Examples of Recursion

1
Examples of Recursion

• Instructor Mainak Chaudhuri
• mainakc_at_cse.iitk.ac.in

2
Sum of natural numbers

• class SumOfNaturalNumbers
• public static void main (String arg)
• int m 3, n 10
• if (m gt n)
• System.out.println (Invalid input!)
• else
• System.out.println (Sum of numbers from
  m to n is Sum(m, n))

3
Sum of natural numbers

• public static int Sum (int m, int n)
• int x, y
• if (mn) return m
• x Sum(m, (mn)/2)
• y Sum((mn)/21, n)
• return (xy)
• // end class

4
GCD revisited

• Recall that gcd (a, b) gcd (a-b, b) assuming a
  gt b.
• Directly defines a recurrence
• public static int gcd (int a, int b)
• if ((a1) (b1)) return 1
• if (ab) return a
• if (a lt b) return gcd (a, b-a)
• if (a gt b) return gcd (a-b, b)

5
GCD revisited

• Refinement gcd (a, b) gcd (a-nb, b) for any
  positive integer n such that a gt nb, in
  particular n a/b, assuming a gt b
• public static int gcd (int a, int b)
• if (0a) return b
• if (0b) return a
• if (a lt b) return gcd (a, ba)
• if (a gt b) return gcd (ab, b)

6
Towers of Hanoi

• Three pegs, one with n disks of decreasing
  diameter two other pegs are empty
• Task move all disks to the third peg under the
  following constraints
• Can move only the topmost disk from one peg to
  another in one step
• Cannot place a smaller disk below a larger one
• An example where recursion is much easier to
  formulate than a loop-based solution

7
Towers of Hanoi

• We want to write a recursive method THanoi (n, 1,
  2, 3) which moves n disks from peg 1 to peg 3
  using peg 2 for intermediate transfers
• The first step is to formulate the algorithm
• Observation THanoi (n, 1, 2, 3) ? THanoi (n-1,
  1, 3, 2) followed by transferring the largest
  disk to peg 3 from peg 1 and calling THanoi (n-1,
  2, 1, 3)
• Stopping condition n 1

8
Towers of Hanoi

• class TowersOfHanoi
• public static void main (String arg)
• int n 10
• THanoi(n, 1, 2, 3)

9
Towers of Hanoi

• public static void THanoi (int n, int source,
  int extra, int destination)
• if (n gt 1)
• THanoi (n-1, source, destination,
  extra)
• System.out.println (Move disk n
  from peg source to peg destination)
• if (n gt 1)
• THanoi (n-1, extra, source,
  destination)
• // How many moves needed? 2n-1
• // end class

10
Towers of Hanoi

• Total number of method calls
• Let Tn be the number of method calls to solve for
  n disks
• Tn 2Tn-1 1 for n gt 1 T1 1
• Use generating function to solve the recurrence
  (worked out in class on board)

11
Fibonacci series

• Number of method calls
• Refer to the program in the last lecture
• Let Tn be the number of method calls to find the
  nth Fibonacci number
• Tn Tn-1 Tn-2 1 for n gt 2 T1 T2 1
• Use generating function to solve the recurrence
• Observe that Tn 2Fn 1 where Fn is the nth
  Fibonacci number
• Tn (21-n/v5)((1v5)n (1-v5)n) 1
• The number (1v5)/2 is called the golden ratio,
  which is lim n?8 (Fn1/Fn)