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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 basic principle of Karatsuba's algorithm is divide-and-conquer, using a formula that allows one to compute the product of two large numbers and using three multiplications of smaller numbers, each with about half as many digits as or , plus some additions and digit shifts.
In computer science, a dichotomic search is a search algorithm that operates by selecting between two distinct alternatives (dichotomies [1] or polychotomies [2] when they are more than two) at each step. It is a specific type of divide and conquer algorithm. A well-known example is binary search. [3]
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 ...
Quicksort is a divide-and-conquer algorithm. It works by selecting a 'pivot' element from the array and partitioning the other elements into two sub-arrays, according to whether they are less than or greater than the pivot. For this reason, it is sometimes called partition-exchange sort. [4] The sub-arrays are then sorted recursively.
Merge sort parallelizes well due to the use of the divide-and-conquer method. Several different parallel variants of the algorithm have been developed over the years. Some parallel merge sort algorithms are strongly related to the sequential top-down merge algorithm while others have a different general structure and use the K-way merge method
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.
[Increase i] If p=1, the algorithm terminates unsuccessfully. Otherwise set (i, p, q) ← (i + q, p − q, 2q − p) (which moves p and q two positions back in the Fibonacci sequence); and return to Step 2. The two variants of the algorithm presented above always divide the current interval into a larger and a smaller subinterval.