Analysis of Algorithms

Dynamic Programming

Dynamic Programming

A dynamic programming algorithm solves every sub problem just once and then

Saves its answer in a table (array), there byAvoiding the work of recomputing the answer every

time the sub problem is encountered.Dynamic programming is typically applied to

optimization problems.What is an optimization problem?

There can be may possible solutions.Each solution has a value andWe wish to find a solution with the optimal (minimum or

maximum) value

Toward Dynamic ProgrammingProblems with divide and conquerRecursive Fibonnacci numbers algorithm with

exponential complexity.When sub problems are not independentFib1(N){ if (N =< 1)

return 1;else

return Fib(N-1) + Fib(N-2)}

Dynamic ProgrammingBottom-up approach with

Memoization Instead of computing again and again, we can store the computed

results in an array. The stored values can be used for any future use.

Assume, we want to compute Fib(n) with max. value of n is MAXVALUE.#define MAXVALUE 100;Fib2(n){

int F[MAXVALUE];for(i = 1; i <= n; i++)

F[i] = 1;for(i = 2; i <= n; i++)F[i] = F[i-1] + F[i-2];return F[n];

}

Dynamic ProgrammingBottom-up Approach with

MemoizationSometimes after developing an algorithm using

array , we are able to revise the algorithm to save originally allocated space

Fib2(n){ int Fn = 1, Fn1 = 1, Fn2 = 1;

for(I = 2; I <= n; I++){ Fn = Fn1 + Fn2;

Fn2 = Fn1;Fn1 = Fn;

}return Fn;

}

Dynamic Programming

The steps in development of dynamic programming algorithm areEstablish a recursive property that gives the

solution to an instance of a problemSolve an instance of the problem in Bottom-

up fashion by solving smaller instances first.

Example of Binomial Coefficient

The binomial coefficient is given by

n. k 0for )!(!

!

knk

n

k

n

For values of n and k that are not small,

We can not compute the binomial coefficient directly from this definition because n! is very large

Example of Binomial Coefficient

We can eliminate the need to compute n! or k! by using following recursive property.

n k or 0 k 1

n k 0 1

1

1

k

n

k

n

k

n

This suggests the following divide-and-conquer algorithm.

Example of Binomial Coefficient Using Divide-and-

ConquerInt bin(int n, int k)

{

if (k = 0 or n = k )

return 1;

else

return(bin(n-1, k-1) + bin(n-1, k))

}

Example of Binomial Coefficient Using Divide-and-

ConquerThis algorithm is very similar to the

divide-and-conquer algorithm of computing nth fabnacci term.

Therefore this algorithm is very inefficient due to the same reasons.

Example of Binomial Coefficient Using Dynamic

ProgrammingWe will use the recursive definition to

construct our solution in an array B.Where B[i, j] will contain

j

i

Solve an instance of the problem in a bottom-up fashion by computing the rows in B in sequence starting with the first row.

`

Int bin(int n, int k){

int i, j;int B[0..n, 0..k];for i = 0 to n

for j = 0 to minimum(i, k) if( j = 0 or j = i)B[i, j] = 1;elseB[i, j] = B[i-1, j-1] + B[i-1, j];

return B[n, k]}

Complexity Analysis

number of passesi 0 1 2 3 k k+1 nj 1 2 3 4 k+1 k+1 k+1total number of passes1+2+3++k+k+1+k+1…….+k+1

(k+1 is repeacted for n-k+1 times)

k(k+1)/2+(k+1)(n-k+1)

=((2n-k+2)(k+1))/2 є Ө(nk)

Dynamic Programming

Dynamic programming is similar to divide and conquer in that we find a recursive property.

But instead of blindly using recursion, we use the recursive property to iteratively solve the instances in sequence starting with the smallest instance

And we solve each instance ONCE