Recurrence Relations

1
Lecture 4

• Recurrence Relations

2
Examples

• Binary Search recurrence relation p. 21 in the
  book
• Towers of Hanoi recurrence relation ch. 2, p.
  18-21 in the book
• Merge Sort recurrence relation ch. 9.3, p.
  215-218 in the book
• Master Theorem

3
Administrative

• Irenas office hours Friday 230-430
• If time allows, at the end of class Ill take
  questions from the homework
• Do some limits using lHospital Rule, taking the
  derivative of a product, ln x
• More proofs by induction (ex. 2 ch 1 book)

4
Analysis of Recursive AlgorithmsRecurrence
Relations

• Binary Search worst case analysis
• T(n) T(n/2)1, T(1)1
• Assume n2k (i.e. k log2n)
• T(2k)T(2 k-1)1T(2k-2) 2
• T(2k-i)i T(20)kk1
• log 2n 1 O(log n)
• T(n) O(log n)

5
Towers of Hanoi

• Show algorithm. End of the World legend.
• T(n)2T(n-1)1 and T(0)0
• T(n)2T(n-1)12(2T(n-2)1)1
• 22 T(n-2)(2 1)
• 22 (2 T(n-3) 1) 2 1 23 T(n-3) (22 2
  1)
• . 2i T(n-i) (2i-1 2 1)
• 2n T(n-n) (2n-1 2 1)
• 2n T(0) 2n - 1 2n - 1 O(2n)

6
Merge Sort

• Input array A of n numbers
• Output array in sorted order
• Algorithm
• divide the array in 2 (left and right)
• recursively sort left and right arrays
• merge the tow sorted arrays into a single one

7
Analysis of Merge Sort

• T(n) 2T(n/2) n
• 2(2T(n/22) n/2) n
• 22 T(n/22) n n 2 2 T(n/ 22) 2n
  
• 22 (2 T((n/22)/2) n/22) 2n
• 23 T(n/23) 3n
• 2k T(n/2k) kn.
• For n 2k (k log2 n) and T(1) 0 we get
• T(n)n log2 n O(n log n)

8
Administrative

• Irenas office hours Friday 230-430
• Questions from the homework
• Do some limits using lHospital Rule, taking the
  derivative of a product, ln x
• More proofs by induction (ex. 2 ch 1 book)