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Cycle Detection
In computer science, cycle detection or cycle finding is the algorithmic problem of finding a cycle in a sequence of iterated function values. For any function that maps a finite set to itself, and any initial value in , the sequence of iterated function values : x_0,\ x_1=f(x_0),\ x_2=f(x_1),\ \dots,\ x_i=f(x_),\ \dots must eventually use the same value twice: there must be some pair of distinct indices and such that . Once this happens, the sequence must continue periodically, by repeating the same sequence of values from to . Cycle detection is the problem of finding and , given and . Several algorithms for finding cycles quickly and with little memory are known. Robert W. Floyd's tortoise and hare algorithm moves two pointers at different speeds through the sequence of values until they both point to equal values. Alternatively, Brent's algorithm is based on the idea of exponential search. Both Floyd's and Brent's algorithms use only a constant number of memory ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical disciplines (including the design and implementation of Computer architecture, hardware and Computer programming, software). Computer science is generally considered an area of research, academic research and distinct from computer programming. Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of computational problem, problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and for preventing Vulnerability (computing), security vulnerabilities. Computer graphics (computer science), Computer graphics and computational geometry address the generation of images. Progr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deadlock
In concurrent computing, deadlock is any situation in which no member of some group of entities can proceed because each waits for another member, including itself, to take action, such as sending a message or, more commonly, releasing a lock. Deadlocks are a common problem in multiprocessing systems, parallel computing, and distributed systems, because in these contexts systems often use software or hardware locks to arbitrate shared resources and implement process synchronization. In an operating system, a deadlock occurs when a process or thread enters a waiting state because a requested system resource is held by another waiting process, which in turn is waiting for another resource held by another waiting process. If a process remains indefinitely unable to change its state because resources requested by it are being used by another process that itself is waiting, then the system is said to be in a deadlock. In a communications system, deadlocks occur mainly due t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pointer (computer Programming)
In computer science, a pointer is an object in many programming languages that stores a memory address. This can be that of another value located in computer memory, or in some cases, that of memory-mapped computer hardware. A pointer ''references'' a location in memory, and obtaining the value stored at that location is known as ''dereferencing'' the pointer. As an analogy, a page number in a book's index could be considered a pointer to the corresponding page; dereferencing such a pointer would be done by flipping to the page with the given page number and reading the text found on that page. The actual format and content of a pointer variable is dependent on the underlying computer architecture. Using pointers significantly improves performance for repetitive operations, like traversing iterable data structures (e.g. strings, lookup tables, control tables and tree structures). In particular, it is often much cheaper in time and space to copy and dereference pointers th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are sufficiently large, no efficient non-quantum integer factorization algorithm is known. However, it has not been proven that such an algorithm does not exist. The presumed difficulty of this problem is important for the algorithms used in cryptography such as RSA public-key encryption and the RSA digital signature. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing. In 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé and Paul Zimmermann factored a 240-digit (795-bit) number (RSA-240) utilizing approximately 900 core-years of computing power. The researchers estimated that a 1024-bit RSA ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pollard's Rho Algorithm
Pollard's rho algorithm is an algorithm for integer factorization. It was invented by John Pollard in 1975. It uses only a small amount of space, and its expected running time is proportional to the square root of the smallest prime factor of the composite number being factorized. Core ideas The algorithm is used to factorize a number n = pq, where p is a non-trivial factor. A polynomial modulo n, called g(x) (e.g., g(x) = (x^2 + 1) \bmod n), is used to generate a pseudorandom sequence. It is important to note that g(x) must be a polynomial. A starting value, say 2, is chosen, and the sequence continues as x_1 = g(2), x_2 = g(g(2)), x_3 = g(g(g(2))), etc. The sequence is related to another sequence \. Since p is not known beforehand, this sequence cannot be explicitly computed in the algorithm. Yet, in it lies the core idea of the algorithm. Because the number of possible values for these sequences is finite, both the \ sequence, which is mod n, and \ sequence will eventually ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Space Complexity
The space complexity of an algorithm or a computer program is the amount of memory space required to solve an instance of the computational problem as a function of characteristics of the input. It is the memory required by an algorithm until it executes completely. Similar to time complexity, space complexity is often expressed asymptotically in big O notation, such as O(n), O(n\log n), O(n^\alpha), O(2^n), etc., where is a characteristic of the input influencing space complexity. Space complexity classes Analogously to time complexity classes DTIME(f(n)) and NTIME(f(n)), the complexity classes DSPACE(f(n)) and NSPACE(f(n)) are the sets of languages that are decidable by deterministic (respectively, non-deterministic) Turing machines that use O(f(n)) space. The complexity classes PSPACE and NPSPACE allow f to be any polynomial, analogously to P and NP. That is, :\mathsf = \bigcup_ \mathsf(n^c) and :\mathsf = \bigcup_ \mathsf(n^c) Relationships between classes The space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. 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 graph. A directed circuit is a non-empty directed trail with a vertex sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rho (letter)
Rho (uppercase Ρ, lowercase ρ or ; el, ρο or el, ρω, label=none) is the 17th letter of the Greek alphabet. In the system of Greek numerals it has a value of 100. It is derived from Phoenician letter res . Its uppercase form uses the same glyph, Ρ, as the distinct Latin letter P; the two letters have different Unicode encodings. Uses Greek Rho is classed as a liquid consonant (together with Lambda and sometimes the nasals Mu and Nu), which has important implications for morphology. In both Ancient and Modern Greek, it represents a alveolar trill , alveolar tap , or alveolar approximant . In polytonic orthography, a rho at the beginning of a word is written with a rough breathing, equivalent to ''h'' ( ''rh''), and a double rho within a word is written with a smooth breathing over the first rho and a rough breathing over the second ( ''rrh''). That apparently reflected an aspirated or voiceless pronunciation in Ancient Greek, which led to the various Greek-deri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reachability
In graph theory, reachability refers to the ability to get from one vertex to another within a graph. A vertex s can reach a vertex t (and t is reachable from s) if there exists a sequence of adjacent vertices (i.e. a walk) which starts with s and ends with t. In an undirected graph, reachability between all pairs of vertices can be determined by identifying the connected components of the graph. Any pair of vertices in such a graph can reach each other if and only if they belong to the same connected component; therefore, in such a graph, reachability is symmetric (s reaches t iff t reaches s). The connected components of an undirected graph can be identified in linear time. The remainder of this article focuses on the more difficult problem of determining pairwise reachability in a directed graph (which, incidentally, need not be symmetric). Definition For a directed graph G = (V, E), with vertex set V and edge set E, the reachability relation of G is the transitive closure ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Directed Graph
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pair where * ''V'' is a set whose elements are called '' vertices'', ''nodes'', or ''points''; * ''A'' is a set of ordered pairs of vertices, called ''arcs'', ''directed edges'' (sometimes simply ''edges'' with the corresponding set named ''E'' instead of ''A''), ''arrows'', or ''directed lines''. It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, which are usually called ''edges'', ''links'' or ''lines''. The aforementioned definition does not allow a directed graph to have multiple arrows with the same source and target nodes, but some authors consider a broader definition that allows directed graphs to have such multiple arcs (namely, they allow the arc set to be a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Graph
In graph theory, a pseudoforest is an undirected graphThe kind of undirected graph considered here is often called a multigraph or pseudograph, to distinguish it from a simple graph. in which every connected component has at most one cycle. That is, it is a system of vertices and edges connecting pairs of vertices, such that no two cycles of consecutive edges share any vertex with each other, nor can any two cycles be connected to each other by a path of consecutive edges. A pseudotree is a connected pseudoforest. The names are justified by analogy to the more commonly studied trees and forests. (A tree is a connected graph with no cycles; a forest is a disjoint union of trees.) Gabow and Tarjan. attribute the study of pseudoforests to Dantzig's 1963 book on linear programming, in which pseudoforests arise in the solution of certain network flow problems.. Pseudoforests also form graph-theoretic models of functions and occur in several algorithmic problems. Pseudoforests a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''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 of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
