Īnother notable example is the algorithm invented by Anatolii A. Another ancient decrease and conquer algorithm is the Euclidean algorithm to compute the greatest common divisor of two numbers by reducing the numbers to smaller and smaller equivalent subproblems, which dates to several centuries BC.Īn early example of a divide-and-conquer algorithm with multiple subproblems is Gauss's 1805 description of what is now called the Cooley–Tukey fast Fourier transform (FFT) algorithm, although he did not analyze its operation count quantitatively and FFTs did not become widespread until they were rediscovered over a century later.Īn early two-subproblem D&C algorithm that was specifically developed for computers and properly analyzed is the merge sort algorithm, invented by John von Neumann in 1945. While a clear description of the algorithm on computers appeared in 1946 in an article by John Mauchly, the idea of using a sorted list of items to facilitate searching dates back at least as far as Babylonia in 200 BC. Īn important application of divide and conquer is in optimization, where if the search space is reduced ("pruned") by a constant factor at each step, the overall algorithm has the same asymptotic complexity as the pruning step, with the constant depending on the pruning factor (by summing the geometric series) this is known as prune and search.Įarly examples of these algorithms are primarily decrease and conquer – the original problem is successively broken down into single subproblems, and indeed can be solved iteratively.īinary search, a decrease and conquer algorithm where the subproblems are of roughly half the original size, has a long history. The name decrease and conquer has been proposed instead for the single-subproblem class. Therefore, some authors consider that the name "divide and conquer" should be used only when each problem may generate two or more subproblems. Under this broad definition, however, every algorithm that uses recursion or loops could be regarded as a "divide and conquer algorithm". These algorithms can be implemented more efficiently than general divide-and-conquer algorithms in particular, if they use tail recursion, they can be converted into simple loops. The name "divide and conquer" is sometimes applied to algorithms that reduce each problem to only one sub-problem, such as the binary search algorithm for finding a record in a sorted list (or its analog in numerical computing, the bisection algorithm for root finding). This approach is known as the merge sort algorithm. ![]() Problems of sufficient simplicity are solved directly.įor example, to sort a given list of n natural numbers, split it into two lists of about n/2 numbers each, sort each of them in turn, and interleave both results appropriately to obtain the sorted version of the given list (cf. Its basic idea is to decompose a given problem into two or more similar, but simpler, subproblems, to solve them in turn, and to compose their solutions to solve the given problem. The divide and conquer paradigm is often used to find the optimal solution of a problem. Upper half: splitting into sublists mid: a one-element list is trivially sorted lower half: composing sorted sublists. The correctness of a divide and conquer algorithm is usually proved by mathematical induction, and its computational cost is often determined by solving recurrence relations.ĭivide-and-conquer approach to sort the list (38,27,43,3,9,82,10) in increasing order. These divide and conquer complications are seen when optimizing the calculation of a Fibonacci number with efficient double recursion. As when proving a theorem by induction, it is often necessary to replace the original problem with a more general or complicated problem in order to initialize the recursion, and there is no systematic method for finding the proper generalization. Understanding and designing divide and conquer algorithms is a complex skill that requires a good understanding of the nature of the underlying problem to be solved. the Karatsuba algorithm), finding the closest pair of points, syntactic analysis (e.g., top-down parsers), and computing the discrete Fourier transform ( FFTs). This divide and conquer technique is the basis of efficient algorithms for all kinds of problems, such as sorting (e.g., quicksort, merge sort), multiplying large numbers (e.g. The solutions to the sub-problems are then combined to give a solution to the original problem. A divide and conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved directly. ![]() In computer science, divide and conquer is an algorithm design paradigm based on multi-branched recursion.
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