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Dynamic Connectivity
In computing and graph theory, a dynamic connectivity structure is a data structure that dynamically maintains information about the connected components of a graph. The set ''V'' of vertices of the graph is fixed, but the set ''E'' of edges can change. The three cases, in order of difficulty, are: * Edges are only added to the graph (this can be called ''incremental connectivity''); * Edges are only deleted from the graph (this can be called ''decremental connectivity''); * Edges can be either added or deleted (this can be called ''fully dynamic connectivity''). After each addition/deletion of an edge, the dynamic connectivity structure should adapt itself such that it can give quick answers to queries of the form "is there a path between ''x'' and ''y''?" (equivalently: "do vertices ''x'' and ''y'' belong to the same connected component?"). Incremental connectivity If edges can only be added, then the dynamic connectivity problem can be solved by a disjoint-set data structure. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computer, computing machinery. It includes the study and experimentation of algorithmic processes, and the development of both computer hardware, hardware and software. Computing has scientific, engineering, mathematical, technological, and social aspects. Major computing disciplines include computer engineering, computer science, cybersecurity, data science, information systems, information technology, and software engineering. The term ''computing'' is also synonymous with counting and calculation, calculating. In earlier times, it was used in reference to the action performed by Mechanical computer, mechanical computing machines, and before that, to Computer (occupation), human computers. History The history of computing is longer than the history of computing hardware and includes the history of methods intended for pen and paper (or for chalk and slate) with or without the aid of tables. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Partition Refinement
In the design of algorithms, partition refinement is a technique for representing a partition of a set as a data structure that allows the partition to be refined by splitting its sets into a larger number of smaller sets. In that sense it is dual to the union-find data structure, which also maintains a partition into disjoint sets but in which the operations merge pairs of sets. In some applications of partition refinement, such as lexicographic breadth-first search, the data structure maintains as well an ordering on the sets in the partition. Partition refinement forms a key component of several efficient algorithms on graphs and finite automata, including DFA minimization, the Coffman–Graham algorithm for parallel scheduling, and lexicographic breadth-first search of graphs. Data structure A partition refinement algorithm maintains a family of disjoint sets . At the start of the algorithm, this family contains a single set of all the elements in the data structure. At e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamic Problem (algorithms)
Dynamic problems in computational complexity theory are problems stated in terms of changing input data. In its most general form, a problem in this category is usually stated as follows: * Given a class of input objects, find efficient algorithms and data structures to answer a certain query about a set of input objects each time the input data is modified, i.e., objects are inserted or deleted. Problems in this class have the following measures of complexity: * Space the amount of memory space required to store the data structure; * Initialization time time required for the initial construction of the data structure; * Insertion time time required for the update of the data structure when one more input element is added; * Deletion time time required for the update of the data structure when an input element is deleted; * Query time time required to answer a query; * Other operations specific to the problem in question The overall set of computations for a dynamic problem is ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamic Connectivity
In computing and graph theory, a dynamic connectivity structure is a data structure that dynamically maintains information about the connected components of a graph. The set ''V'' of vertices of the graph is fixed, but the set ''E'' of edges can change. The three cases, in order of difficulty, are: * Edges are only added to the graph (this can be called ''incremental connectivity''); * Edges are only deleted from the graph (this can be called ''decremental connectivity''); * Edges can be either added or deleted (this can be called ''fully dynamic connectivity''). After each addition/deletion of an edge, the dynamic connectivity structure should adapt itself such that it can give quick answers to queries of the form "is there a path between ''x'' and ''y''?" (equivalently: "do vertices ''x'' and ''y'' belong to the same connected component?"). Incremental connectivity If edges can only be added, then the dynamic connectivity problem can be solved by a disjoint-set data structure. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spanning Forest
In the mathematical field of graph theory, a spanning tree ''T'' of an undirected graph ''G'' is a subgraph that is a tree which includes all of the vertices of ''G''. In general, a graph may have several spanning trees, but a graph that is not connected will not contain a spanning tree (see about spanning forests below). If all of the edges of ''G'' are also edges of a spanning tree ''T'' of ''G'', then ''G'' is a tree and is identical to ''T'' (that is, a tree has a unique spanning tree and it is itself). Applications Several pathfinding algorithms, including Dijkstra's algorithm and the A* search algorithm, internally build a spanning tree as an intermediate step in solving the problem. In order to minimize the cost of power networks, wiring connections, piping, automatic speech recognition, etc., people often use algorithms that gradually build a spanning tree (or many such trees) as intermediate steps in the process of finding the minimum spanning tree. The Interne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Tour Tree
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential discoveries in many other branches of mathematics, such as analytic number theory, complex analysis, and infinitesimal calculus. He also introduced much of modern mathematical terminology and Mathematical notation, notation, including the notion of a mathematical function. He is known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory. Euler has been called a "universal genius" who "was fully equipped with almost unlimited powers of imagination, intellectual gifts and extraordinary memory". He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Kingdom of Prussia, Prussia. Euler is credited for popularizing the Greek letter \pi (lowercase Pi (letter), pi) to denote Pi, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Link-cut Tree
A link/cut tree is a data structure for representing a 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 ''O''(log(''n'')) amortized time. Moreover, we can quickly adjust the collection of link/cut trees to changes in the represented forest. In particular, we can adjust it to merge (link) and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Without Loss Of Generality
''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicate the assumption that what follows is chosen arbitrarily, narrowing the premise to a particular case, but does not affect the validity of the proof in general. The other cases are sufficiently similar to the one presented that proving them follows by essentially the same logic. As a result, once a proof is given for the particular case, it is trivial to adapt it to prove the conclusion in all other cases. In many scenarios, the use of "without loss of generality" is made possible by the presence of symmetry. For example, if some property ''P''(''x'',''y'') of real numbers is known to be symmetric in ''x'' and ''y'', namely that ''P''(''x'',''y'') is equivalent to ''P''(''y'',''x''), then in proving that ''P''(''x'',''y'') holds for every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Depth-first Search
Depth-first search (DFS) is an algorithm for traversing or searching tree or graph data structures. The algorithm starts at the root node (selecting some arbitrary node as the root node in the case of a graph) and explores as far as possible along each branch before backtracking. Extra memory, usually a stack, is needed to keep track of the nodes discovered so far along a specified branch which helps in backtracking of the graph. A version of depth-first search was investigated in the 19th century by French mathematician Charles Pierre Trémaux as a strategy for solving mazes. Properties The time and space analysis of DFS differs according to its application area. In theoretical computer science, DFS is typically used to traverse an entire graph, and takes time where , V, is the number of vertices and , E, the number of edges. This is linear in the size of the graph. In these applications it also uses space O(, V, ) in the worst case to store the stack of vertices on t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') which are connected by ''Glossary of graph theory terms#edge, edges'' (also called ''arcs'', ''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 (mathematics), set of vertices (also called nodes or points); * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In graph theory, a tree is an undirected graph in which any two vertices are connected by path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. A directed tree, oriented tree,See .See . polytree,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 arborescence or out-tree� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
