Printing brackets in Matrix Chain Multiplication Problem

Prerequisite : Dynamic Programming | Set 8 (Matrix Chain Multiplication)

Given a sequence of matrices, find the most efficient way to multiply these matrices together. The problem is not actually to perform the multiplications, but merely to decide in which order to perform the multiplications.

We have many options to multiply a chain of matrices because matrix multiplication is associative. In other words, no matter how we parenthesize the product, the result will be the same. For example, if we had four matrices A, B, C, and D, we would have: 

(ABC)D = (AB)(CD) = A(BCD) = ....

However, the order in which we parenthesize the product affects the number of simple arithmetic operations needed to compute the product, or the efficiency. For example, suppose A is a 10 × 30 matrix, B is a 30 × 5 matrix, and C is a 5 × 60 matrix. Then,  

(AB)C = (10×30×5) + (10×5×60) = 1500 + 3000 = 4500 operations
A(BC) = (30×5×60) + (10×30×60) = 9000 + 18000 = 27000 operations.

Clearly the first parenthesization requires less number of operations.

Given an array p[] which represents the chain of matrices such that the ith matrix Ai is of dimension p[i-1] x p[i]. We need to write a function MatrixChainOrder() that should return the minimum number of multiplications needed to multiply the chain. 

Input:  p[] = {40, 20, 30, 10, 30}  
Output: Optimal parenthesization is  ((A(BC))D)
        Optimal cost of parenthesization is 26000
There are 4 matrices of dimensions 40x20, 20x30, 
30x10 and 10x30. Let the input 4 matrices be A, B, 
C and D.  The minimum number of  multiplications are 
obtained by putting parenthesis in following way
(A(BC))D --> 20*30*10 + 40*20*10 + 40*10*30

Input: p[] = {10, 20, 30, 40, 30} 
Output: Optimal parenthesization is (((AB)C)D)
        Optimal cost of parenthesization is 30000
There are 4 matrices of dimensions 10x20, 20x30, 
30x40 and 40x30. Let the input 4 matrices be A, B, 
C and D.  The minimum number of multiplications are 
obtained by putting parenthesis in following way
((AB)C)D --> 10*20*30 + 10*30*40 + 10*40*30

Input: p[] = {10, 20, 30}  
Output: Optimal parenthesization is (AB)
        Optimal cost of parenthesization is 6000
There are only two matrices of dimensions 10x20 
and 20x30. So there is only one way to multiply 
the matrices, cost of which is 10*20*30

This problem is mainly an extension of previous post. In the previous post, we have discussed algorithm for finding optimal cost only. Here we need print parenthesization also.

The idea is to store optimal break point for every subexpression (i, j) in a 2D array bracket[n][n]. Once we have bracket array us constructed, we can print parenthesization using below code. 

// Prints parenthesization in subexpression (i, j)
printParenthesis(i, j, bracket[n][n], name)
{
    // If only one matrix left in current segment
    if (i == j)
    {
        print name;
        name++;
        return;
    }

    print "(";

    // Recursively put brackets around subexpression
    // from i to bracket[i][j].
    printParenthesis(i, bracket[i][j], bracket, name);

    // Recursively put brackets around subexpression
    // from bracket[i][j] + 1 to j.
    printParenthesis(bracket[i][j]+1, j, bracket, name);

    print ")";
}

Below is the implementation of the above steps.

Optimal Parenthesization is : ((A(BC))D)
Optimal Cost is : 26000

Time Complexity: O(n³) 
Auxiliary Space: O(n²)

Another Approach:

This solution try to solve the problem using Recursion using permutations.

Let's take example:  {40, 20, 30, 10, 30}
n = 5

Let’s divide that into a Matrix

[ [40, 20], [20, 30], [30, 10], [10, 30] ]

[ A , B , C , D ]

it contains 4 matrices i.e. (n - 1)

We have 3 combinations to multiply  i.e.  (n-2)

AB    or    BC    or     CD

Algorithm:

1) Given array of matrices with length M, Loop through  M – 1 times

2) Merge consecutive matrices in each loop

for (int i = 0; i < M - 1; i++) {
   int cost =  (matrices[i][0] * 
                 matrices[i][1] * matrices[i+1][1]);
   
   // STEP - 3
   // STEP - 4
}

3) Merge the current two matrices into one, and remove merged matrices list from list.

If  A, B merged, then A, B must be removed from the List

and NEW matrix list will be like
newMatrices = [  AB,  C ,  D ]

We have now 3 matrices, in any loop
Loop#1:  [ AB,  C,   D ]
Loop#2:  [ A,   BC,  D ]
Loop#3   [ A,   B,   CD ]

4) Repeat: Go to STEP – 1  with  newMatrices as input M — recursion

5) Stop recursion, when we get 2 matrices in the list.

Workflow

Matrices are reduced in following way, 

and cost’s must be retained and summed-up during recursion with previous values of each parent step.

[ A, B , C, D ]

[(AB), C, D ]
 [ ((AB)C), D ]--> [ (((AB)C)D) ] 
 - return & sum-up total cost of this step.
 [ (AB),  (CD)] --> [ ((AB)(CD)) ] 
 - return .. ditto..

 [ A, (BC), D ]
 [ (A(BC)), D ]--> [ ((A(BC))D) ] 
  - return
 [ A, ((BC)D) ]--> [ (A((BC)D)) ] 
  - return
    
 [ A, B, (CD) ]
 [ A, (B(CD)) ]--> [ (A(B(CD))) ] 
  - return
 [ (AB), (CD) ]--> [ ((AB)(CD)) ] 
  - return .. ditto..

on return i.e. at final step of each recursion, check if  this value smaller than of any other.

Below is JAVA,c# and Javascript implementation of above steps.

matrices = 
[[40 20] [20 30] [30 10] [10 30] ]
Final labels: 
((A(BC))D)
Final Cost:
26000

Time Complexity : O(n²)

Auxiliary Space:O(n²)

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