# How do you find the optimal Parenthesization in a matrix chain multiplication?

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## How do you find the optimal Parenthesization in a matrix chain multiplication?

Step1: Structure of an optimal parenthesization: Our first step in the dynamic paradigm is to find the optimal substructure and then use it to construct an optimal solution to the problem from an optimal solution to subproblems. Ai Ai+1…. Aj. If i < j then any parenthesization of the product Ai Ai+1 ……

## What is time complexity of matrix chain multiplication?

An be the given sequence of n matrices, generally matrix chain multiplication algorithm is used to obtain its-product with minimum cost(lowest cost). However the matrix chain multiplication is a dynamic programming paradigm and takes O(n3) computational complexity.

**What will be the time complexity for matrix chain multiplication by brute force method?**

As before, if we have n matrices to multiply, it will take O(n) time to generate each of the O(n2) costs and entries in the best matrix for an overall complexity of O(n3) time at a cost of O(n2) space.

### How many ways can 4 matrix can be Parenthesized for chain multiplication?

There are many options because matrix multiplication is associative. In other words, no matter how the product is parenthesized, the result obtained will remain the same. For example, for four matrices A, B, C, and D, there are five possible options: ((AB)C)D = (A(BC))D = (AB)(CD) = A((BC)D) = A(B(CD)).

### What is matrix chain multiplication explain it with example?

Example: We are given the sequence {4, 10, 3, 12, 20, and 7}. The matrices have size 4 x 10, 10 x 3, 3 x 12, 12 x 20, 20 x 7. We need to compute M [i,j], 0 ≤ i, j≤ 5. We initialize the diagonal element with equal i,j value with ‘0’. …

**What is currently the best algorithm for matrix multiplication?**

In linear algebra, the Strassen algorithm, named after Volker Strassen, is an algorithm for matrix multiplication. It is faster than the standard matrix multiplication algorithm for large matrices, with a better asymptotic complexity, although the naive algorithm is often better for smaller matrices.

## What is the running time of naïve simple basic matrix multiplication algorithm?

Naive Approach (Iterative) Thus the running time of this square matrix multiplication algorithm is O(n^{3}) . This algorithm looks so natural and trivial that it is very hard to imagine that there is a better way to multiply matrices.