|
Akra–Bazzi Method
In computer science, the Akra–Bazzi method, or Akra–Bazzi theorem, is used to analyze the asymptotic behavior of the mathematical recurrence relation, recurrences that appear in the analysis of divide and conquer algorithm, divide and conquer algorithms where the sub-problems have substantially different sizes. It is a generalization of the master theorem (analysis of algorithms), master theorem for divide-and-conquer recurrences, which assumes that the sub-problems have equal size. It is named after mathematicians Mohamad Akra and Louay Bazzi. Formulation The Akra–Bazzi method applies to recurrence formulas of the form :T(x)=g(x) + \sum_^k a_i T(b_i x + h_i(x))\qquad \textx \geq x_0. The conditions for usage are: * sufficient base cases are provided * a_i and b_i are constants for all i * a_i > 0 for all i * 0 0, and can thus be computed using the Akra–Bazzi method to be \Theta(n \log n). See also * Master theorem (analysis of algorithms) * Asymptotic complexity ... [...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] |
Recurrence Relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divide And Conquer Algorithm
In computer science, divide and conquer is an algorithm design paradigm. A divide-and-conquer algorithm recursively breaks down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem. The divide-and-conquer technique is the basis of efficient algorithms for many problems, such as sorting (e.g., quicksort, merge sort), multiplying large numbers (e.g., the Karatsuba algorithm), finding the closest pair of points, syntactic analysis (e.g., top-down parsers), and computing the discrete Fourier transform (FFT). Designing efficient divide-and-conquer algorithms can be difficult. As in mathematical induction, it is often necessary to generalize the problem to make it amenable to a recursive solution. The correctness of a divide-and-conquer algorithm is usually proved by mathematical induction, and its computational cost is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code execution through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a Heuristic (computer science), heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result. As an effective method, an algorithm ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Master Theorem (analysis Of Algorithms)
In the analysis of algorithms, the master theorem for divide-and-conquer recurrences provides an asymptotic analysis (using Big O notation) for recurrence relations of types that occur in the analysis of many divide and conquer algorithms. The approach was first presented by Jon Bentley, Dorothea Blostein (née Haken), and James B. Saxe in 1980, where it was described as a "unifying method" for solving such recurrences. The name "master theorem" was popularized by the widely used algorithms textbook ''Introduction to Algorithms'' by Cormen, Leiserson, Rivest, and Stein. Not all recurrence relations can be solved with the use of this theorem; its generalizations include the Akra–Bazzi method. Introduction Consider a problem that can be solved using a recursive algorithm such as the following: procedure p(input ''x'' of size ''n''): if ''n'' 1). Crucially, a and b must not depend on n. The theorem below also assumes that, as a base case for the recurrence, T(n)=\Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mohamad Akra
Muhammad ( ar, مُحَمَّد; 570 – 8 June 632 CE) was an Arab religious, social, and political leader and the founder of Islam. According to Islamic doctrine, he was a prophet divinely inspired to preach and confirm the monotheistic teachings of Adam, Abraham, Moses, Jesus, and other prophets. He is believed to be the Seal of the Prophets within Islam. Muhammad united Arabia into a single Muslim polity, with the Quran as well as his teachings and practices forming the basis of Islamic religious belief. Muhammad was born approximately 570CE in Mecca. He was the son of Abdullah ibn Abd al-Muttalib and Amina bint Wahb. His father Abdullah was the son of Quraysh tribal leader Abd al-Muttalib ibn Hashim, and he died a few months before Muhammad's birth. His mother Amina died when he was six, leaving Muhammad an orphan. He was raised under the care of his grandfather, Abd al-Muttalib, and paternal uncle, Abu Talib. In later years, he would periodically seclude himse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Louay Bazzi
Luay, Luayy or Louay (Arabic: لُؤَيّ ''lū’ayy'', ), also spelled Loai, Loay or Luai, is an Arabic male given name from the noun ''la’y'' (لَأْي), meaning "Strong, Patient, a Sheild and/or Protector". Notable people with the name include: Given name * Luay Hamza Abbas, Iraqi writer * Louay Almokdad, Syrian-British businessman and politician * Luai al-Atassi, Syrian military leader * Louay Chanko, Syrian footballer * Louay Kayyali, Syrian modern artist * Luay Nakhleh, Palestinian-Israeli-American professor of computer science * Louay M. Safi, Syrian-American scholar of Islam * Luay Salah, Iraqi footballer * Loai al-Saqa, Syrian al-Qaeda member * Loay Taleb, Syrian footballer Family name * Ka'b ibn Lu'ayy, ancestor of the Islamic prophet Muhammad Muhammad ( ar, مُحَمَّد; 570 – 8 June 632 Common Era, CE) was an Arab religious, social, and political leader and the founder of Islam. According to Muhammad in Islam, Islamic doctrine, he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for ''Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rates: d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floor Function
In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or equal to , denoted or . For example, , , , and . Historically, the floor of has been–and still is–called the integral part or integer part of , often denoted (as well as a variety of other notations). Some authors may define the integral part as if is nonnegative, and otherwise: for example, and . The operation of truncation generalizes this to a specified number of digits: truncation to zero significant digits is the same as the integer part. For an integer, . Notation The ''integral part'' or ''integer part'' of a number ( in the original) was first defined in 1798 by Adrien-Marie Legendre in his proof of the Legendre's formula. Carl Friedrich Gauss introduced the square bracket notation in hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ceiling Function
In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or equal to , denoted or . For example, , , , and . Historically, the floor of has been–and still is–called the integral part or integer part of , often denoted (as well as a variety of other notations). Some authors may define the integral part as if is nonnegative, and otherwise: for example, and . The operation of truncation generalizes this to a specified number of digits: truncation to zero significant digits is the same as the integer part. For an integer, . Notation The ''integral part'' or ''integer part'' of a number ( in the original) was first defined in 1798 by Adrien-Marie Legendre in his proof of the Legendre's formula. Carl Friedrich Gauss introduced the square bracket notation in his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Merge Sort
In computer science, merge sort (also commonly spelled as mergesort) is an efficient, general-purpose, and comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the order of equal elements is the same in the input and output. Merge sort is a divide-and-conquer algorithm that was invented by John von Neumann in 1945. A detailed description and analysis of bottom-up merge sort appeared in a report by Goldstine and von Neumann as early as 1948. Algorithm Conceptually, a merge sort works as follows: #Divide the unsorted list into ''n'' sublists, each containing one element (a list of one element is considered sorted). #Repeatedly merge sublists to produce new sorted sublists until there is only one sublist remaining. This will be the sorted list. Top-down implementation Example C-like code using indices for top-down merge sort algorithm that recursively splits the list (called ''runs'' in this example) into sublists until sublist size i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Master Theorem (analysis Of Algorithms)
In the analysis of algorithms, the master theorem for divide-and-conquer recurrences provides an asymptotic analysis (using Big O notation) for recurrence relations of types that occur in the analysis of many divide and conquer algorithms. The approach was first presented by Jon Bentley, Dorothea Blostein (née Haken), and James B. Saxe in 1980, where it was described as a "unifying method" for solving such recurrences. The name "master theorem" was popularized by the widely used algorithms textbook ''Introduction to Algorithms'' by Cormen, Leiserson, Rivest, and Stein. Not all recurrence relations can be solved with the use of this theorem; its generalizations include the Akra–Bazzi method. Introduction Consider a problem that can be solved using a recursive algorithm such as the following: procedure p(input ''x'' of size ''n''): if ''n'' 1). Crucially, a and b must not depend on n. The theorem below also assumes that, as a base case for the recurrence, T(n)=\Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
