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Level Ancestor Problem
In graph theory and theoretical computer science, the level ancestor problem is the problem of preprocessing a given rooted tree ''T'' into a data structure that can determine the ancestor of a given node at a given distance from the root of the tree. More precisely, let ''T'' be a rooted tree with ''n'' nodes, and let ''v'' be an arbitrary node of ''T''. The level ancestor query LA(''v'',''d'') requests the ancestor of node ''v'' at depth ''d'', where the depth of a node ''v'' in a tree is the number of edges on the shortest path from the root of the tree to node ''v''. It is possible to solve this problem in constant time per query, after a preprocessing algorithm that takes O(''n'') and that builds a data structure that uses O(''n'') storage space. Jump pointer algorithm The jump pointer algorithm pre-processes a tree in O(''n'' log ''n'') time and answers level ancestor queries in O(log ''n'') time. The jump pointer algorithm associates up to log&nbs ... [...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] |
Recursion
Recursion occurs when the definition of a concept or process depends on a simpler or previous version of itself. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function (mathematics), function being defined is applied within its own definition. While this apparently defines an infinite number of instances (function values), it is often done in such a way that no infinite loop or infinite chain of references can occur. A process that exhibits recursion is ''recursive''. Video feedback displays recursive images, as does an infinity mirror. Formal definitions In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties: * A simple ''base case'' (or cases) — a terminating scenario that does not use recursion to produce an answer * A ''recursive step'' — a set of rules that reduce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lowest Common Ancestor
In graph theory and computer science, the lowest common ancestor (LCA) (also called least common ancestor) of two nodes and in a Tree (graph theory), tree or directed acyclic graph (DAG) is the lowest (i.e. deepest) node that has both and as descendants, where we define each node to be a descendant of itself (so if has a direct connection from , is the lowest common ancestor). The LCA of and in is the shared ancestor of and that is located farthest from the root. Computation of lowest common ancestors may be useful, for instance, as part of a procedure for determining the distance between pairs of nodes in a tree: the distance from to can be computed as the distance from the root to , plus the distance from the root to , minus twice the distance from the root to their lowest common ancestor . In a tree data structure where each node points to its parent, the lowest common ancestor can be easily determined by finding the first intersection of the paths from and to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Range Query (computer Science)
In computer science, the range query problem consists of efficiently answering several queries regarding a given interval of elements within an array. For example, a common task, known as range minimum query, is finding the smallest value inside a given range within a list of numbers. Definition Given a function f that accepts an array, a range query f_q(l, r) on an array a= _1,..,a_n/math> takes two indices l and r and returns the result of f when applied to the subarray _l, \ldots, a_r/math>. For example, for a function \operatorname that returns the sum of all values in an array, the range query \operatorname_q(l, r) returns the sum of all values in the range , r/math>. Solutions Prefix sum array Range sum queries may be answered in constant time and linear space by pre-computing an array of same length as the input such that for every index , the element is the sum of the first elements of . Any query may then be computed as follows: \operatorname_q(l, r) = p_r - p_. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Tour
In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends on the same vertex. They were first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. The problem can be stated mathematically like this: :Given the graph in the image, is it possible to construct a path (or a cycle; i.e., a path starting and ending on the same vertex) that visits each edge exactly once? Euler proved that a necessary condition for the existence of Eulerian circuits is that all vertices in the graph have an even degree, and stated without proof that connected graphs with all vertices of even degree have an Eulerian circuit. The first complete proof of this latter claim was published posthumously in 1873 by Carl Hierholzer. This is known as Euler's Theorem: :A con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Method Of Four Russians
In computer science, the Method of Four Russians or "The Four-Russians speedup," is a technique for speeding up algorithms involving Boolean matrices, or more generally algorithms involving matrices in which each cell may take on only a bounded number of possible values. Idea The main idea of the method is to partition the matrix into small square blocks of size for some parameter , and to use a lookup table to perform the algorithm quickly within each block. The index into the lookup table encodes the values of the matrix cells on the upper left of the block boundary prior to some operation of the algorithm, and the result of the lookup table encodes the values of the boundary cells on the lower right of the block after the operation. Thus, the overall algorithm may be performed by operating on only blocks instead of on matrix cells, where is the side length of the matrix. In order to keep the size of the lookup tables (and the time needed to initialize them) sufficiently sm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jagged Array
In computer science, a jagged array, also known as a ragged array or irregular array is an array data structure, array of arrays of which the member arrays can be of different lengths, producing rows of jagged edges when visualized as output. In contrast, two-dimensional arrays are always rectangular so jagged arrays should not be confused with Array data type, multidimensional arrays, but the former is often used to emulate the latter. Arrays of arrays in languages such as Java, PHP, Python (multidimensional lists), Ruby, C#.NET, Visual Basic.NET, Perl, JavaScript, Objective-C, Swift, and Atlas Autocode are implemented as Iliffe vectors. Examples In C Sharp (programming language), C# and Java (programming language), Java jagged arrays can be created with the following code: int[][] c; c = new int[2][]; // creates 2 rows c[0] = new int[5]; // 5 columns for row 0 c[1] = new int[3]; // create 3 columns for row 1 In C (programming language), C and C++, a jagged array can be cr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (data Structure)
In computer science, a tree is a widely used abstract data type that represents a hierarchical tree structure with a set of connected nodes. Each node in the tree can be connected to many children (depending on the type of tree), but must be connected to exactly one parent, except for the ''root'' node, which has no parent (i.e., the root node as the top-most node in the tree hierarchy). These constraints mean there are no cycles or "loops" (no node can be its own ancestor), and also that each child can be treated like the root node of its own subtree, making recursion a useful technique for tree traversal. In contrast to linear data structures, many trees cannot be represented by relationships between neighboring nodes (parent and children nodes of a node under consideration, if they exist) in a single straight line (called edge or link between two adjacent nodes). Binary trees are a commonly used type, which constrain the number of children for each parent to at most two. Whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path (graph Theory)
In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A directed path (sometimes called dipath) in a directed graph is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction. Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See e.g. , , or . cover more advanced algorithmic topics concerning paths in graphs. Definitions Walk, trail, and path * A walk is a finite or infinite sequence of edges which joins a sequence of vertices. : Let be a graph. A finite walk is a sequence of edges for which there is a sequence of vertices such that ''Φ''(''e''''i'') = for . is the ''vertex sequence'' of the walk. The walk is ''closed'' if ''v''1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science
Theoretical computer science is a subfield of computer science and mathematics that focuses on the Abstraction, abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely. The Association for Computing Machinery, ACM's Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following description: History While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved. Information theory was added to the field with A Mathematical Theory of Communication, a 1948 mathematical theory of communication by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of Hebbian learning, learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of neural networks and para ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Array Data Structure
In computer science, an array is a data structure consisting of a collection of ''elements'' (value (computer science), values or variable (programming), variables), of same memory size, each identified by at least one ''array index'' or ''key'', a collection of which may be a tuple, known as an index tuple. An array is stored such that the position (memory address) of each element can be computed from its index tuple by a mathematical formula. The simplest type of data structure is a linear array, also called a one-dimensional array. For example, an array of ten 32-bit (4-byte) integer variables, with indices 0 through 9, may be stored as ten Word (data type), words at memory addresses 2000, 2004, 2008, ..., 2036, (in hexadecimal: 0x7D0, 0x7D4, 0x7D8, ..., 0x7F4) so that the element with index ''i'' has the address 2000 + (''i'' × 4). The memory address of the first element of an array is called first address, foundation address, or base address. Because the mathematical conc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (graph Theory)
In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices). In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another. From the point of view of graph theory, vertices are treated as featureless and indivisible objects, although they may have additional structure depending on the application from which the graph arises; for instance, a semantic network is a graph in which the vertices represent concepts or classes of objects. The two vertices forming an edge are said to be the endpoints of this edge, and the edge is said to be incident to the vertices. A vertex ''w'' is said to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
