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.
ThisThe divide-and-conquer technique is the basis of efficient algorithms for all kinds ofmany problems, such as [[sorting algorithm|sorting]] (e.g., [[quicksort]], [[merge sort]]), [[multiplication algorithm|multiplying large numbers]] (e.g., the [[Karatsuba algorithm]]), finding the [[Closest pair of points problem|closest pair of points]], [[syntactic analysis]] (e.g., [[top-down parser]]s), and computing the [[discrete Fourier transform]] ([[fast Fourier transform|FFT]]).<ref>{{cite book |last1=Blahut |first1=Richard |title=Fast Algorithms for Signal Processing |publisher=Cambridge University Press |isbn=978-0-511-77637-3 |pages=139–143}}</ref>
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 recursive solution. The correctness of a divide-and-conquer algorithm is usually proved by [[mathematical induction]], and its computational cost is often determined by solving [[recurrence relation]]s.▼
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. As when proving a [[theorem]] by [[Mathematical induction|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.{{Clarify|date=October 2017}} These divide-and-conquer complications are seen when optimizing the calculation of a [[Fibonacci number#Matrix form|Fibonacci number with efficient double recursion]].{{Why|date=October 2017}}{{Citation needed|date=October 2017}}
▲The correctness of a divide-and-conquer algorithm is usually proved by [[mathematical induction]], and its computational cost is often determined by solving [[recurrence relation]]s.
