In the following implementation of the divide-and-conquer algorithm I have used the partitioning of the matrices using index calculations. How to partition the matrices into A, B and C? It can be done by creating 12 new n/2 x n/2 matrices by copying entries or by partitioning the matrices by index calculations. The equation 4.9 mentioned in the algorithm is as follows: B, with the assumption that n is an exact power of 2 in each of the n x n matrices. The divide-and-conquer algorithm to compute the matrix product C = A. + SQUARE-MATRIX-MULTIPLY-RECURSIVE(A22, B22) + SQUARE-MATRIX-MULTIPLY-RECURSIVE(A22, B21)Ĭ22 = SQUARE-MATRIX-MULTIPLY-RECURSIVE(A21, B12) + SQUARE-MATRIX-MULTIPLY-RECURSIVE(A12, B22)Ĭ21 = SQUARE-MATRIX-MULTIPLY-RECURSIVE(A21, B11) + SQUARE-MATRIX-MULTIPLY-RECURSIVE(A12, B21)Ĭ12 = SQUARE-MATRIX-MULTIPLY-RECURSIVE(A11, B12) There are various ways available to solve any computer problem, but the mentioned are a good example of divide and conquer approach.The following algorithm for square matrix multiplication is from the Introduction to Algorithms, Third edition: SQUARE-MATRIX-MULTIPLY-RECURSIVE (A, B)Įlse partition A, B, and C as in equations (4.9)Ĭ11 = SQUARE-MATRIX-MULTIPLY-RECURSIVE(A11, B11) The following computer algorithms are based on divide-and-conquer programming approach − They must be handled using the pointers available in the nodes of the list. The repository contains a report, code, and a jupyter file. Various searching algorithms can also be performed on the linked list data structures with a slightly different technique as linked lists are not indexed linear data structures. This is an implementation of matrix multiplication algorithm with python. These nodes are then combined (or merged) in recursively until the final solution is achieved. Then, the nodes in the list are sorted (conquered). Like arrays, linked lists are also linear data structures that store data sequentially.Ĭonsider the merge sort algorithm on linked list following the very popular tortoise and hare algorithm, the list is divided until it cannot be divided further. Linked Lists as InputĪnother data structure that can be used to take input for divide and conquer algorithms is a linked list (for example, merge sort using linked lists). Since arrays are indexed and linear data structures, sorting algorithms most popularly use array data structures to receive input. Then, the subproblems are sorted (the conquer step) and are merged to form the solution of the original array back (the combine step). In the input for a sorting algorithm below, the array input is divided into subproblems until they cannot be divided further. In algorithms that require input to be in the form of a list, like various sorting algorithms, array data structures are most commonly used. There are various ways in which various algorithms can take input such that they can be solved using the divide and conquer technique. Their usage is explained as Arrays as Input Two major data structures used are − arrays and linked lists. ![]() ![]() There are several ways to give input to the divide and conquer algorithm design pattern. The common procedure for the divide and conquer design technique is as follows −ĭivide − We divide the original problem into multiple sub-problems until they cannot be divided further.Ĭonquer − Then these subproblems are solved separately with the help of recursionĬombine − Once solved, all the subproblems are merged/combined together to form the final solution of the original problem. These sub-problems are solved first and the solutions are merged together to form the final solution. The same technique is applied on algorithms.ĭivide and conquer approach breaks down a problem into multiple sub-problems recursively until it cannot be divided further. To do that, the first step is to section the hair in smaller strands to make the combing easier than combing the hair altogether. I get that why the original big O for this problem is 192 for the total amount of. The passed matrices must be nxn matrices (square matrices), where n is a power of 2. On my homework, we have a problem regarding divide a conquer for matrix multiplication where if you are multiplying a (4x12) by (12x4) the original total of multiplications is 192, but can be further dropped to 147. A divide and conquer algorithm which runs in O (N3). Consider an instance where we need to brush a type C curly hair and remove all the knots from it. Strassen's Algorithm for Non-Square Matrices. To understand the divide and conquer design strategy of algorithms, let us use a simple real world example.
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