1. Quicksort analysis Bubble sort
Sorting
Quicksort analysis
Bubble sort
November 20, 2017
Hassan Khosravi / Geoffrey Tien

2. Hassan Khosravi / Geoffrey Tien
Quicksort analysis
How long does Quicksort take to run?
Let's consider the best and the worst case
These differ because the partitioning algorithm may not always do a good job
Let's look at the best case first
Each time a sub-array is partitioned the pivot is the exact midpoint of the slice (or as close as it can get)
So it is divided in half
What is the running time?
November 20, 2017
Hassan Khosravi / Geoffrey Tien

3. Hassan Khosravi / Geoffrey Tien
Quicksort best case
Running time
Each sub-array is divided in half in each partition
Each time a series of sub-arrays are partitioned 𝑛 (approximately) comparisons are made
The process ends once all the sub-arrays left to be partitioned are of size 1
How many times does 𝑛 have to be divided in half before the result is 1?
log 2 𝑛 times
Quicksort performs 𝑛∙ log 2 𝑛 operations in the best case
Same complexity as Merge sort
November 20, 2017
Hassan Khosravi / Geoffrey Tien

4. Hassan Khosravi / Geoffrey Tien
Quicksort worst case
Every partition step ends with no values on one side of the pivot
The array has to be partitioned 𝑛 times, not log 2 𝑛 times
𝑛 comparisons in the first partition step…
𝑛−1 comparisons in the second step…
𝑛−2 comparisons in the third step… …
𝑛+ 𝑛−1 + 𝑛−2 +…+2+1= 𝑛(𝑛−1) 2
So in the worst case Quicksort performs around 𝑛 2 operations
The worst case usually occurs when the array is nearly sorted (in either direction)
As bad as selection sort!
November 20, 2017
Hassan Khosravi / Geoffrey Tien

5. Quicksort average case
With a large array we would have to be very, very unlucky to get the worst case
Unless there was some reason for the array to already be partially sorted
The average case is much more like the best case than the worst case
There is an easy way to fix a partially sorted array to that it is ready for Quicksort
Randomize the positions of the array elements!
What is the complexity of performing a random scramble of the array?
November 20, 2017
Hassan Khosravi / Geoffrey Tien

6. Quicksort average case
Intuition
Let’s assume that pivot choice is random
Half the time the pivot will be from the centre half of the array
Thus at worst the split will be 𝑛 4 and 3𝑛 4
We can apply this to the notion of a good split
Every "good" split: 2 partitions of size 𝑛/4 and 3𝑛/4
Or, divides 𝑛 by 4/3
Expected number of partitions is at most 2∙ log 4/3 𝑛
𝑂 log 𝑛
Given 𝑛 comparisons at each partitioning step, we have Θ 𝑛 log 𝑛
Same best-case complexity as Merge sort, but in practice, Quicksort tends to be slightly faster. Why?
November 20, 2017
Hassan Khosravi / Geoffrey Tien

7. Merge sort vs Quicksort
and Quicksort summary
If Quicksort worst case is so bad, why use it?
worst case is exceedingly rare and can be easily avoided
in practice, faster than Merge sort.
can be sorted in-place using 𝑂 log 𝑛 stack space
also highly parallelizable
Name
Best
Average
Worst
Stable
Memory
Selection sort
𝑂 𝑛 2
challenging
𝑂 1
Insertion sort
𝑂 𝑛
Yes
Merge sort
𝑂 𝑛 log 𝑛
Quicksort
𝑂 log 𝑛
November 20, 2017
Hassan Khosravi / Geoffrey Tien

8. Hassan Khosravi / Geoffrey Tien
Bubble sort
Bubble sort is a simple sorting algorithm with a good best case performance, but generally poor performance in average and worst case
Scans the array from left to right
checks adjacent pairs and swaps if out of order
If any swap was performed during a scan, re-scan the array
The next largest item is put into place during each scan, so each scan does not need to go all the way to the end of the array
Array is sorted if a scan can complete without performing any swaps
November 20, 2017
Hassan Khosravi / Geoffrey Tien

9. Hassan Khosravi / Geoffrey Tien
Bubble sort
Example
swapped =
TRUE
FALSE 6 5 3 1 8 7 2 4 i
swapped =
FALSE
TRUE 5 3 1 6 7 2 4 8 i
swapped =
TRUE
FALSE 3 1 5 6 2 4 7 8 i
swapped =
TRUE
FALSE 1 3 5 2 4 6 7 8 i
swapped =
FALSE
TRUE 1 3 2 4 5 6 7 8 i
swapped =
TRUE
FALSE 1 2 3 4 5 6 7 8
Completed scan with no swaps i
November 20, 2017
Hassan Khosravi / Geoffrey Tien

10. Hassan Khosravi / Geoffrey Tien
Bubble sort algorithm
#define TRUE 1
#define FALSE 0
void bubbleSort(int arr[], int size) {
int i, j, temp;
int swapped = FALSE;
for (i = 0; i < n; i++) {
swapped = FALSE; // reset flag for this scan iteration
for (j = 0; j < n-i-1; j++) {
if (arr[j] > arr[j+1]) {
temp = arr[j+1];
arr[j+1] = arr[j];
arr[j] = temp;
swapped = TRUE; // a swap occurred during this scan }
if (!swapped)
return;
November 20, 2017
Hassan Khosravi / Geoffrey Tien

11. Hassan Khosravi / Geoffrey Tien
Bubble sort analysis
Inner loop increments and swaps
Outer loop repeats if a swap occurred
if no swap occurred, algorithm terminates
Best case – everything already in order!
An item out of place can only move to the left by one position per outer loop iteration
if the smallest item is at the rightmost position, it requires 𝑛−1 outer loop iterations – worst case!
Average case is more similar to worst case
𝑂 𝑛
𝑂 𝑛 2
November 20, 2017
Hassan Khosravi / Geoffrey Tien

12. When do we use Bubble sort
Rarely, in practice
When we want to show some "bad" (but easy to implement) sorting algorithms for comparison in a second-year computer science class
but it is far from the worst, e.g. Bogosort
Name
Best
Average
Worst
Stable
Memory
Selection sort
𝑂 𝑛 2
challenging
𝑂 1
Insertion sort
𝑂 𝑛
Yes
Merge sort
𝑂 𝑛 log 𝑛
Quicksort
𝑂 log 𝑛
Bubble sort
Heapsort
𝑂 𝑛 log 𝑛 No
𝑂 1
November 20, 2017
Hassan Khosravi / Geoffrey Tien

13. Hassan Khosravi / Geoffrey Tien
Exercise
Suppose we implemented Merge sort, to split the array into 3 (roughly) equally-sized subarrays, and the Merge function repeatedly copies in the smallest of the three values in each subarray index
Perform a recurrence analysis to see how the (asymptotic) running time will be affected.
For the following sequence of values, perform a single round of Quicksort partition using the implementation described in these notes; identify which element becomes the pivot value
9, 6, 7, 17, 11, -1, -5, 19, 1, 2
November 20, 2017
Hassan Khosravi / Geoffrey Tien

14. Readings for this lesson
Thareja
Chapter 14.11, 14.7 (Quicksort, Bubble sort)
Next class:
Thareja Chapter 15.1 – 15.3 (Hash tables, hash functions)
November 20, 2017
Hassan Khosravi / Geoffrey Tien