1. Algorithm Design and Analysis (ADA)
, Semester
2.5. Insertion Sort
Objective
asymptotic analysis of insertion sort

2. Overview
1. What is Sorting? 2. Insertion Sort

3. 1. What is Sorting?
Input: sequence <a1, a2, …, an> of numbers. Output: permutation <a'1, a'2, …, a'n> such that a'1 ≤ a'2 ≤ … ≤ a'n Example: Input: Output:

4. non-obvious applications
Sorting is Essential
Sort a list of names.
Organize an MP3 library.
Display Google PageRank results.
List RSS feed in reverse chronological order.
Find the median.
Find the closest pair.
Binary search in a database.
Identify statistical outliers.
Find duplicates in a mailing list.
Data compression.
Computer graphics.
Computational biology.
Supply chain management.
Load balancing on a parallel computer.
. . .
obvious applications
problems become easy once items are in sorted order
non-obvious applications

5. Different Sorting Needs
Applications have different sorting needs:
Stable?
Parallel?
Deterministic?
Keys all distinct?
Multiple key types?
Linked list or arrays?
Large or small items?
Is your array randomly ordered?
Need guaranteed performance?

6. Many Different Sorting Algorithms
Internal sorts
Insertion sort, selection sort, bubblesort, shaker sort
Quicksort, mergesort, heapsort, samplesort, shellsort
Solitaire sort, red-black sort, splaysort, , ...
External sorts
Poly-phase mergesort, cascade-merge, oscillating sort
String/radix sorts
Distribution, MSD, LSD, 3-way string quicksort
Parallel sorts
Bitonic sort, Batcher even-odd sort
Smooth sort, cube sort, column sort
GPUsort

7. 2. Insertion Sort
sorted i j
key A: 1
length

8. Example of Insertion Sort8 2 4 9 3 6

9. Example of Insertion Sort8 2 4 9 3 6

10. Example of Insertion Sort8 2 4 9 3 6 2 8 4 9 3 6

11. Example of Insertion Sort8 2 4 9 3 6 2 8 4 9 3 6

12. Example of Insertion Sort8 2 4 9 3 6 2 8 4 9 3 6 2 4 8 9 3 6

13. Example of Insertion Sort8 2 4 9 3 6 2 8 4 9 3 6 2 4 8 9 3 6

14. Example of Insertion Sort8 2 4 9 3 6 2 8 4 9 3 6 2 4 8 9 3 6 2 4 8 9 3 6

15. Example of Insertion Sort8 2 4 9 3 6 2 8 4 9 3 6 2 4 8 9 3 6 2 4 8 9 3 6

16. Example of Insertion Sort8 2 4 9 3 6 2 8 4 9 3 6 2 4 8 9 3 6 2 4 8 9 3 6 2 3 4 8 9 6

17. Example of Insertion Sort8 2 4 9 3 6 2 8 4 9 3 6 2 4 8 9 3 6 2 4 8 9 3 6 2 3 4 8 9 6

18. Example of Insertion Sort8 2 4 9 3 6 2 8 4 9 3 6 2 4 8 9 3 6 2 4 8 9 3 6 2 3 4 8 9 6 2 3 4 6 8 9
done

19. Insertion Sort Java Code
void insertionSort(int[] A) // A[0 .. n-1] {
(1) for (int j = 1; j < A.length; j++) { // start with 1 (not 2)
(2) int key = A[j];
(3) int i = j - 1;
while((i >= 0) && (A[i] < key)) {
A[i+1] = A[i];
(6) i--;
(7) }
(8) A[i+1] = key;
(9) } }

20. Insertion Sort Structure Tree
for 1-9
block 1-9 2 3
while 4-7 8
block 4-7 5 6

21. Lines 2, 3, 5, 6, 8: each is O(1) Block of 4-7 For of 4-7 is:
= O(1) + O(1) = O(1)
For of 4-7 is:
= O( (n-1) * 1) = O(n-1)
= O(n), simplified
Block of 1-9
= O(1) + O(1) + O(n) + O(1) = O(n)
For of 1-8 is:
= O( (n-1) * n) = O(n2 - n)
= O(n2), simplified
this is the hard part –
assume the worse case where the loop has to move the most elements)

22. Analyzing Insertion Sort
What can T() be?
Best case -- inner loop body never executed
 T(n) is a linear function
Worst case -- inner loop body executed for all previous elements
 T(n2) is a quadratic function
Average case
tricky