1. Recursion

2. The process of solving a problem by reducing it smaller versions of itself is called recursion
Recursion is a powerful way to solve some certain problems for which the solution otherwise will be very complicated
Example: Factorail , Fibonacci numbers and Towers of Hanoi

3. Factorial
n!=n*(n-1)! if n>0 and 0!=1 Recursive definition: A definition in which something is defined in terms of a smaller version of itself int fact (int num) { if(num = = 0) return 1; else return num * fact(num-1); }

4. Direct and Indirect Recursion
A function is directly recursive if it calls itself
A function that calls another function and eventually results in original function call is said to be indirectly recursive
if function A calls B and B calls A, then function A is recursive
if function A calls B and B calls C and C calls D and D calls A, then A function is recursive

5. Infinite Recursion
If every recursive call results in another recursive call, then the recursive function (algorithm) is said to have infinite recursion
As a theory infinite recursion executes forever
But , because the memory is finite the recursive function will execute till the system runs out of memory and results in abnormal termination of program

6. How to design Recursive program
understand the problem requirements
determine the limiting conditions
identify the base case and provide a direct solution to each base case
identify the general cases and provide a solution to each general case in terms of smaller versions of itself

7. Largest element in array
Pseudo code:
Base case: the size of the arr is 1
the only element in the arr is the largest
General case: The size of the arr is greater then 1
to find the largest element in arr[a]…arr[b]
- find the largest element in arr[a+1]…arr[b] and call it max
- compare the elements arr[a] and max
if (arr[a]>=max )
the largest element in arr[a]…arr[b] is arr[a]
otherwise
the largest element in arr[a]…arr[b] is max

8. int largest(const int arr[], int lowerIndex, int upperIndex) { int max; if (lowerIndex == upperIndex) return arr[lowerIndex]; else max=largest(arr, lowerIndex+1, upperIndex); if (arr[lowerIndex]>= max) return max; }

9. Fibonacci numbers
Fibonacci number is the sum of the first two numbers of Fibonacci.
Fn= Fn-1 + Fn-2
In mathematics, the Fibonacci numbers or Fibonacci series or Fibonacci sequence are the numbers in the following integer sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …; F0 F1 F2 F3 F4 F5 F6 F7 F8 F9
F10
F11
F12
F13
F14
F15
F16
F17
F18
F19
F20 1 2 3 5 8 13 21 34 55 89
144
233
377
610
987
1597
2584
4181
6765

10. Applications computer algorithms: biological settings:
Fibonacci search technique
Fibonacci heap data structure
Fibonacci cubes used for interconnecting parallel and distributed systems
biological settings:
branching in trees
phyllotaxis (the arrangement of leaves on a stem)
fruit sprouts of a pineapple
flowering of artichoke
an uncurling fern
arrangement of a pine cone

11. Applications

12. Function of Fibonacci numbers (C++)
int FibNum(int first, int second, int nth) {
if (nth == 1)
return first;
else if (nth == 2)
return second;
else
return FibNum(first, second, nth-1) FibNum(first, second, nth-2); }

13. Towers of Hanoi (recursive)
FUNCTION MoveTower(disk, source, dest, spare): IF disk == 0, THEN: move disk from source to dest ELSE: MoveTower(disk - 1, source, spare, dest) // Step1 above // Step 2 above MoveTower(disk - 1, spare, dest, source) // Step 3 above END IF

14. Diagram view how function ToH works

15. main function
#include <iostream> using namespace std; void moveDiscs(int, int, int, int, int); void main() { int count = 0; int NUM_DISCS = 3; // Number of discs to move const int FROM_PEG = 1; // Initial "from" peg const int TO_PEG = 3; // Initial "to" peg const int TEMP_PEG = 2; // Initial "temp" peg moveDiscs(NUM_DISCS, FROM_PEG, TO_PEG, TEMP_PEG, count); cout << "All the pegs are moved!\n"; system("PAUSE"); }

16. moveDiscs Function
void moveDiscs(int num, int fromPeg, int toPeg, int tempPeg, int count) { if (num > 0) moveDiscs(num - 1, fromPeg, tempPeg, toPeg, count); cout << "Move a disc from peg " <<fromPeg << " to peg " << toPeg <<endl; count++;
if (fromPeg == 1 && toPeg == 3) { if (num == 1) if(count == 3) cout<<" *"<<endl<<" *"<<endl<<" *"<<endl; else cout<< "*"<<endl<<"* *"<<endl; }

17. else
cout << " *"<<endl <<" * *"<<endl; }
if (fromPeg==1 && toPeg==2)
cout << "* * *"<<endl;
if (fromPeg ==3 && toPeg==2)
cout << " *"<<endl <<"* *"<<endl;
if (fromPeg==2 && toPeg==1)
cout << "* * *"<<endl;
if (fromPeg==2 && toPeg==3)
cout << " *"<< endl <<"* *"<<endl;
moveDiscs(num-1, tempPeg, toPeg, fromPeg, count); }

18. Output of the program ToH

19. Another function of ToH
void moveDisks(int count, int peg1, int peg3, int peg2) { if (count>0) moveDisks(count-1, peg1, peg2, peg3); cout<<"Move disk"<<count << " from " << peg1 << " to " << peg3 << " . " << endl; moveDisks(count-1, peg2, peg3, peg1); }

20. Binary to Decimal To convert binary number to decimal
find the weight of each bit from right to left
weight of rightmost bit is 0
multiply the bit by 2 to the power of it’s weight and add all of the numbers

21. Binary to Decimal Recursive solution
void binToDec(int binary, int& decimal, int& weight) { int bit; if (binary>0) bit=binary % 10; decimal=decimal+bit*static_cast<int>(pow(2.0,weight)); binary=binary/10; weight++; binToDec(binary, decimal, weight); }

22. Decimal to Binary To convert decimal to binary
divide decimal number by 2 till dividend will be less then 2
the first remainder of division is the left leftmost element of binary number and the last remainder is rightmost element of binary number
write the remainders in their positions

23. Decimal to Binary Recursive solution
void decToBin(int num, int base) { if (num > 0) decToBin(num/base, base); cout << num % base; }