Algorithm Design and Analysis (ADA)
242-535, Semester 1 2014-2015
3. Insertion Sort
• Objective
o asymptotic analysis
of insertion sort
242-535 ADA: 3. Insertion Sort
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Overview
1. What is Sorting?
2. Insertion Sort
242-535 ADA: 3. Insertion Sort
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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:
242-535 ADA: 3. Insertion Sort
8 2 4 9 3 6
2 3 4 6 8 9
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Sorting is Essential
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
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.
...
242-535 ADA: 3. Insertion Sort
obvious applications
problems become easy once
items are in sorted order
non-obvious applications
4
Different Sorting Needs
• Applications have different sorting needs:
o
o
o
o
o
o
o
o
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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?
242-535 ADA: 3. Insertion Sort
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Many Different Sorting Algorithms
• Internal sorts
o Insertion sort, selection sort, bubblesort, shaker sort
o Quicksort, mergesort, heapsort, samplesort, shellsort
o Solitaire sort, red-black sort, splaysort, , ...
• External sorts
o Poly-phase mergesort, cascade-merge, oscillating sort
• String/radix sorts
o Distribution, MSD, LSD, 3-way string quicksort
• Parallel sorts
o Bitonic sort, Batcher even-odd sort
o Smooth sort, cube sort, column sort
o GPUsort
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2. Insertion Sort
INSERTION-SORT (A, n)
⊳ A[1 . . n]
for j ← 2 to n
do key ← A[ j]
i←j–1
while i > 0 and A[i] > key
do A[i+1] ← A[i]
i←i–1
A[i+1] = key
“pseudocode”
1
i
j
A:
sorted
key
n
Example of Insertion Sort
8
2
4
9
3
6
Example of Insertion Sort
8
2
4
9
3
6
Example of Insertion Sort
8
2
4
9
3
6
2
8
4
9
3
6
Example of Insertion Sort
8
2
4
9
3
6
2
8
4
9
3
6
Example of Insertion Sort
8
2
4
9
3
6
2
8
4
9
3
6
2
4
8
9
3
6
Example of Insertion Sort
8
2
4
9
3
6
2
8
4
9
3
6
2
4
8
9
3
6
Example of Insertion Sort
8
2
4
9
3
6
2
8
4
9
3
6
2
4
8
9
3
6
2
4
8
9
3
6
Example of Insertion Sort
8
2
4
9
3
6
2
8
4
9
3
6
2
4
8
9
3
6
2
4
8
9
3
6
Example of Insertion Sort
8
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
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9
6
Example of Insertion Sort
8
2
4
9
3
6
2
8
4
9
3
6
2
4
8
9
3
6
2
4
8
9
3
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2
3
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9
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Example of Insertion Sort
8
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
Insertion Sort Java Code
void insertionSort(int[] A)
// A[0 .. n-1]
{
(1) for (int j = 1; j < num.length; j++) { // start with 1 (not 2)
(2)
int key = num[j];
(3)
int i = j - 1;
(4)
while((i >= 0) && (A[i] < key)) {
(5)
A[i+1] = A[i];
(6)
i--;
(7)
}
(8)
A[i+1] = key;
(9) }
}
242-535 ADA: 3. Insertion Sort
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Insertion Sort Structure Tree
for
1-9
block
1-9
2
while
4-7
3
8
block
4-7
5
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• Lines 2, 3, 5, 6, 8: each is O(1)
• Block of 4-7
= O(1) + O(1) = O(1)
• For of 4-7 is:
this is the hard part –
assume the worse case where
the loop has to move the
most elements)
= 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
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Analyzing Insertion Sort
• What can T() be?
o Best case -- inner loop body never executed
•  T(n) is a linear function
o Worst case -- inner loop body executed for all previous
elements
•  T(n2) is a quadratic function
o Average case
• tricky
242-535 ADA: 3. Insertion Sort
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