 |
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] |
 |
Push–relabel Maximum Flow Algorithm
In mathematical optimization, the push–relabel algorithm (alternatively, preflow–push algorithm) is an algorithm for computing maximum flows in a flow network. The name "push–relabel" comes from the two basic operations used in the algorithm. Throughout its execution, the algorithm maintains a "preflow" and gradually converts it into a maximum flow by moving flow locally between neighboring nodes using ''push'' operations under the guidance of an admissible network maintained by ''relabel'' operations. In comparison, the Ford–Fulkerson algorithm performs global augmentations that send flow following paths from the source all the way to the sink. The push–relabel algorithm is considered one of the most efficient maximum flow algorithms. The generic algorithm has a strongly polynomial time complexity, which is asymptotically more efficient than the Edmonds–Karp algorithm. Specific variants of the algorithms achieve even lower time complexities. The variant based on the h ... [...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] |
 |
Pets Flow
A pet, or companion animal, is an animal kept primarily for a person's company or entertainment rather than as a working animal, livestock, or a laboratory animal. Popular pets are often considered to have attractive/Cuteness, cute appearances, Animal cognition, intelligence, and relatable personalities, but some pets may be taken in on an altruistic basis (such as a stray animal) and accepted by the owner regardless of these characteristics. Two of the most popular pets are dogs and cats. Other animals commonly kept include House rabbit, rabbits; ferrets; domestic pig, pigs; rodents such as gerbils, hamsters, chinchillas, rats, mice, and guinea pigs; birds such as parrots, passerines, and fowls; reptiles such as turtles, lizards, snakes, and iguanas; aquarium, aquatic pets such as fish, freshwater snails, and sea snail, saltwater snails; amphibians such as frogs and salamanders; and arthropod pets such as tarantulas and hermit crabs. Smaller pets include Rodents as pets, rod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
|
Symposium On Foundations Of Computer Science
The IEEE Annual Symposium on Foundations of Computer Science (FOCS) is an academic conference in the field of theoretical computer science. FOCS is sponsored by the IEEE Computer Society. As writes, FOCS and its annual Association for Computing Machinery counterpart STOC (the Symposium on Theory of Computing) are considered the two top conferences in theoretical computer science, considered broadly: they “are forums for some of the best work throughout theory of computing that promote breadth among theory of computing researchers and help to keep the community together.” includes regular attendance at FOCS and STOC as one of several defining characteristics of theoretical computer scientists. Awards The Knuth Prize for outstanding contributions to theoretical computer science is presented alternately at FOCS and STOC. Works of the highest quality presented at the conference are awarded the Best Paper Award. In addition, the Machtey Award is presented to the best studen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
|
 |
Dynamic Trees
A link/cut tree is a data structure for representing a forest (graph theory), forest, a set of rooted trees, and offers the following operations: * Add a tree consisting of a single node to the forest. * Given a node in one of the trees, disconnect it (and its subtree) from the tree of which it is part. * Attach a node to another node as its child. * Given a node, find the root of the tree to which it belongs. By doing this operation on two distinct nodes, one can check whether they belong to the same tree. The represented forest may consist of very deep trees, so if we represent the forest as a plain collection of parent pointer trees, it might take us a long time to find the root of a given node. However, if we represent each tree in the forest as a link/cut tree, we can find which tree an element belongs to in big O notation, ''O''(logarithm, log(''n'')) amortized analysis, amortized time. Moreover, we can quickly adjust the collection of link/cut trees to changes in the rep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
 |
Dinic's Algorithm
Dinic's algorithm or Dinitz's algorithm is a strongly polynomial algorithm for computing the maximum flow in a flow network, conceived in 1970 by Israeli (formerly Soviet) computer scientist Yefim Dinitz. The algorithm runs in O(, V, ^2, E, ) time and is similar to the Edmonds–Karp algorithm, which runs in O(, V, , E, ^2) time, in that it uses shortest augmenting paths. The introduction of the concepts of the ''level graph'' and ''blocking flow'' enable Dinic's algorithm to achieve its performance. History Dinitz invented the algorithm in January 1969, as a master's student in Georgy Adelson-Velsky's group. A few decades later, he would recall: In 1970, Dinitz published a description of the algorithm in Proceedings of the USSR Academy of Sciences, ''Doklady Akademii Nauk SSSR''. In 1974, Shimon Even and (his then Ph.D. student) Alon Itai at the Technion – Israel Institute of Technology, Technion in Haifa were very curious and intrigued by Dinitz's algorithm as well as Alexande ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
 |
Breadth-first Search
Breadth-first search (BFS) is an algorithm for searching a tree data structure for a node that satisfies a given property. It starts at the tree root and explores all nodes at the present depth prior to moving on to the nodes at the next depth level. Extra memory, usually a queue, is needed to keep track of the child nodes that were encountered but not yet explored. For example, in a chess endgame, a chess engine may build the game tree from the current position by applying all possible moves and use breadth-first search to find a win position for White. Implicit trees (such as game trees or other problem-solving trees) may be of infinite size; breadth-first search is guaranteed to find a solution node if one exists. In contrast, (plain) depth-first search (DFS), which explores the node branch as far as possible before backtracking and expanding other nodes, may get lost in an infinite branch and never make it to the solution node. Iterative deepening depth-first search ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
|
Edmonds–Karp Algorithm
In computer science, the Edmonds–Karp algorithm is an implementation of the Ford–Fulkerson method for computing the maximum flow in a flow network in O(, V, , E, ^2) time. The algorithm was first published by Yefim Dinitz in 1970, and independently published by Jack Edmonds and Richard Karp in 1972. Dinitz's algorithm includes additional techniques that reduce the running time to O(, V, ^2, E, ). Algorithm The algorithm is identical to the Ford–Fulkerson algorithm, except that the search order when finding the augmenting path is defined. The path found must be a shortest path that has available capacity. This can be found by a breadth-first search, where we apply a weight of 1 to each edge. The running time of O(, V, , E, ^2) is found by showing that each augmenting path can be found in O(, E, ) time, that every time at least one of the edges becomes saturated (an edge which has the maximum possible flow), that the distance from the saturated edge to the source along t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |