–1895), informally Queen Min, empress of Joseon * Menes or Min (a spelling variant no longer accepted), an early Egyptian pharaoh * Min Hogg (born 1939), British journalist and magazine editor * Min, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge Contraction In A Multigraph
Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by Microsoft * Microsoft Edge Legacy, a discontinued web browser developed by Microsoft * EdgeHTML, the layout engine used in Microsoft Edge Legacy * ThinkPad Edge, a Lenovo laptop computer series marketed from 2010 * Silhouette edge, in computer graphics, a feature of a 3D body projected onto a 2D plane * Explicit data graph execution, a computer instruction set architecture Telecommunication(s) * EDGE (telecommunication), a 2G digital cellular communications technology * Edge Wireless, an American mobile phone provider * Motorola Edge series, a series of smartphones made by Motorola * Samsung Galaxy Note Edge, a phablet made by Samsung * Samsung Galaxy S7 Edge or Samsung Galaxy S6 Edge, smartphones made by Samsung * Ubuntu Edge, a protot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dense Graph
In mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges (where every pair of vertices is connected by one edge). The opposite, a graph with only a few edges, is a sparse graph. The distinction of what constitutes a dense or sparse graph is ill-defined, and is often represented by 'roughly equal to' statements. Due to this, the way that density is defined often depends on the context of the problem. The graph density of simple graphs is defined to be the ratio of the number of edges with respect to the maximum possible edges. For undirected simple graphs, the graph density is: :D = \frac = \frac For directed, simple graphs, the maximum possible edges is twice that of undirected graphs (as there are two directions to an edge) so the density is: :D = \frac = \frac where is the number of edges and is the number of vertices in the graph. The maximum number of edges for an undirected graph is = \frac2, so the maximal density is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galton–Watson Process
The Galton–Watson process, also called the Bienaymé-Galton-Watson process or the Galton-Watson branching process, is a branching stochastic process arising from Francis Galton's statistical investigation of the extinction of family names. The process models family names as patrilineal (passed from father to son), while offspring are randomly either male or female, and names become extinct if the family name line dies out (holders of the family name die without male descendants). Galton's investigation of this process laid the groundwork for the study of branching processes as a subfield of probability theory, and along with these subsequent processes the Galton-Watson process has found numerous applications across population genetics, computer science, and other fields. History There was concern amongst the Victorians that aristocratic surnames were becoming extinct. In 1869, Galton published '' Hereditary Genius'', in which he treated the extinction of different social ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Binary Tree
In computer science and probability theory, a random binary tree is a binary tree selected at random from some probability distribution on binary trees. Different distributions have been used, leading to different properties for these trees. Random binary trees have been used for analyzing the average-case complexity of data structures based on binary search trees. For this application it is common to use random trees formed by inserting nodes one at a time according to a random permutation. The resulting trees are very likely to have logarithmic depth and logarithmic Strahler number. The treap and related balanced binary search trees use update operations that maintain this random structure even when the update sequence is non-random. Other distributions on random binary trees include the uniform discrete distribution in which all distinct trees are equally likely, distributions on a given number of nodes obtained by repeated splitting, binary tries and radix trees for random d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford Stein
Clifford Seth Stein (born December 14, 1965), a computer scientist, is a professor of industrial engineering and operations research at Columbia University in New York, NY, where he also holds an appointment in the Department of Computer Science. Stein is chair of the Industrial Engineering and Operations Research Department at Columbia University. Prior to joining Columbia, Stein was a professor at Dartmouth College in New Hampshire. Stein's research interests include the design and analysis of algorithms, combinatorial optimization, operations research, network algorithms, scheduling, algorithm engineering and computational biology. Stein has published many influential papers in the leading conferences and journals in his fields of research, and has occupied a variety of editorial positions including in the journals ''ACM Transactions on Algorithms'', ''Mathematical Programming'', ''Journal of Algorithms'', '' SIAM Journal on Discrete Mathematics'' and ''Operations Research ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
10 Repetitions Of Karger’s Contraction Procedure
1 (one, unit, unity) is a number, numeral, and glyph. It is the first and smallest positive integer of the infinite sequence of natural numbers. This fundamental property has led to its unique uses in other fields, ranging from science to sports, where it commonly denotes the first, leading, or top thing in a group. 1 is the unit of counting or measurement, a determiner for singular nouns, and a gender-neutral pronoun. Historically, the representation of 1 evolved from ancient Sumerian and Babylonian symbols to the modern Arabic numeral. In mathematics, 1 is the multiplicative identity, meaning that any number multiplied by 1 equals the same number. 1 is by convention not considered a prime number. In digital technology, 1 represents the "on" state in binary code, the foundation of computing. Philosophically, 1 symbolizes the ultimate reality or source of existence in various traditions. In mathematics The number 1 is the first natural number after 0. Each natural number, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree (graph Theory)
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. The degree of a vertex v is denoted \deg(v) or \deg v. The maximum degree of a graph G is denoted by \Delta(G), and is the maximum of G's vertices' degrees. The minimum degree of a graph is denoted by \delta(G), and is the minimum of G's vertices' degrees. In the multigraph shown on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, every vertex has the same degree, and so we can speak of ''the'' degree of the graph. A complete graph (denoted K_n, where n is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum possible degree, n-1. In a signed graph, the number of positive edges connected to the vertex v is called positive deg(v) and the number of connected negative edges is enti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle Graph
In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if the graph is simple) connected in a closed chain. The cycle graph with vertices is called . The number of vertices in equals the number of edges, and every vertex has degree 2; that is, every vertex has exactly two edges incident with it. If n = 1, it is an isolated loop. Terminology There are many synonyms for "cycle graph". These include simple cycle graph and cyclic graph, although the latter term is less often used, because it can also refer to graphs which are merely not acyclic. Among graph theorists, cycle, polygon, or ''n''-gon are also often used. The term ''n''-cycle is sometimes used in other settings. A cycle with an even number of vertices is called an even cycle; a cycle with an odd number of vertices is called an odd cycle. Properties A cycle graph is: * 2-edge colorable, if and only if it has an even n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spanning Tree Interpretation Of Karger’s Algorithm
Spanning may refer to: * Disc spanning, a feature of CD and DVD burning software * File spanning, the ability to package a single file or data stream into separate files of a specified size * Linear spanning, a concept in abstract algebra * Spanning tree, a subgraph which is a tree, containing all the vertices of a graph * Søren Spanning (1951–2020), Danish actor See also * Span (other) * Spanner (other) {{disambig, surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimum Spanning Tree
A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. That is, it is a spanning tree whose sum of edge weights is as small as possible. More generally, any edge-weighted undirected graph (not necessarily connected) has a minimum spanning forest, which is a union of the minimum spanning trees for its connected components. There are many use cases for minimum spanning trees. One example is a telecommunications company trying to lay cable in a new neighborhood. If it is constrained to bury the cable only along certain paths (e.g. roads), then there would be a graph containing the points (e.g. houses) connected by those paths. Some of the paths might be more expensive, because they are longer, or require the cable to be buried deeper; these paths would be represented by edges with larger weight ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kruskal’s Algorithm
Kruskal's algorithm finds a minimum spanning forest of an undirected edge-weighted graph. If the graph is connected, it finds a minimum spanning tree. It is a greedy algorithm that in each step adds to the forest the lowest-weight edge that will not form a cycle. The key steps of the algorithm are sorting and the use of a disjoint-set data structure to detect cycles. Its running time is dominated by the time to sort all of the graph edges by their weight. A minimum spanning tree of a connected weighted graph is a connected subgraph, without cycles, for which the sum of the weights of all the edges in the subgraph is minimal. For a disconnected graph, a minimum spanning forest is composed of a minimum spanning tree for each connected component. This algorithm was first published by Joseph Kruskal in 1956, and was rediscovered soon afterward by . Other algorithms for this problem include Prim's algorithm, Borůvka's algorithm, and the reverse-delete algorithm. Algorithm The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
