1. ITEC 2620M Introduction to Data Structures
Instructor: Prof. Z. Yang
Course Website:
Office: DB 3049

2. Complexity Analysis – Recurrence Relations

3. Key Points of this Lecture
Analysis of recursive algorithms
Recurrence relations
Quicksort, Mergesort
Best, Worst, and Average cases

4. Complexity Analysis
Determine how the processing time of an algorithm grows with input size.
What if the algorithm is recursive?
use recurrence relations
Recurrence relations
The problem is solved through smaller sub-problems.
The time to solve a problem is based on the time to solve smaller sub-problems.
Express the time to solve a problem of size n as T(n).

5. From Recursion to Recurrence Relations
Recursion has two cases
recursive case
base case
Recurrence relation needs two cases
recurring case
for complexity analysis, T(1) is usually 1
eliminate constant factors

6. Quicksort What is the complexity of Quicksort?
Is there a Best, Worst, and Average case?
Can Quicksort run faster or slower based on the input?
yes!
What is the best case?
pivot always splits the sub-groups exactly in half
What is the worst case?
pivot never makes sub-groups
all elements are always to one side or the other
What is the average case?
pivot splits the sub-groups loosely in half

7. Quicksort - Recurrence Relation
Best Case
The time to do a problem of size n is the time to do two sub-problems of size n/2 + n compares (during partition).
The time to do a problem of size 1 is 1 (a constant).
Worst Case
The time to do a problem of size n is the time to do a sub-problem of size (n-1 )+ n compares.
Average Case
O(nlogn)

8. Mergesort What is the complexity of Mergesort?
Is there a Best, Worst, and Average case?
Can Mergesort run faster or slower based on the input? No
Does Mergesort look at the values before splitting? No.
Does merge always process all values? Yes.
Complexity of Mergesort algorithm does not depend on input.
Recurrence relations
The time to do a problem of size n is the time to do two sub-problems of size n/2 + n compares (during merge).
The time to do a problem of size 1 is 1 (a constant).

9. Summary
Mergesort and Quicksort both have O(nlogn) in the best and average cases.
Mergesort is also O(nlogn) in the worst case.
Quicksort is O(n**2) in the worst case.
Worst case is unlikely, and constants are smaller for Quicksort.
Quicksort is the preferred algorithm for many applications
But not all!