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Algorithms: Dynamic Programming - Matrix Chain Multiplication with C Program Source Code





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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 















Basic Data Structures and Algorithms



Sorting- at a glance

 Bubble Sort One of the most elementary sorting algorithms to implement - and also very inefficient. Runs in quadratic time. A good starting point to understand sorting in general, before moving on to more advanced techniques and algorithms. A general idea of how the algorithm works and a the code for a C program.

Insertion Sort - Another quadratic time sorting algorithm - an example of dynamic programming. An explanation and step through of how the algorithm works, as well as the source code for a C program which performs insertion sort.

Selection Sort - Another quadratic time sorting algorithm - an example of a greedy algorithm. An explanation and step through of how the algorithm works, as well as the source code for a C program which performs selection sort.

Shell Sort- An inefficient but interesting algorithm, the complexity of which is not exactly known.

Merge Sort An example of a Divide and Conquer algorithm. Works in O(n log n) time. The memory complexity for this is a bit of a disadvantage.

Quick Sort In the average case, this works in O(n log n) time. No additional memory overhead - so this is better than merge sort in this regard. A partition element is selected, the array is restructured such that all elements greater or less than the partition are on opposite sides of the partition. These two parts of the array are then sorted recursively.

Heap Sort- Efficient sorting algorithm which runs in O(n log n) time. Uses the Heap data structure.

Binary Search Algorithm- Commonly used algorithm used to find the position of an element in a sorted array. Runs in O(log n) time.

Basic Data Structures and Algorithms


 Stacks Last In First Out data structures ( LIFO ). Like a stack of cards from which you pick up the one on the top ( which is the last one to be placed on top of the stack ). Documentation of the various operations and the stages a stack passes through when elements are inserted or deleted. C program to help you get an idea of how a stack is implemented in code.

Queues First in First Out data structure (FIFO). Like people waiting to buy tickets in a queue - the first one to stand in the queue, gets the ticket first and gets to leave the queue first. Documentation of the various operations and the stages a queue passes through as elements are inserted or deleted. C Program source code to help you get an idea of how a queue is implemented in code.

Single Linked List A self referential data structure. A list of elements, with a head and a tail; each element points to another of its own kind.

Double Linked List- A self referential data structure. A list of elements, with a head and a tail; each element points to another of its own kind in front of it, as well as another of its own kind, which happens to be behind it in the sequence.

Circular Linked List Linked list with no head and tail - elements point to each other in a circular fashion.

 Binary Search Trees A basic form of tree data structures. Inserting and deleting elements in them. Different kind of binary tree traversal algorithms.

 Heaps A tree like data structure where every element is lesser (or greater) than the one above it. Heap formation, sorting using heaps in O(n log n) time.

 Height Balanced Trees - Ensuring that trees remain balanced to optimize complexity of operations which are performed on them.

Graphs

 Depth First Search - Traversing through a graph using Depth First Search in which unvisited neighbors of the current vertex are pushed into a stack and visited in that order.

Breadth First Search - Traversing through a graph using Breadth First Search in which unvisited neighbors of the current vertex are pushed into a queue and then visited in that order.

Minimum Spanning Trees: Kruskal Algorithm- Finding the Minimum Spanning Tree using the Kruskal Algorithm which is a greedy technique. Introducing the concept of Union Find.

Minumum Spanning Trees: Prim's Algorithm- Finding the Minimum Spanning Tree using the Prim's Algorithm.

Dijkstra Algorithm for Shortest Paths- Popular algorithm for finding shortest paths : Dijkstra Algorithm.

Floyd Warshall Algorithm for Shortest Paths- All the all shortest path algorithm: Floyd Warshall Algorithm

Bellman Ford Algorithm - Another common shortest path algorithm : Bellman Ford Algorithm.

Dynamic Programming A technique used to solve optimization problems, based on identifying and solving sub-parts of a problem first.

Integer Knapsack problemAn 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.

 Elementary cases : Fractional Knapsack Problem, Task Scheduling - Elementary problems in Greedy algorithms - Fractional Knapsack, Task Scheduling. Along with C Program source code.

Data Compression using Huffman TreesCompression using Huffman Trees. A greedy technique for encoding information.


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