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### Matrix Chain Multiplication

Dynamic Programming solves problems by combining the solutions to subproblems just like the divide and conquer method. Dynamic programming method is used to solve the problem of multiplication of a chain of matrices so that the fewest total scalar multiplications are performed.

Given a chain (A1, A2, A3, A4….An) of n matrices, we wish to compute the product. A product of matrices is fully parenthesized if it is either a single matrix or the product of fully parenthesized matrix products, surrounded by parenthesis. Since, matrix multiplication is associative all parenthesizations yield the same product. But, the way we parenthesize a chain of matrices have an impact on the cost of evaluating the product. We divide a chain of matrices to be multiplied into two optimal sub-chains, and then the optimal parenthesizations of the sub-chains must be composed of optimal chains.

### Algorithm - this has been covered in the tutorial document:

A block of memory cache is used to store the result of the function for specific values. If cache[i][j] = -1 then we do not know the result. If it is some number then that denotes the return value of multiply (i,j). We store it to avoid computing the same again.

Input Format: First integer must be the number of matrices. It has to be followed by rows of first matrix, columns of first matrix, columns for second matrix, columns for third matrix and so on.

Complete Tutorial with Examples :

## Matrix Chain Multiplication - C Program Source Code

`#include<string.h>#include<stdio.h>#include<limits.h>int min(int a,int b){        return a < b ? a:b;}/*d[i] is used to store the  dimension of the matrix ith matrix has dimension d[i-1] * d[i].  So for no loss of generality we will put d[0] to be number of rows of the first matrix and we   start the index from 1 */int d[100100]; /* Cache is used to store the result of the function for specific values. If cache[i][j] = -1 then we   do not know the result. If it is some number then that denotes the return value of multiply(i,j).   We store it to avoid computing the same again*/int cache[1024][1024];int multiply(int from,int to){        if(from==to)return 0;        if(cache[from][to]!=-1)        {                return cache[from][to];        }        int iter,result = INT_MAX;        /*We put the paranthesis at every possible step and we take the one for which computation           is minimum */        for(iter=from;iter<to;iter++)        {                /* Update the result every time */                result= min(result,multiply(from,iter) + multiply(iter+1,to) + d[from-1]*d[iter]*d[to]);        }        return result;}/* Input Format: First integer must be the number of matrices. It has to be followed by    rows of first matrix, columns of first matrix, columns for second matrix, columns for third matrix,... */int main(){        /*Initialising cache to -1 */        memset(cache, -1,sizeof(cache));        int number_of_matrices;         scanf("%d",&number_of_matrices);        scanf("%d",&d[0]);        int iter;        for(iter=1;iter<=number_of_matrices;iter++)        {                scanf("%d",&d[iter]);        }        printf("%d\n",multiply(1,number_of_matrices));}`
Related Tutorials (common examples of Dynamic Programming):

 Integer Knapsack problem An elementary problem, often used to introduce the concept of dynamic programming. Matrix Chain Multiplication Given a long chain of matrices of various sizes, how do you parenthesize them for the purpose of multiplication - how do you chose which ones to start multiplying first? Longest Common Subsequence Given two strings, find the longest common sub sequence between them.

Some Important Data Structures and Algorithms, at a glance:

 Arrays : Popular Sorting and Searching Algorithms Bubble Sort Insertion Sort Selection Sort Shell Sort Merge Sort Quick Sort Heap Sort Binary Search Algorithm Basic Data Structures  and Operations on them Stacks Queues Single Linked List Double Linked List Circular Linked List
 Tree Data Structures Binary Search Trees Heaps Height Balanced Trees Graphs and Graph Algorithms Depth First Search Breadth First Search Minimum Spanning Trees: Kruskal Algorithm Minumum Spanning Trees: Prim's Algorithm Dijkstra Algorithm for Shortest Paths Floyd Warshall Algorithm for Shortest Paths Bellman Ford Algorithm Popular Algorithms in Dynamic Programming Dynamic Programming Integer Knapsack problem Matrix Chain Multiplication Longest Common Subsequence Greedy Algorithms Elementary cases : Fractional Knapsack Problem, Task Scheduling Data Compression using Huffman Trees

Basic Data Structures and Algorithms

Sorting- at a glance