Dynamic Programming

Fibonacci numbers-example- Defined by Recursion F0 = 0 F1 = 1

F n = Fn-1 + F n-2 n >= 2

F2 = 0+1; F3 = 1+1 =2; F 4= 1+2 = 3

F5 = 2+3 = 5; ……..

Calculating Fi

Option(1) --Directly implement Recursion int fib(int n) int f1,, f2 , f

If(n < 2) f = n; else f1 = fib(n-1)i

f2 = fib(n-2)i

f = f 1+ f2i

f = f I

very inefficient : how many calls ?

For example: for f6 ?

Dynamic programming / Fibonacci numbers Call structure of Recursion

4 7

23 2

2

1

6

1 1 0

5

1 0

4

0112

23

01

01

1st call

2nd

3rd

4th

5th

6th 7th

8th 10th 11th 14th 15th

16th

20th21st

24th 25th call

23rd

17th

18th

9th13th 19th 22nd

12th

Contd..

Very efficient: runtime at least (2n/2) calculates the same numbers several times.

IMPROVE: --- save previous results in a memory and recall when

needed. (n) calls

Full Binary Tree at least to depth n/2 2n/2 nodes.

recall computed values f[0] = 0; f[1] = 1; for (i=2; i< = n; i++) f[i] = f[i-1] + f[i-2];

Ω

Θ

Dynamic Programming Sub problem Graph

Decompose the main problem into smaller sub problems of the same kind.

Solve smaller problems recursively save the intermediate solution.

Combine the solutions.

Sub problem Graph

-- vertices are : instances

-- Directed edge : 1 3

1 directly calls 7

6 5 4 3 2 1 0

Structure of calls

Contd..

we can try to calculate in advance those instances which are needed for a given solution.

Those are exactly the nodes which are reachable from a given starting state(S).

The Depth-First search tree of the sub problem graph gives exactly those nodes which are reachable from S

perform dfs get reachable nodes Calculate sub problems and record solutions to dictionary

solution.

Dynamic programming version of the Fibonacci function.

Example 10.3

fibDPwrap(n) Dict soln = create(n); return fibDP(soln, n);fibDP(soln, k)int fib, f1, f2;

If(k<2) fib = k;

else if(member(soln, k-1) = = false) f1 = fibDP(soln, k-1); else f1 = retrieve(soln, k-1);if(member(soln, k-2) = = false) f2 = fibDP(soln, k-2); else f2 = retrieve(soln, k-2); fib = f1 + f2;Store(soln, k, fib); return fib;