Presentation on theme: "Dynamic Programming Fibonacci numbers-example- Defined by Recursion F 0 = 0 F 1 = 1 F n = F n-1 + F n-2 n >= 2 F 2 = 0+1; F 3 = 1+1 =2; F 4 = 1+2 = 3 F."— Presentation transcript:
Dynamic Programming Fibonacci numbers-example- Defined by Recursion F 0 = 0 F 1 = 1 F n = F n-1 + F n-2 n >= 2 F 2 = 0+1; F 3 = 1+1 =2; F 4 = 1+2 = 3 F 5 = 2+3 = 5; ……..
Calculating F i Option(1) --Directly implement Recursion int fib(int n) int f 1,, f 2, f If(n < 2) f = n; else f 1 = fib(n-1) i f 2 = fib(n-2) i f = f 1 + f 2i f = f I very inefficient : how many calls ? For example: for f 6 ?
Dynamic programming / Fibonacci numbers Call structure of Recursion
Contd.. Very efficient: runtime at least (2 n/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 2 n/2 nodes. recall computed values f = 0; f = 1; for (i=2; i< = n; i++) f[i] = f[i-1] + f[i-2];
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 6543210 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;