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Dynamic Programming: Steps

- The seven steps in the development of a dynamic programming algorithm are as follows:
- Establish a recursive property that gives the solution to an instance of the problem.
- Develop a recursive algorithm as per recursive property
- See if same instance of the problem is being solved again an again in recursive calls
- Develop a memoized recursive algorithm
- See the pattern in storing the data in the memory
- Convert the memoized recursive algorithm into iterative algorithm
- Optimize the iterative algorithm by using the storage as required (storage optimization)

Exercise 1

- Initialise a two dimensional array A[1..n, 1..n] using following definition with an efficient algorithm.
- A[i, j] = i + j if i = j
- A[i, j] = Min(A[i, j-1], A[i+1, j])+ (i+j) if i < j
- A[i, j] = Max(A[i-1, j], A[i, j+1])+ (i *j) if i > j
- Where i is for row index and j is for column index.

Quiz

Develop a recursive algorithm for the following definition, where P[] is a one-dimensional array having indices from i-1 to j,and m is a two-dimensional array of size ixj

Solution for the Quiz

- RMC(p, i, j)
- {
- if i = j
- return 0
- m[i, j] =
- for(k = i; k <= j-1; k++)
- { q = RMC(p, i, k) + RMC(p, k+1, j) + pi-1pkpj
- if(q < m[i, j])
- m[i, j] = q
- }
- return m[i, j];
- }

Quiz

Convert the algorithm of the previous Quiz into an iterative algorithm

Quiz

Develop a recursive algorithm for the following definition,

0 if i = 0 or j = 0

C(i, j) = c(i-1, j-1) + 1 if i, j > 0 and xi = yj

max (c(i, j-1), c(i-1, j)) if i, j > 0 and xi yj

X =

Y =

i varies from m to 0 and j varies from n to 0

Quiz

- Develop a memoized recursive algorithm for the following definition

0 if i = 0 or j = 0

C(i, j) = c(i-1, j-1) + 1 if i, j > 0 and xi = yj

max (c(i, j-1), c(i-1, j)) if i, j > 0 and xi yj

X =

Y =

i varies from m to 0 and j varies from n to 0

Quiz

Convert the algorithm of the previous Quiz into an iterative algorithm

Third Practice Problem – Quiz

Develop a recursive and memoized recursive algorithm for the following recursive definition

P is an array of j elements

A[i, i] = Pi

A[i, i-1] and A[j+1, j] are defined to be 0

Home Exercise

- Convert the iterative algorithm of the quizzes into one which consume memory as per requirement of the problem

Matrix Multiplications

- We can multiply two matrices A and B if

Columns of A = Rows of B I.e.

order of A = p * q and order of B = q * r

void Matrix_Multiply(A, B)

{ If (columns of A != Rows of B)

error

else {for(I = 1; I <= n; I++)

for(j = 1; j <= N; j++)

{C[ I ][ j ] = 0;

for(k = 1; k <= N; k++)

C[ I ][ j ] += A[ I ][k ] * B[ k ][ j ]

}

}

}

No. of Scalar Multiplications = p * q * r

Matrix Chain Multiplication

- Problem arises when we want to multiply a chain of matrices. For example

A = 50 * 10

B = 10 * 40

C = 40 * 30

D = 30 * 5

- We want to compute A * B * C * D.
- In which sequence should we call matrix multiply?
- Matrix multiplication is associative.

Matrix Chain Multiplication

A = 50 * 10, B = 10 * 40, C = 40 * 30,D = 30 * 5

Matrix Chain Multiplication Problem

- Given a chain
of n matrices. - Multiply all the matrices with minimum cost
- First Compute the order in which matrices should be multiplied
- Or in other words
- We have to find an optimum sequence or an optimum parenthesization of matrix chain multiplication.
- Then by using that order multiply the matrices.

Matrix Chain Multiplication Problem Bottom Up Approach with Memoization

Matrix-Chain-Multiply(A [ ] , s[ ], i, j)

// A is an array of matrices

// s is an array of order, s[i,j] represents parenthesization position for // matrix i to j

{ if (j > i){

X = Matrix-Chain-Multiply(A, s, i, s[i, j]);

Y = Matrix-Chain-Multiply(A, s, s[i, j]+1, j);

return Matrix-Multiply(X, Y);}

else return Ai

}

The problem is to compute s[i,j] for a given chain of matrices from i to j.

Formulation of the problem

- Fully parenthesize the product A1 * A2….An in a way that minimizes the number of scalar multiplications.

Matrix Chain Multiplication Problem

- Ai..j is a notation for the matrix that results from evaluating the product AiAi+1…Aj.
- An optimal parenthesization of the product A1A2….An splits the product between Ak and Ak+1 for some integer k in the range 1<=k
- For some value of k we first
- Compute the matrices A1..k and Ak+1..n and then
- Multiply them together to produce the final product A1..n.

Matrix Chain Multiplication Problem

- The cost of this optimal parenthesization is thus the
- Cost of computing the matrix A1..k, plus
- The cost of computing Ak+1..n, plus
- The cost of multiplying them together.

Matrix Chain Multiplication Problem

- Parenthesization of the “prefix” sub chain A1A2…Ak within this optimal parenthesization of A1A2…An must be
- An optimal parenthesization of A1A2…Ak.
- Why?
- A similar observation holds for the parenthesization of the sub chain Ak+1Ak+2…An
- Thus, an optimal solution to an instance of the matrix-chain multiplication problem contains within it optimal solutions to sub problem instances.

Matrix Chain Multiplication Problem

A Recursive Solution

- Define the value of an optimal solution recursively in terms of the optimal solutions to subproblems.
- For this problem, we pick as our subproblems
- the problems of determining the minimum cost of a parenthesization of

AiAi+1….Aj, for 1 <= i <= j <=n

A Recursive Solution

- Let m[i, j] be the minimum number of scalar multiplications needed to compute the matrix Ai..j;
- The cost of cheapest way to compute A1..n would thus be m[1, n]

A Recursive Solution

- We can define m[i, j] recursively as follows:
- If i = j, the chain consists of just one matrix Ai..i=Ai, so
- No scalar multiplications are necessary to compute the product. Thus
- m[i, i] = 0 for i = 1, 2, …..,n.
- To compute m[i, j] when i < j, we take advantage of the structure of an optimal solution from step 1.
- Assume that optimal parenthesization splits the product Ai, Ai+1,….Aj between Ak and Ak+1, where i<=k < j.

A Recursive Solution

- Now m[i, j] is equal to the minimum cost for
- Computing the subproducts A1..k, and
- Computing the subproducts Ak+1..j, plus
- The cost of multiplying these two matrices together.
- For i = 1, 2, ….., n, matrix Ai has dimensions Pi-1*Pi
- Since computing the matrix product A1..kAk+1..j takes pi-1pkpj scalar multiplications, we obtain
- m[i, j] = m[i, k] + m[k+1, j] + pi-1pkpj
- But we don’t know the value of k for an optimum solution.
- There are only j-1 possible values for k, namely k=i,i+1, …j-1.

A Recursive Solution

- Since, the optimal parenthesization must use one of these values for k, we need only check them all to find the best.
- Thus, our recursive definition for the minimum cost of parenthesizing the product Ai, Ai+1, … Aj becomes

Matrix Chain Multiplication Problem

- Based on this recurrence, we can easily write the recursive algorithm to compute the minimum cost m[1..n] for multiplying A1, A2, ….,An.

Recursive-Matrix-Chain(p, i, j)

{

- if i = j
- return 0
- MinCost =
- for(k = i; k <= j-1; k++)

{ q = Recursive-Matrix-Chain(p, i, k) + Recursive-Matrix- Chain(p, k+1, j) + pi-1pkpj

- if(q < MinCost)
- MinCost = q

}

- return MinCost;

}

Matrix Chain Multiplication Problem

- A recursive algorithm may encounter each sub problem many times in different branches of its recursion tree.
- Instead of computing the same solution again and again, we can use the memoization strategy.

Matrix Chain Multiplication ProblemTop Down Recursive Approach with Memoization

Memoized-Matrix-Chain(p)

{

n = length[p] – 1;

for(i= 1; i <= n; i++)

for(j = i; j <= n; j++)

m[i, j] =

return LookUp-Chain(p, 1, n);

}

Matrix Chain Multiplication ProblemTop Down Recursive Approach with Memoization

LookUp-Chain(p, i, j)

{ if( m[i, j] < )

return m[i, j]

if(i == j)

m[i, j] = 0;

else

{ for(k = i; k <= j-1; k++)

{ q = LookUp-Chain(p, I, k) + LookUp-Chain(p, k+1, j) + pi-1pkpj

if(q < m[i, j])

{ m[i, j] = q;

s[i, j] = k;

}

}

}

}

Matrix Chain Multiplication ProblemBottom Up Heuristic Approach

- To keep track of how to construct an optimal solution, let us define
- s[i, j] to be a value of k at which we can split the product Ai,Ai+1, … Aj to obtain an optimal parenthesization.i.e.
- s[i, j] equals a value k such that m[i, j] = m[i, k] + m[k+1, j] + pi-1pkpj.

Matrix Chain Multiplication ProblemBottom Up Approach with Memoization

Matrix-Chain-Order(p)

- { n = length(p) –1
- for(i = 1; i <= n; i++) m[i, i] = 0;
- for(l = 2; l <= n; l ++)
- for(i = l; i<=n-l+1; i++)
- { j = i + l – 1; m[i, j] =
- for(k = i; k<= j-1; k++)
- { q = m[i, k] + m[k+1, j] + pi-1pkpj
- if (q < m[i, j])
- { m[i, j] = q;
- s[i, j] = k;
- }
- }
- }
- return m and s;

}

Matrix Chain Multiplication ProblemBottom Up Approach with Memoization

Matrix-Chain-Multiply(A , s, i, j)

{

if (j > i)

{

X = Matrix-Chain-Multiply(A, s, i, s[i, j]);

Y = Matrix-Chain-Multiply(A, s, s[i, j]+1, j);

return Matrix-Multiply(X, Y);

}

else

return Ai

}

Longest Common Subsequence (LCS) Problem

- Given two sequences of symbols, X and Y, determine the longest subsequence of symbols that appears in both X and Y.
- Example:

Y =

- LCS of X, Y?

Longest Common Subsequence (LCS) ProblemBrute Force Approach

- Enumerate all subsequences of X.
- Each subsequence of X corresponds to a subset of indices {1, 2, ….,m} of X.
- There are 2m subsequences of X, so this approach requires exponential time.

j

0

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0

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yj

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xi

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B

7

Longest Common Subsequence (LCS)LCS – Algorithm: Constructing an LCS

- Table b[1..m, 1..n] can be used to construct an LCS.
- Initial invocation is Print_LCS(b, X, length[X], length[Y]).

Print_LCS(b[], X[], i, j)

{ if(i = = 0 or j = = 0)

return 0;

if(b[i, j] = = “” // common elements

{ Print_LCS(b, X, I-1, j-1);

print xi;

}

else if (b[I, j] = = “”)

Print_LCS(b, X, I-1, j);

else Print_LCS(b, X, I, j-1);

}

- Devise a memoized version of LCS_Length();

Longest Common Subsequence (LCS) ProblemDynamic Programming Approach

- Characterize the LCS problem
- The LCS problem has an optimal substructure property.
- Can we formulate optimal substructure?
- ith preifx of a sequence X is Xi, e.g.,
- X4 of sequence X = is

Longest Common Subsequence (LCS) ProblemDynamic Programming Approach

Optimal Substructure of an LCS:

Let X =

Y =

Z =

- If xm = yn then zk = xm = yn and zk-1 is an LCS of Xm-1 and Yn-1.
- If xm yn then zk xm implies that Z is an LCS of Xm-1 and Y.
- If xm yn then zk yn implies that Z is an LCS of X and Yn-1

Longest Common Subsequence (LCS) ProblemDynamic Programming Approach

Step 1: Characterizing an LCS

- LCS problem has an optimal substructure property. I.e.
- There are either one or two subproblems to examine when finding an LCS of

X =

- if xm = yn
- find an LCS of Xm-1 and Yn-1 appending xm and yn (xm = yn) to this LCS yields an LCS of X and Y.
- If xm yn
- solve two subproblems:
- Finding an LCS of Xm-1 and Y
- Finding an LCS of X and Yn-1
- Longer of I and II is an LCS of X and Y.
- This shows overlapping subproblmes property in LCS.

Longest Common Subsequence (LCS) ProblemDynamic Programming Approach

Step II:Define a recursive solution

- Establish a recurrence for the cost of an optimal solution.
- Let c[i, j] to be the length of the longest subsequences (LCS) of Xi and Yj.

0 if i = 0 or j = 0

c[i, j] = c[i-1, j-1] + 1 if i, j > 0 and xi = yj

max(c[i, j-1], c[i-1, j]) if i, j > 0 and xi yj

- Write a recursive algorithm to compute the length of an LCS of two sequences?

Longest Common Subsequence (LCS) ProblemDynamic Programming Approach

Step III:Devise an algorithm to compute the length of an LCS and then construct the LCS.

- Development of an algorithm
- Input two sequences X and Y and a two dimensional table c[0..m, 0..n]
- According to the definition

for(i = 1; I <= m; I++)

{ for(j = 1; j <= n; j++)

{ if (X[i] = = Y[j])

c[i, j] = c[i-1, j-1] + 1

else if (c[i-1, j] >= c[i, j-1])

c[i, j] = c[i-1, j];

else c[i, j] = c[i, j-1];

}

}

/* It means to compute c[i, j]

We need to know

c[i-1, j-1], c[i, j-1] and c[i-1, j] */

j

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Longest Common Subsequence (LCS) ProblemDynamic Programming Approach ExampleLongest Common Subsequence (LCS) ProblemDynamic Programming Approach Example

- LCS – Algorithm: Computing the length

LCS-Length(X, Y)

{ m = length[X];

n = length[Y];

for(i = 1; i <= m; i++) c[i, 0] = 0;

for(j = 0; j <= n; j++) c[0, j] = 0;

for(i = 1; i <= m; I++)

{ for(j = 1; j <= n; j++)

{ if(X[i] = = Y[i]

{ c[i, j] = c[i-1, j-1] + 1; b[I, j] = “” }

else if(c[i-1, j] >= c[i, j-1])

{ c[i, j] = c[i-1, j]; b[I, j] = “” }

else

{ c[i, j] = c[i, j-1]; b[I, j] = “” }

}

} return c and b; }

Complexity: O(m*n)

Longest Common Subsequence (LCS) ProblemDynamic Programming Approach Example

- LCS – Algorithm: Constructing an LCS
- Table b[1..m, 1..n] returned by LCS-length() can be used to construct an LCS.
- Initial invocation is Print_LCS(b, X, length[X], length[Y]).

Print_LCS(b, X, i, j)

{ if(i = = 0 or j = = 0)

return 0;

if(b[i, j] = = “”

{ Print_LCS(b, X, I-1, j-1);

print xi;

}

else if (b[I, j] = = “”)

Print_LCS(b, X, I-1, j);

else Print_LCS(b, X, I, j-1);

}

- Devise a memoized version of LCS_Length();

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