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Kadane's Algorithm
In computer science, the maximum sum subarray problem, also known as the maximum segment sum problem, is the task of finding a contiguous subarray with the largest sum, within a given one-dimensional array A ...nof numbers. Formally, the task is to find indices i and j with 1 \leq i \leq j \leq n , such that the sum : \sum_^j A is as large as possible. (Some formulations of the problem also allow the empty subarray to be considered; by convention, the sum of all values of the empty subarray is zero.) Each number in the input array A could be positive, negative, or zero. For example, for the array of values minus;2, 1, −3, 4, −1, 2, 1, −5, 4 the contiguous subarray with the largest sum is , −1, 2, 1 with sum 6. Some properties of this problem are: # If the array contains all non-negative numbers, then the problem is trivial; a maximum subarray is the entire array. # If the array contains all non-positive numbers, then a solution is any subarray of si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Subarray Visualization
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the ''local'' or ''relative'' extrema), or on the entire domain (the ''global'' or ''absolute'' extrema). Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions. As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. Definition A real-valued function ''f'' defined on a domain ''X'' has a global (or absolute) maximum point at ''x''∗, if for all ''x'' in ''X''. Similarly, the function has a global (or absolute) minimum point at ''x''∗, if for all ''x'' in ''X''. The value of the function a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shortest Path Problem
In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The problem of finding the shortest path between two intersections on a road map may be modeled as a special case of the shortest path problem in graphs, where the vertices correspond to intersections and the edges correspond to road segments, each weighted by the length of the segment. Definition The shortest path problem can be defined for graphs whether undirected, directed, or mixed. It is defined here for undirected graphs; for directed graphs the definition of path requires that consecutive vertices be connected by an appropriate directed edge. Two vertices are adjacent when they are both incident to a common edge. A path in an undirected graph is a sequence of vertices P = ( v_1, v_2, \ldots, v_n ) \in V \times V \times \cdots \times V such that v_i is adjacent to v_ for 1 \l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimization Algorithms And Methods
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems of sorts arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries. In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a define ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Computer Journal
''The Computer Journal'' is a peer-reviewed scientific journal covering computer science and information systems. Established in 1958, it is one of the oldest computer science research journals. It is published by Oxford University Press on behalf of BCS, The Chartered Institute for IT. The authors of the best paper in each annual volume receive the Wilkes Award from BCS, The Chartered Institute for IT. Editors-in-chief The following people have been editor-in-chief: * 1958–1969 Eric N. Mutch * 1969–1992 Peter Hammersley * 1993–2000 C. J. van Rijsbergen * 2000–2008 Fionn Murtagh * 2008–2012 Erol Gelenbe * 2012–2016 Fionn Murtagh * 2016–2020 Steve Furber Stephen Byram Furber (born 21 March 1953) is a British computer scientist, mathematician and hardware engineer, currently the ICL Professor of Computer Engineering in the Department of Computer Science at the University of Manchester, UK. A ... * 2021–present Tom Crick References External links Offic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Communications Of The ACM
''Communications of the ACM'' is the monthly journal of the Association for Computing Machinery (ACM). It was established in 1958, with Saul Rosen as its first managing editor. It is sent to all ACM members. Articles are intended for readers with backgrounds in all areas of computer science and information systems. The focus is on the practical implications of advances in information technology and associated management issues; ACM also publishes a variety of more theoretical journals. The magazine straddles the boundary of a science magazine, trade magazine, and a scientific journal. While the content is subject to peer review, the articles published are often summaries of research that may also be published elsewhere. Material published must be accessible and relevant to a broad readership. From 1960 onward, ''CACM'' also published algorithms, expressed in ALGOL. The collection of algorithms later became known as the Collected Algorithms of the ACM. See also * ''Journal of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subset Sum Problem
The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset S of integers and a target-sum T, and the question is to decide whether any subset of the integers sum to precisely T''.'' The problem is known to be NP. Moreover, some restricted variants of it are NP-complete too, for example: * The variant in which all inputs are positive. * The variant in which inputs may be positive or negative, and T=0. For example, given the set \, the answer is ''yes'' because the subset \ sums to zero. * The variant in which all inputs are positive, and the target sum is exactly half the sum of all inputs, i.e., T = \frac(a_1+\dots+a_n) . This special case of SSP is known as the partition problem. SSP can also be regarded as an optimization problem: find a subset whose sum is at most ''T'', and subject to that, as close as possible to ''T''. It is NP-hard, but there are several algorithms that can solve it reasonably quickly in pra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamic Programming
Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. In both contexts it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a recursive manner. While some decision problems cannot be taken apart this way, decisions that span several points in time do often break apart recursively. Likewise, in computer science, if a problem can be solved optimally by breaking it into sub-problems and then recursively finding the optimal solutions to the sub-problems, then it is said to have '' optimal substructure''. If sub-problems can be nested recursively inside larger problems, so that dynamic programming methods are applicable, then there is a relation between the value of the larger problem and the values of the sub-problems.Cormen, T. H.; Leiserson, C. E.; R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Python (programming Language)
Python is a high-level, general-purpose programming language. Its design philosophy emphasizes code readability with the use of significant indentation. Python is dynamically-typed and garbage-collected. It supports multiple programming paradigms, including structured (particularly procedural), object-oriented and functional programming. It is often described as a "batteries included" language due to its comprehensive standard library. Guido van Rossum began working on Python in the late 1980s as a successor to the ABC programming language and first released it in 1991 as Python 0.9.0. Python 2.0 was released in 2000 and introduced new features such as list comprehensions, cycle-detecting garbage collection, reference counting, and Unicode support. Python 3.0, released in 2008, was a major revision that is not completely backward-compatible with earlier versions. Python 2 was discontinued with version 2.7.18 in 2020. Python consistently r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop Invariant
In computer science, a loop invariant is a property of a program loop that is true before (and after) each iteration. It is a logical assertion, sometimes checked within the code by an assertion call. Knowing its invariant(s) is essential in understanding the effect of a loop. In formal program verification, particularly the Floyd-Hoare approach, loop invariants are expressed by formal predicate logic and used to prove properties of loops and by extension algorithms that employ loops (usually correctness properties). The loop invariants will be true on entry into a loop and following each iteration, so that on exit from the loop both the loop invariants and the loop termination condition can be guaranteed. From a programming methodology viewpoint, the loop invariant can be viewed as a more abstract specification of the loop, which characterizes the deeper purpose of the loop beyond the details of this implementation. A survey article covers fundamental algorithms from many a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Born Kadane
Joseph "Jay" Born Kadane (born January 10, 1941) is the Leonard J. Savage University Professor of Statistics, Emeritus in the Department of Statistics and Social and Decision Sciences at Carnegie Mellon University. Kadane is one of the early proponents of Bayesian statistics, particularly the subjective Bayesian philosophy. Education and career Kadane was born in Washington, DC and raised in Freeport on Long Island, Kadane, prepared at Phillips Exeter Academy, earned an A.B. in mathematics from Harvard College and a Ph.D. in statistics from Stanford in 1966, under the supervision of Professor Herman Chernoff. While in graduate school, Kadane worked for the Center for Naval Analyses (CNA). Upon finishing, he accepted a joint appointment at the Yale statistics department and the Cowles Foundation. In 1968, he left Yale and served as an analyst at CNA for three years. In 1971, he moved to Pittsburgh to join Morris H. DeGroot at Carnegie Mellon University. He became the second ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kadane Run −2,1,−3,4,−1,2,1,−5,4
Joseph "Jay" Born Kadane (born January 10, 1941) is the Leonard J. Savage University Professor of Statistics, Emeritus in the Department of Statistics and Social and Decision Sciences at Carnegie Mellon University. Kadane is one of the early proponents of Bayesian statistics, particularly the subjective Bayesian philosophy. Born in Washington, DC and raised in Freeport on Long Island, Kadane, prepared at Phillips Exeter Academy, earned an A.B. in mathematics from Harvard College and a Ph.D. in statistics from Stanford in 1966, under the supervision of Professor Herman Chernoff. While in graduate school, Kadane worked for the Center for Naval Analyses (CNA). Upon finishing, he accepted a joint appointment at the Yale statistics department and the Cowles Foundation. In 1968, he left Yale and served as an analyst at CNA for three years. In 1971, he moved to Pittsburgh to join Morris H. DeGroot at Carnegie Mellon University. He became the second tenured professor in the Department ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Vision
Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the human visual system can do. Computer vision tasks include methods for acquiring, processing, analyzing and understanding digital images, and extraction of high-dimensional data from the real world in order to produce numerical or symbolic information, e.g. in the forms of decisions. Understanding in this context means the transformation of visual images (the input of the retina) into descriptions of the world that make sense to thought processes and can elicit appropriate action. This image understanding can be seen as the disentangling of symbolic information from image data using models constructed with the aid of geometry, physics, statistics, and learning theory. The scientific discipline of computer vision is concerned with the theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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