242-535 ADA: 3. Insertion Sort

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• Objectiveo asymptotic analysis

of insertion sort

Algorithm Design and Analysis (ADA)

242-535, Semester 1 2014-2015

3. Insertion Sort

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1. What is Sorting?2. Insertion Sort

Overview

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Input: sequence <a1, a2, …, an> of numbers.

Output: permutation <a'1, a'2, …, a'n> suchthat a'1 ≤ a'2 ≤ … ≤ a'n

Example:Input: 8 2 4 9 3 6Output: 2 3 4 6 8 9

1. What is Sorting?

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o Sort a list of names.o Organize an MP3 library.o Display Google PageRank results.o List RSS feed in reverse chronological order.o Find the median. o Find the closest pair.o Binary search in a database.o Identify statistical outliers.o Find duplicates in a mailing list.o Data compression.o Computer graphics. o Computational biology.o Supply chain management.o Load balancing on a parallel computer.o . . .

Sorting is Essentialobvious applications

problems become easy once items are in sorted order

non-obvious applications

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

Different Sorting Needs

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• Internal sortso Insertion sort, selection sort, bubblesort, shaker sorto Quicksort, mergesort, heapsort, samplesort, shellsorto Solitaire sort, red-black sort, splaysort, , ...

• External sortso Poly-phase mergesort, cascade-merge, oscillating sort

• String/radix sortso Distribution, MSD, LSD, 3-way string quicksort

• Parallel sortso Bitonic sort, Batcher even-odd sorto Smooth sort, cube sort, column sorto GPUsort

Many Different Sorting Algorithms

2. Insertion SortINSERTION-SORT (A, n) ⊳ A[1 . . n]

for j ← 2 to ndo key ← A[ j]

i ← j – 1while i > 0 and A[i] > key

do A[i+1] ← A[i]i ← i – 1

A[i+1] = key

“pseudocode”

sorted

i j

keyA:

1 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 4 8 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 6

2 3 4 8 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 6

2 3 4 8 9 6

2 3 4 6 8 9 done

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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) }}

Insertion Sort Java Code

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Insertion Sort Structure Tree

for1-9

3

block1-9

while4-72

5

8

block4-7

6

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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:= 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 wherethe loop has to move the most elements)

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• 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

Analyzing Insertion Sort