1. Andy Wang Object Oriented Programming in C++ COP 3330
Recursion
Andy Wang
Object Oriented Programming in C++
COP 3330

2. Recursion A recursive function is a function that calls itself
Many tasks can be done with either recursion or with iteration
Iteration involves a loop, but not recursive calls
Make sure the recursion stops at some point
Infinite recursion will result in a crash, when stack frames overrun the limit

3. One example: Factorials
n! = n*(n – 1)*(n – 2)…*1
Iteratively, we can define a function as follows
unsigned long Factorial(int n) {
unsigned long f = 1;
for (int j = n; j >= 1; j--)
f *= j;
return f; }

4. One example: Factorials
But, notice that
n! = n*(n – 1)*(n – 2)…*1 = n*(n – 1)!
We can define a recursive function
unsigned long Factorial(int n) {
if (n <= 1)
return 1;
else
return (n*Factorial(n – 1)); }

5. One example: Factorials
The recursive function will end when n = 1
Each recursive call will call the function with a smaller n
So, which is better?
The performance is similar in this case
Recursive version uses more memory
Each recursive call takes one stack frame

6. Is Recursion Better? Depends! Let’s check out the Fibonnaci example
A Fibonnaci number is defined recursively
First and the second elements are both 1
Nth element is the sum of elements n – 1 and n – 2
Fib(n) = Fib(n – 1) + Fib(n – 2)

7. Is Recursion Better? Iterative Fibonnaci function int Fib(int n) {
int n1 = 1, n2 = 1, n3;
int j = 2;
while (j < n) {
n3 = n1 + n2;
n2 = n3;
n1 = n2;
j++; }
return n;

8. Is Recursion Better? Recursive Fibonnaci function int Fib(int n) {
if (n <= 0) return 0;
else if (n == 1) return 1;
else return Fib(n – 1) + Fib(n – 2); }

9. Which One is More Efficient?
Try them out!
on/fib.cpp
Fib 10, 20, 30, 40, 50…
Iterative version is definitely faster

10. Recursion vs. Iteration
Any recursive solution can be done iteratively
A recursive solution is often easier to code
Recursion does involve extra overhead
Function calls, stack frames
Iterative solutions are typically faster

11. Sorting Algorithms Many solutions
You’ve probably seen bubble sort and selection sort
Two sort algorithms (often written with recursion) are much faster
Quick sort
Merge sort

12. GCD Example
r19-5.cpp
Compute the greatest common divisor

13. Pr19-5.cpp
#include <iostream> using namespace std; int gcd(int x, int y) { if (x % y == 0) return y; else return gcd(y, x % y); } int main() { int num1, num2; cout << “Enter two integers: “; cin >> num1 >> num2; cout << “The GDC of “ << num1 << “ and “ << num2 << “ is “ << gcd(num1, num2) << endl; return 0;

14. More Recursion Examples
13-01.cpp: write a number vertically using recursion
13-02.cpp: write a number vertically using iteration
13-03.cpp: raise a number to a power of n (recursion)
13-06.cpp: binary search (recursion)
13-08.cpp: binary search (iteration)

15. 13-01.cpp
#include <iostream> using namespace std; void writeVertical(int n) { if (n < 10) { cout << n << endl; } else { writeVertical(n/10); cout << (n % 10) << endl; }

16. 13-01.cpp
int main() { cout << “writeVertical(3):” << endl; writeVertical(3); cout << “writeVertical(12):” << endl; writeVertical(12); cout << “writeVertical(123):” << endl; writeVertical(123); return 0; }

17. 13-02.cpp
Power of ten that has the same number of digits as n (nsTens = 1000 if n = 2345)
#include <iostream> using namespace std; void writeVertical(int n) { int nsTens = 1; int leftEndPiece = n; while (leftEndPiece > 9) { leftEndPiece = leftEndPiece / 10; nsTens = nsTens*10; } for (int powerOf10 = nsTens; powerOf10 > 0; powerOf10 = powerOf10/10) { cout << (n/powerOf10) << endl; n = n % powerOf10;

18. 13-02.cpp
int main() { cout << “writeVertical(3):” << endl; writeVertical(3); cout << “writeVertical(12):” << endl; writeVertical(12); cout << “writeVertical(123):” << endl; writeVertical(123); return 0; }

19. 13-03.cpp
#include <iostream> #include <cstdlib> using namespace std; int power(int x, int n) { if (n < 0) { cout << “Illegal argument to power.\n”; exit(1); } if (n > 0) return (power(x, n – 1)* x); else // n == 0 return 1;

20. 13-03.cpp
int main() { for (int n = 0; n < 4; n++) { cout << “3 to the power “ << n << “ is “ << power(3, n) << endl; } return 0;

21. 13-06.cpp
#include <iostream> using namespace std; const int ARRAY_SIZE = 10; void search(const int a[], int first, int last, int key, bool &found, int &location) { int mid; if (first > last) found = false else { mid = (first + last/2) if (key == a[mid]) { found = true; location = mid; } else if (key < a[mid] search(a, first, mid – 1, key, found, location); else if (key > a[mid] search(a, mid + 1, last, key, found, location); }

22. 13-06.cpp
int main() { int a[ARRAY_SIZE]; cout << “Array contains:\n”; for (int j = 0; j < ARRAY_SIZE; J++) { a[j] = 3*j; cout << a[j] << ‘ ‘; } cout << endl; …

23. 13-06.cpp
int key, location; bool found; cout << “Enter number to be located: “; cin >> key; search(a, 0, ARRAY_SIZE – 1, key, found, location); if (found) cout << key << “ is in index location “ << location << endl; else cout << key << “ is not in the array.” << endl; return 0; }

24. 13-06.cpp
#include <iostream> using namespace std; const int ARRAY_SIZE = 10; void search(const int a[], int lowEnd, int highEnd, int key, int first = lowend, last = highEnd, mid; found = false; while ((first <= last) && !found) { mid = (first + last)/2 if (key == a[mid]) { found = true; location = mid; } else if (key < a[mid]) last = mid – 1; } else if (key > a[mid]) first = mid + 1; }

25. 13-08.cpp
int main() { int a[ARRAY_SIZE]; cout << “Array contains:\n”; for (int j = 0; j < ARRAY_SIZE; J++) { a[j] = 3*j; cout << a[j] << ‘ ‘; } cout << endl; …

26. 13-08.cpp
int key, location; bool found; cout << “Enter number to be located: “; cin >> key; search(a, 0, ARRAY_SIZE – 1, key, found, location); if (found) cout << key << “ is in index location “ << location << endl; else cout << key << “ is not in the array.” << endl; return 0; }