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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 .
The master theorem always yields asymptotically tight bounds to recurrences from divide and conquer algorithms that partition an input into smaller subproblems of equal sizes, solve the subproblems recursively, and then combine the subproblem solutions to give a solution to the original problem. The time for such an algorithm can be expressed ...
In computer science, the Akra–Bazzi method, or Akra–Bazzi theorem, is used to analyze the asymptotic behavior of the mathematical recurrences that appear in the analysis of divide and conquer algorithms where the sub-problems have substantially different sizes.
Merge sort is a divide-and-conquer algorithm that was invented by John von Neumann in 1945. [3] A detailed description and analysis of bottom-up merge sort appeared in a report by Goldstine and von Neumann as early as 1948.
Quicksort is a type of divide-and-conquer algorithm for sorting an array, based on a partitioning routine; the details of this partitioning can vary somewhat, so that quicksort is really a family of closely related algorithms. Applied to a range of at least two elements, partitioning produces a division into two consecutive non empty sub-ranges ...
The Karatsuba algorithm is a fast multiplication algorithm. It was discovered by Anatoly Karatsuba in 1960 and published in 1962. [ 1 ] [ 2 ] [ 3 ] It is a divide-and-conquer algorithm that reduces the multiplication of two n -digit numbers to three multiplications of n /2-digit numbers and, by repeating this reduction, to at most n log 2 3 ...
This technique can be used to improve the efficiency of many eigenvalue algorithms, but it has special significance to divide-and-conquer. For the rest of this article, we will assume the input to the divide-and-conquer algorithm is an real symmetric tridiagonal matrix . The algorithm can be modified for Hermitian matrices.
Master theorem (analysis of algorithms), analyzing the asymptotic behavior of divide-and-conquer algorithms; Ramanujan's master theorem, providing an analytic expression for the Mellin transform of an analytic function; MacMahon master theorem (MMT), in enumerative combinatorics and linear algebra; Glasser's master theorem in integral calculus