Recursion
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This document discusses recursion, which is a fundamental concept in computer science and mathematics where a function calls itself. It provides examples of common recursive definitions like factorials and Fibonacci numbers. It explains how recursive programs work by dividing the problem into smaller subproblems until a base case is reached. The document also introduces the "divide and conquer" paradigm where most recursive programs make two recursive calls on halves of the input to solve problems more efficiently than iterative approaches.
![Recurrences
• To compute F (N) in linear time
#define max 25
int fibonacci (int N)
{
int i, F[max];
F[0] = 1; F[1] = 1;
for(i = 2; i <= max; i++)
F[i] = F[i - 1] + F[i - 2];
return F[N];
}](https://image.slidesharecdn.com/recursion-130411041429-phpapp02/85/Recursion-7-320.jpg)
Recursion
- 1. Recursion Kasun Ranga Wijeweera (Email: krw19870829@gmail.com) Department of Statistics and Computer Science Faculty of Science University of Peradeniya
- 2. What is Recursion? • A fundamental concept in computer science and mathematics • A recursive program is one that calls itself • There must be a termination condition
- 3. Recurrences • Recursive definitions of functions are quite common in mathematics • The simplest type, involving integer arguments are called recurrence relations • Most familiar such a function is the factorial function, defined by the formula N ! = N * (N - 1) !, for N >=1 with 0 ! = 1
- 4. Recurrences • The corresponding simple recursive program is as follows int factorial (int N) { if (N == 0) return 1; return N * factorial (N - 1); } • Problem: factorial (-1) leads to an infinite loop
- 5. Recurrences • A second well-known recurrence relation is the one that defines the Fibonacci numbers F (N) = F (N - 1) + F (N - 2), for N >= 2 with F (0) = F (1) =1 • This defines the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610,…
- 6. Recurrences • The corresponding recursive program is as follows int fibonacci (int N) { if (N <= 1) return 1; return fibonacci (N - 1) + fibonacci (N - 2); } • The running time of this program is exponential
- 7. Recurrences • To compute F (N) in linear time #define max 25 int fibonacci (int N) { int i, F[max]; F[0] = 1; F[1] = 1; for(i = 2; i <= max; i++) F[i] = F[i - 1] + F[i - 2]; return F[N]; }
- 8. Divide and Conquer • Most of the recursive programs use two recursive calls, each operating on about half the input • This is so called “divide and conquer” paradigm • Used to achieve significant economics • They normally do not reduce to trivial loops like the factorial program • They normally do not lead to excessive re-computing as Fibonacci program, because the input is divided without overlap
- 9. Divide and Conquer: Example • Let us consider the task of drawing the markings for each inch on a ruler • There is a mark at the 1/2 point • Slightly shorter marks at 1/4 intervals • Still shorter marks at 1/8 intervals, etc
- 10. Divide and Conquer: Example • Suppose the desired resolution is 1/(2^n) • Put a mark at every point between 0 and (2^n), end points not included • A procedure mark (x, h) is used to mark h units high at x position • The middle mark should be n units high • The marks in the left and right halves should be n-1 units high
- 11. Divide and Conquer: Example • The relevant “divide and conquer” recursive program is as follows rule (int a, int r, int h) { int m = (a + r) / 2; if (h > 0) { mark (m, h); rule (a, m, h-1); rule (m, r, h-1); } }
- 12. Any Questions?
- 13. Thank You!