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Min-cut
In graph theory, a minimum cut or min-cut of a graph is a cut (a partition of the vertices of a graph into two disjoint subsets) that is minimal in some metric. Variations of the minimum cut problem consider weighted graphs, directed graphs, terminals, and partitioning the vertices into more than two sets. The weighted min-cut problem allowing both positive and negative weights can be trivially transformed into a weighted maximum cut problem by flipping the sign in all weights. __TOC__ Without terminal nodes The minimum cut problem in undirected, weighted graphs limited to non-negative weights can be solved in polynomial time by the Stoer-Wagner algorithm. In the special case when the graph is unweighted, Karger's algorithm provides an efficient randomized method for finding the cut. In this case, the minimum cut equals the edge connectivity of the graph. A generalization of the minimum cut problem without terminals is the minimum -cut, in which the goal is to partition th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Max-flow Min-cut Theorem
In computer science and optimization theory, the max-flow min-cut theorem states that in a flow network, the maximum amount of flow passing from the ''source'' to the ''sink'' is equal to the total weight of the edges in a minimum cut, i.e., the smallest total weight of the edges which if removed would disconnect the source from the sink. For example, imagine a network of pipes carrying water from a reservoir (the source) to a city (the sink). Each pipe has a capacity representing the maximum amount of water that can flow through it per unit of time. The max-flow min-cut theorem tells us that the maximum amount of water that can reach the city is limited by the smallest total capacity of any set of pipes that, if cut, would completely isolate the reservoir from the city. This smallest total capacity is the min-cut. So, if there's a bottleneck in the pipe network, represented by a small min-cut, that bottleneck will determine the overall maximum flow of water to the city. This is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Cut (graph Theory)
In graph theory, a cut is a partition of the vertices of a graph into two disjoint subsets. Any cut determines a cut-set, the set of edges that have one endpoint in each subset of the partition. These edges are said to cross the cut. In a connected graph, each cut-set determines a unique cut, and in some cases cuts are identified with their cut-sets rather than with their vertex partitions. In a flow network, an s–t cut is a cut that requires the ''source'' and the ''sink'' to be in different subsets, and its ''cut-set'' only consists of edges going from the source's side to the sink's side. The ''capacity'' of an s–t cut is defined as the sum of the capacity of each edge in the ''cut-set''. Definition A cut is a partition of of a graph into two subsets and . The cut-set of a cut is the set of edges that have one endpoint in and the other endpoint in . If and are specified vertices of the graph , then an – cut is a cut in which belongs to the set and belongs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Maxflow
In optimization theory, maximum flow problems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate. The maximum flow problem can be seen as a special case of more complex network flow problems, such as the circulation problem. The maximum value of an s-t flow (i.e., flow from source s to sink t) is equal to the minimum capacity of an s-t cut (i.e., cut severing s from t) in the network, as stated in the max-flow min-cut theorem. History The maximum flow problem was first formulated in 1954 by T. E. Harris and F. S. Ross as a simplified model of Soviet railway traffic flow. In 1955, Lester R. Ford, Jr. and Delbert R. Fulkerson created the first known algorithm, the Ford–Fulkerson algorithm.Ford, L.R., Jr.; Fulkerson, D.R., ''Flows in Networks'', Princeton University Press (1962). In their 1955 paper, Ford and Fulkerson wrote that the problem of Harris and Ross is formulated as follows (see p. 5):Consider a rail network conn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Gomory–Hu Tree
In combinatorial optimization, the Gomory–Hu tree of an undirected graph with capacities is a weighted tree that represents the minimum ''s''-''t'' cuts for all ''s''-''t'' pairs in the graph. The Gomory–Hu tree can be constructed in maximum flow computations. It is named for Ralph E. Gomory and T. C. Hu. Definition Let G = (V_G, E_G, c) be an undirected graph with c(u,v) being the capacity of the edge (u,v) respectively. : Denote the minimum capacity of an - cut by \lambda_ for each s, t \in V_G . : Let T = (V_G, E_T) be a tree, and denote the set of edges in an - path by P_ for each s,t \in V_G. Then is said to be a Gomory–Hu tree of , if for each s, t \in V_G : \lambda_ = \min_ c(S_e, T_e), where # S_e, T_e \subseteq V_G are the two connected components of T \setminus \, and thus (S_e, T_e) forms an - cut in . # c(S_e, T_e) is the capacity of the (S_e,T_e) cut in . Algorithm Gomory–Hu Algorithm :''Input'': A weighted undirected graph G = ((V_G,E_G), c) : ''Outp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Flow Network
In graph theory, a flow network (also known as a transportation network) is a directed graph where each edge has a capacity and each edge receives a flow. The amount of flow on an edge cannot exceed the capacity of the edge. Often in operations research, a directed graph is called a network, the vertices are called nodes and the edges are called arcs. A flow must satisfy the restriction that the amount of flow into a node equals the amount of flow out of it, unless it is a source, which has only outgoing flow, or sink, which has only incoming flow. A flow network can be used to model traffic in a computer network, circulation with demands, fluids in pipes, currents in an electrical circuit, or anything similar in which something travels through a network of nodes. As such, efficient algorithms for solving network flows can also be applied to solve problems that can be reduced to a flow network, including survey design, airline scheduling, image segmentation, and the matching prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Segmentation-based Object Categorization
The image segmentation problem is concerned with partitioning an image into multiple regions according to some homogeneity criterion. This article is primarily concerned with graph theoretic approaches to image segmentation applying graph partitioning via minimum cut or maximum cut. Segmentation-based object categorization can be viewed as a specific case of spectral clustering applied to image segmentation. Applications of image segmentation * Image compression ** Segment the image into homogeneous components, and use the most suitable compression algorithm for each component to improve compression. * Medical diagnosis ** Automatic segmentation of MRI images for identification of cancerous regions. * Mapping and measurement ** Automatic analysis of remote sensing data from satellites to identify and measure regions of interest. * Transportation ** Partition a transportation network makes it possible to identify regions characterized by homogeneous traffic states. Segm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Maximum Cut
In a graph, a maximum cut is a cut whose size is at least the size of any other cut. That is, it is a partition of the graph's vertices into two complementary sets and , such that the number of edges between and is as large as possible. Finding such a cut is known as the max-cut problem. The problem can be stated simply as follows. One wants a subset of the vertex set such that the number of edges between and the complementary subset is as large as possible. Equivalently, one wants a bipartite subgraph of the graph with as many edges as possible. There is a more general version of the problem called weighted max-cut, where each edge is associated with a real number, its weight, and the objective is to maximize the total weight of the edges between and its complement rather than the number of the edges. The weighted max-cut problem allowing both positive and negative weights can be trivially transformed into a weighted minimum cut problem by flipping the sign in all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
K-edge-connected Graph
In graph theory, a connected graph is -edge-connected if it remains connected whenever fewer than edges are removed. The edge-connectivity of a graph is the largest for which the graph is -edge-connected. Edge connectivity and the enumeration of -edge-connected graphs was studied by Camille Jordan in 1869. Formal definition Let G = (V, E) be an arbitrary graph. If the subgraph G' = (V, E \setminus X) is connected for all X \subseteq E where , X, < k, then ''G'' is said to be ''k''-edge-connected. The edge connectivity of is the maximum value ''k'' such that ''G'' is ''k''-edge-connected. The smallest set ''X'' whose removal disconnects ''G'' is a minimum cut in ''G''. The edge connectivity version of Menger's theorem provides an alternative and equivalent character ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Vertex Separator
In graph theory, a vertex subset is a vertex separator (or vertex cut, separating set) for nonadjacent Vertex (graph theory), vertices and if the Graph partition, removal of from the Graph (discrete mathematics), graph separates and into distinct connected component (graph theory), connected components. Examples Consider a grid graph with rows and columns; the total number of vertices is . For instance, in the illustration, , , and . If is odd, there is a single central row, and otherwise there are two rows equally close to the center; similarly, if is odd, there is a single central column, and otherwise there are two columns equally close to the center. Choosing to be any of these central rows or columns, and removing from the graph, partitions the graph into two smaller connected subgraphs and , each of which has at most vertices. If (as in the illustration), then choosing a central column will give a separator with r \leq \sqrt vertices, and similarly if the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cycle (graph Theory)
In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal. A graph without cycles is called an ''acyclic graph''. A directed graph without directed cycles is called a '' directed acyclic graph''. A connected graph without cycles is called a ''tree''. Definitions Circuit and cycle * A circuit is a non-empty trail in which the first and last vertices are equal (''closed trail''). : Let be a graph. A circuit is a non-empty trail with a vertex sequence . * A cycle or simple circuit is a circuit in which only the first and last vertices are equal. * ''n'' is called the length of the circuit resp. length of the cycle. Directed circuit and directed cycle * A directed circuit is a non-empty directed trail in which the first and last vertices are equal (''closed directed trail''). : Let be a directed grap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
