Algorithm of Matrix Chain Multiplication

MATRIX-CHAIN-ORDER (p)
 1. n   length[p]-1
 2. for i ← 1 to n
 3. do m [i, i] ← 0
 4. for l ← 2 to n    // l is the chain length
 5. do for i ← 1 to n-l + 1
 6. do j ← i+ l -1
 7. m[i,j] ←  ∞
 8. for k  ← i to j-1
 9. do q  ← m [i, k] + m [k + 1, j] + p_i-1 p_k p_j	
 10. If q < m [i,j]
 11. then m [i,j] ← q
 12. s [i,j] ← k
 13. return m and s.

We will use table s to construct an optimal solution.

Step 1: Constructing an Optimal Solution:

PRINT-OPTIMAL-PARENS (s, i, j)
 1. if i=j
 2. then print "A"
 3. else print "("
 4. PRINT-OPTIMAL-PARENS (s, i, s [i, j])
 5. PRINT-OPTIMAL-PARENS (s, s [i, j] + 1, j)
 6. print ")"

Analysis: There are three nested loops. Each loop executes a maximum n times.

l, length, O (n) iterations.
i, start, O (n) iterations.
k, split point, O (n) iterations

Body of loop constant complexity

Total Complexity is: O (n³)

Algorithm with Explained Example

Question: P [7, 1, 5, 4, 2}

Solution: Here, P is the array of a dimension of matrices.

So here we will have 4 matrices:

A₁_7x1  A₂_1x5  A₃_5x4  A₄_4x2
i.e.
First Matrix A₁ have dimension 7 x 1
Second Matrix A₂ have dimension 1 x 5
Third Matrix A₃ have dimension 5 x 4
Fourth Matrix A₄ have dimension 4 x 2

Let say,
From P = {7,	1,	5,	4,	2} - (Given)
And P is the Position
p₀ = 7,	p₁ =1,	p₂ = 5,	p₃ = 4,	p₄=2.
Length of array P = number of elements in P
∴length (p)= 5
From step 3
Follow the steps in Algorithm in Sequence
According to Step 1 of Algorithm Matrix-Chain-Order

Step 1:

 n ← length [p]-1
	   Where n is the total number of elements
     And length [p] = 5
∴ n = 5 - 1 = 4
n = 4 
Now we construct two tables m and s.
Table m has dimension [1.....n, 1.......n]
Table s has dimension [1.....n-1, 2.......n]

Now, according to step 2 of Algorithm

for i ← 1 to n
this means: for i ← 1 to 4 (because n =4)
for  i=1
m [i, i]=0
m [1, 1]=0
Similarly for i = 2, 3, 4
m [2, 2] = m [3,3] = m [4,4] = 0
i.e. fill all the diagonal entries "0" in the table m
Now,
l ← 2 to n
l ← 2 to 4    (because n =4 )

Case 1:

1. When l - 2

    for (i ← 1 to n - l + 1)
	    i ← 1 to 4 - 2 + 1
	    i ← 1 to 3

When i = 1
  do   j ← i + l - 1
	         j ← 1 + 2 - 1
	         j ← 2
     i.e. j = 2
Now, m [i, j] ← ∞	 
	i.e. m [1,2] ← ∞ 
Put ∞ in m [1, 2] table
	  for k ← i to j-1
	       k ← 1 to 2 - 1
	       k ← 1 to 1
            k = 1
Now q ← m [i, k] + m [k + 1, j] + p_i-1 p_k p_j
			for l = 2
			     i = 1
			     j =2
			    k = 1
	    q ← m [1,1] + m [2,2] + p₀x p₁x p₂
             and m [1,1] = 0
		  for i ← 1 to 4
     ∴ q ← 0 + 0 + 7 x 1 x 5
	    q ← 35
We have m [i, j] = m [1, 2] = ∞ 
Comparing q with m [1, 2]
		q < m [i, j]
		i.e. 35 < m [1, 2]
		35 < ∞
		True
	then, m [1, 2 ] ← 35  (∴ m [i,j] ← q)
		s [1, 2] ← k
	and the value of k = 1
		s [1,2 ] ← 1
Insert "1" at dimension s [1, 2] in table s. And 35 at m [1, 2]

2. l remains 2

    L = 2
		    i ← 1 to n - l + 1
		    i ← 1 to 4 - 2 + 1
		    i ← 1 to 3
		for i = 1 done before
		Now value of i becomes 2
		i = 2
j ←  i + l - 1
		 j ←  2 + 2 - 1
		 j ←  3
		j = 3
m [i , j]  ← ∞
i.e. m [2,3] ← ∞
	Initially insert ∞ at m [2, 3]
Now, for k ← i to j - 1
		k ← 2 to 3 - 1
		k ← 2 to 2
		i.e. k =2
		Now, q ← m [i, k] + m [k + 1, j] + p_i-1 p_k p_j
		For l =2
			     i = 2
			     j = 3
			     k = 2
		q ← m [2, 2] + m [3, 3] + p₁x p₂ x p₃
		              q ← 0 + 0 + 1 x 5 x 4
					  q ← 20
					  Compare q with m [i ,j ]
		If q < m [i,j]
	i.e. 20 < m [2, 3]
20 < ∞        
True
Then m [i,j ] ← q
	     m [2, 3 ] ← 20
and s [2, 3] ← k
	and k = 2
s [2,3] ← 2

3. Now i become 3

  i = 3
  l = 2
j ← i + l - 1
j ← 3 + 2 - 1
j ← 4
j = 4
Now, m [i, j ] ← ∞
          m [3,4 ] ← ∞
Insert ∞ at m [3, 4] 
  for k ← i to j - 1
	       k ← 3 to 4 - 1
	       k ← 3 to 3
	i.e. k = 3
Now, q ← m [i, k] + m [k + 1, j] + p_i-1 p_k p_j
i = 3
				l = 2
				j = 4
				k = 3
q ← m [3, 3] + m [4,4] + p₂  x p₃ x p₄
q ← 0 + 0 + 5 x 2 x 4
q   40
Compare q with m [i, j]
		If q < m [i, j]
		   40 < m [3, 4]
		   40 < ∞ 
		True
Then, m [i,j] ← q
	      m [3,4] ← 40
and s [3,4] ← k
	  s [3,4] ← 3

Case 2: l becomes 3

      L = 3
	      for i = 1 to n - l + 1
	           i = 1 to 4 - 3 + 1
		 i = 1 to 2
When i = 1
	j ← i + l - 1
	j ← 1 + 3 - 1
         j ← 3
	j = 3
Now, m [i,j]   ← ∞
          m [1, 3]  ←  ∞
for k ← i to j - 1
     k ← 1 to 3 - 1
     k ← 1 to 2

Now we compare the value for both k=1 and k = 2. The minimum of two will be placed in m [i,j] or s [i,j] respectively.

Now from above

Value of q become minimum for k=1    
		∴ m [i,j] ← q
		    m [1,3] ← 48
Also m [i,j] > q
i.e. 48 < ∞
∴ m [i , j] ← q
   m [i, j] ← 48
and s [i,j] ← k
i.e. m [1,3] ← 48
	  s [1, 3] ←  1
Now i become 2
	i = 2
	 l = 3
then j ← i + l -1
	   j ←  2 + 3 - 1
	   j ← 4
	   j = 4	
	   so m [i,j] ← ∞
m [2,4] ← ∞
Insert initially ∞ at m [2, 4]
      for k   ← i to j-1
	      k  ← 2 to 4 - 1
	      k  ← 2 to 3

Here, also find the minimum value of m [i,j] for two values of k = 2 and k =3

  But 28 < 	∞
   So m [i,j] ← q
      And q  ← 28
  m [2, 4]  ← 28
  and   s [2, 4]  ← 3
i.e. It means in s table at s [2,4] insert 3 and at m [2,4] insert 28.

Case 3: l becomes 4

L = 4
	   For i ← 1 to n-l + 1
		i ← 1 to 4 - 4 + 1
		i ← 1
		i = 1
	do j  ← i + l - 1
	     j ← 1 + 4 - 1
	     j ← 4
	    j = 4
Now m [i,j]  ←  ∞
        m [1,4] ←  ∞
for k  ← i to j -1
     k ← 1 to 4 - 1
     k  ← 1 to 3
When k = 1
q  ←  m [i, k] + m [k + 1, j] + p_i-1 p_k p_j                
q  ← m [1,1] + m [2,4] + p₀xp₄x p₁
q ← 0 + 28 + 7 x 2 x 1
q  ← 42
Compare q and m [i, j]
m [i,j] was ∞
i.e. m [1,4] 
if q < m [1,4]
   42< ∞
    True 
Then m [i,j] ← q
	m [1,4] ← 42
and s [1,4]  1	? k =1
When k = 2
	L = 4, i=1, j = 4
q ← m [i, k] + m [k + 1, j] + p_i-1 p_k p_j                
q ← m [1, 2] + m [3,4] + p₀ xp₂ xp₄
q ← 35 + 40 + 7 x 5 x 2
q  ← 145
Compare q and m [i,j]
Now m [i, j]
      i.e. m [1,4] contains 42.
So if q < m [1, 4]
But 145 less than or not equal to m [1, 4]
   So 145 less than or not equal to 42.
So no change occurs.
When k = 3
	l = 4
	i = 1
	j = 4
q  ←  m [i, k] + m [k + 1, j] + p_i-1 p_k p_j                
q ← m [1, 3] + m [4,4] + p₀ xp₃ x p₄
q ← 48 + 0 + 7 x 4 x 2
q  ← 114
Again q less than or not equal to m [i, j]
	i.e. 114 less than or not equal to m [1, 4]
	       114 less than or not equal to 42