Lec-32 Recursion -Recursion in Computer Science

Lec-32 Recursion -Recursion in Computer Science

• 1.
  A mathematical look We are familiar with f(x) = 3x+5  How about f(x) = 3x+5 if x > 10 or f(x) = f(x+2) -3 otherwise
• 2.
  Calculate f(5) f(x) =3x+5 if x > 10 or f(x) = f(x+2) -3 otherwise  f(5) = f(7)-3  f(7) = f(9)-3  f(9) = f(11)-3  f(11) = 3(11)+5 = 38 But we have not determined what f(5) is yet!
• 3.
  Calculate f(5) f(x) =3x+5 if x > 10 or f(x) = f(x+2) -3 otherwise  f(5) = f(7)-3 = 29  f(7) = f(9)-3 = 32  f(9) = f(11)-3 = 35  f(11) = 3(11)+5 = 38 Working backwards we see that f(5)=29
• 4.
  Series of calls f(5) f(7) f(9) f(11)
• 5.
  Recursion occurs whena function/procedure calls itself. Many algorithms can be best described in terms of recursion. Example: Factorial function The product of the positive integers from 1 to n inclusive is called "n factorial", usually denoted by n!: n! = 1 * 2 * 3 .... (n-2) * (n-1) * n Recursion Recursion
• 6.
  Recursive Definition Recursive Definition ofthe Factorial Function n! = 1, if n = 0 n * (n-1)! if n > 0 5! = 5 * 4! 4! = 4 * 3! 3! = 3 * 2! 2! = 2 * 1! 1! = 1 * 0! = 5 * 24 = 120 = 4 * 3! = 4 * 6 = 24 = 3 * 2! = 3 * 2 = 6 = 2 * 1! = 2 * 1 = 2 = 1 * 0! = 1
• 7.
  The Fibonacci numbersare a series of numbers as follows: fib(1) = 1 fib(2) = 1 fib(3) = 2 fib(4) = 3 fib(5) = 5 ... fib(n) = 1, n <= 2 fib(n-1) + fib(n-2), n > 2 Recursive Definition Recursive Definition of the Fibonacci Numbers fib(3) = 1 + 1 = 2 fib(4) = 2 + 1 = 3 fib(5) = 2 + 3 = 5
• 8.
  int BadFactorial(n){ int x= BadFactorial(n-1); if (n == 1) return 1; else return n*x; } What is the value of BadFactorial(2)? Recursive Definition Recursive Definition We must make sure that recursion eventually stops, otherwise it runs forever:
• 9.
  Using Recursion Properly UsingRecursion Properly For correct recursion we need two parts: 1. One (ore more) base cases that are not recursive, i.e. we can directly give a solution: if (n==1) return 1; 2. One (or more) recursive cases that operate on smaller problems that get closer to the base case(s) return n * factorial(n-1); The base case(s) should always be checked before the recursive calls.