Sorting

CSE 225 Data Structures and Algorithms

Sorting
Outline
 We are going to look at three simple sorting techniques:
Bubble Sort, Selection Sort and Insertion Sort.
 We will write code for Bubble Sort, Selection Sort and
Insertion Sort. We will analyze the running time for
each of the above.

2
Bubble Sort
 Compare each element (except the last one) with its
neighbor to the right
 If they are out of order, swap them
 This puts the largest element at the very end
 The last element is now in the correct and final place
 Compare each element (except the last two) with its
neighbor to the right
 If they are out of order, swap them
 This puts the second largest element next to last
 The last two elements are now in their correct and final places
 Compare each element (except the last three) with its
neighbor to the right
 Continue as above until you have no unsorted elements on the left

3
Example of Bubble Sort

7 2 8 5 4 2 7 5 4 8 2 5 4 7 8 2 4 5 7 8
2 7 8 5 4 2 7 5 4 8 2 5 4 7 8 2 4 5 7 8

2 7 8 5 4 2 5 7 4 8 2 4 5 7 8 (done)
2 7 5 8 4 2 5 4 7 8
2 7 5 4 8

4
Code for Bubble Sort
 void bubbleSort(int a[], int size) {
int outer, inner;
for (outer = size - 1; outer > 0; outer--) { // counting down
for (inner = 0; inner < outer; inner++) { // bubbling up
if (a[inner] > a[inner + 1]) { // if out of order...
int temp = a[inner]; // ...then swap
a[inner] = a[inner + 1];
a[inner + 1] = temp;
}
}
}
}

5
Analysis of Bubble Sort
 for (outer = size - 1; outer > 0; outer--) {
for (inner = 0; inner < outer; inner++) {
if (a[inner] > a[inner + 1]) {
// code for swap omitted
} } }
 Let n = size of the array
 The outer loop is executed n-1 times (call it n, that’s close enough)
 Each time the outer loop is executed, the inner loop is executed
 Inner loop executes n-1 times at first, linearly dropping to just once

 On average, inner loop executes about n/2 times for each execution

of the outer loop

 In the inner loop, the comparison is always done (constant time), the

swap might be done (also constant time)

 Result is n * n/2, that is, O(n2/2) = O(n2)

6
Another sort: Selection Sort
 Given an array of length n,
 Search elements 0 through n-1 and select the smallest
 Swap it with the element in location 0
 Search elements 1 through n-1 and select the smallest
 Swap it with the element in location 1
 Search elements 2 through n-1 and select the smallest
 Swap it with the element in location 2
 Search elements 3 through n-1 and select the smallest
 Swap it with the element in location 3
 Continue in this fashion until there’s nothing left to search

7
Example and analysis of Selection Sort

7 2 8 5 4
 The Selection Sort might swap an
array element with itself--this is
2 7 8 5 4
harmless, and not worth checking for
 Analysis:
2 4 8 5 7  The outer loop executes n-1 times
 The inner loop executes about n/2
2 4 5 8 7 times on average (from n to 2 times)
 Work done in the inner loop is constant
2 4 5 7 8 (swap two array elements)
 Time required is roughly (n-1)*(n/2)
 You should recognize this as O(n2)
8
Code for Selection Sort
void selectionSort(int a[], int size) {
int outer, inner, min;
for (outer = 0; outer < size - 1; outer++) { // outer counts down
min = outer;
for (inner = outer + 1; inner < size; inner++) {
if (a[inner] < a[min]) {
min = inner;
}
}
// a[min] is least among a[outer]..a[size - 1]
int temp = a[outer];
a[outer] = a[min];
a[min] = temp;
}
}

9
One step of insertion sort

sorted next to be inserted

3 4 7 12 14 14 20 21 33 38 10 55 9 23 28 16
temp
less than 10 10
3 4 7 10 12 14 20
12 14 14 21 21 38
20 33 33 10
38 55 9 23 28 16

sorted

10
Code for Insertion Sort
void insertionSort(int a[], int size) {
int inner, outer;

for (outer = 1; outer < size; outer++) {

int temp = a[outer];
inner = outer;
while (inner > 0 && a[inner - 1] >= temp) {
a[inner] = a[inner - 1];
inner--;
}
a[inner] = temp;
}
}

11
Insertion Sort – Analysis
 Running time depends on not only the size of the array
but also the contents of the array.
 Best-case:  O(n)
 Array is already sorted in ascending order.
 Inner loop will not be executed.
 Worst-case:  O(n2)
 Array is in reverse order:
 Inner loop is executed i-1 times, for i = 2,3, …, n
 Average-case:  O(n2)
 We have to look at all possible initial data organizations.
 So, Insertion Sort is O(n2)
Summary
 Bubble Sort, Selection Sort, and Insertion Sort are all O(n2)
 As we will see later, we can do much better than this with
somewhat more complicated sorting algorithms
 Within O(n2),
 Bubble Sort is very slow, and should probably never be used for anything
 Selection Sort is intermediate in speed
 Insertion Sort is usually the fastest of the three--in fact, for small arrays
(say, 10 or 15 elements), insertion sort is faster than more complicated
sorting algorithms
 Selection Sort and Insertion Sort are “good enough” for small
arrays

13
Divide and Conquer

 Recursive in structure
 Divide the problem into sub-problems that are similar
to the original but smaller in size
 Conquer the sub-problems by solving them recursively.
If they are small enough, just solve them in a
straightforward manner.
 Combine the solutions to create a solution to the
original problem
An Example: Merge Sort

Sorting Problem: Sort a sequence of n elements into

non-decreasing order.

 Divide: Divide the n-element sequence to be

sorted into two subsequences of n/2 elements each
 Conquer: Sort the two subsequences recursively
using merge sort.
 Combine: Merge the two sorted subsequences to
produce the sorted answer.
Merge Sort – Example
Original Sequence Sorted Sequence
18 26 32 6 43 15 9 1 1 6 9 15 18 26 32 43

18 26 32 6 43 15 9 1 6 18 26 32 1 9 15 43
43

18 26 32 6 43 15 9 1 18 26 6 32 15 43 1 9

18 26 32 6 43 15 9 1 18 26 32 6 43 15 9 1

18 26 32 6 43 15 9 1
Merge-Sort (A, p, r)

INPUT: a sequence of n numbers stored in array A

OUTPUT: an ordered sequence of n numbers

MergeSort (A, p, r) // sort A[p..r] by divide & conquer

1 if p < r
2 then q  (p+r)/2
3 MergeSort (A, p, q)
4 MergeSort (A, q+1, r)
5 Merge (A, p, q, r) // merges A[p..q] with A[q+1..r]

Initial Call: MergeSort(A, 1, n)

Procedure Merge

Merge(A, p, q, r)
1 Copy two sorted sub arrays of A into L and R
2 i1
3 j1 Input: Array containing
4 for k p to r sorted subarrays A[p..q] and
5 do if L[i]  R[j] A[q+1..r].
6 then A[k]  L[i]
Output: Merged sorted
7 ii+1
subarray in A[p..r].
8 else A[k]  R[j]
9 jj+1
Merging two sorted arrays

20 12
13 11
7 9
2 1
Merging two sorted arrays

20 12
13 11
7 9
2 1

1
Merging two sorted arrays

20 12 20 12
13 11 13 11
7 9 7 9
2 1 2

1
Merging two sorted arrays

20 12 20 12
13 11 13 11
7 9 7 9
2 1 2

1 2
Merging two sorted arrays

20 12 20 12 20 12
13 11 13 11 13 11
7 9 7 9 7 9
2 1 2

1 2
Merging two sorted arrays

20 12 20 12 20 12
13 11 13 11 13 11
7 9 7 9 7 9
2 1 2

1 2 7
Merging two sorted arrays

20 12 20 12 20 12 20 12
13 11 13 11 13 11 13 11
7 9 7 9 7 9 9
2 1 2

1 2 7
Merging two sorted arrays

20 12 20 12 20 12 20 12
13 11 13 11 13 11 13 11
7 9 7 9 7 9 9
2 1 2

1 2 7 9
Merging two sorted arrays

20 12 20 12 20 12 20 12 20 12
13 11 13 11 13 11 13 11 13 11
7 9 7 9 7 9 9
2 1 2

1 2 7 9
Merging two sorted arrays

20 12 20 12 20 12 20 12 20 12
13 11 13 11 13 11 13 11 13 11
7 9 7 9 7 9 9
2 1 2

1 2 7 9 11
Merging two sorted arrays

20 12 20 12 20 12 20 12 20 12 20 12
13 11 13 11 13 11 13 11 13 11 13
7 9 7 9 7 9 9
2 1 2

1 2 7 9 11
Merging two sorted arrays

20 12 20 12 20 12 20 12 20 12 20 12
13 11 13 11 13 11 13 11 13 11 13
7 9 7 9 7 9 9
2 1 2

1 2 7 9 11 12
Merging two sorted arrays

20 12 20 12 20 12 20 12 20 12 20 12
13 11 13 11 13 11 13 11 13 11 13
7 9 7 9 7 9 9
2 1 2

1 2 7 9 11 12

Time = (n) to merge a total of n elements

(linear time).