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Matrix chain multiplication using Dynamic programming

Pictorial representation of Dynamic programming

Dynamic Programming is a common approach used by the programmers to reduce the time complexity to a certain extent(if possible) so that the time of computation can be improved.

Now since we have gone through the process of Dynamic Programming let us see the problem we are going to solve using Dynamic Programming.

Suppose we have a chain of matrices that are required to be computed.

Let us suppose the matrix chain be as follows

Hence, as shown, we have 4 matrices with each dimension mentioned.

Now there can be multiple possibilities of multiplication.

For example CASE 1: Suppose we first perform A34=A3 *A4, where the resultant matrix is A34. Now to this matrix, we multiply A2. Thus, A234=A2*A34. To the resultant matrix, we finally multiply A1.

CASE 2: Suppose in this case we multiply A1 and A2. Thus, A12=A1*A2. Now we multiply A3 and A4. Thus, we get A34=A3*A4. Now we multiply the two resultants to retrieve our resultant. Result= A12 * A34.

In total there are several combinations possible. Now if the chain is small then we can try each and every process naively and find the resultant. But this cannot be a permanent solution.

Suppose if the chain is of 3 matrices, there are only 2 combinations

Suppose if the chain is of 4 matrices (as shown above), there are 3 different combinations. Now if there are many more matrices in the chain then combinations also increase exponentially.

Relation between combinations and number of matrices in the chain.

Now that we are through with the understanding of the problem, let us try to resolve it.

We note that we can solve it by storing the resultant number of multiplications of two matrices in a data structure.

Actually, we need two different 2-D arrays. one to store the number of multiplication 2 matrices need to undergo in order to form a pair and the second one to store the position where brackets need to be put.

Thus, let us derive the formula

Suppose we are storing the resultant number of multiplications in an array A

Before we move to the algorithm formula let’s first see the way of finding the number of multiplication and number of additions 2 matrices need to undergo in order to give the result.

Suppose we have two matrices namely A and B of dimensions mxn and nxr respectively.

Then the number of multiplication to give AB=m.n.r

The number of additions in this =m.r

Well now going back to our above example. Or let us bring that fresh again