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Understanding Insertion Sort: Analysis and Applications

This document provides an overview of the Insertion Sort algorithm, which is a fundamental sorting technique used in computer science. It begins with an explanation of sorting and proceeds to detail the mechanics of Insertion Sort, including pseudocode and elucidative examples. The text explores various applications of sorting algorithms, emphasizing the significance of sorting in different fields such as data analysis, computer graphics, and supply chain management. Additionally, the document discusses performance analysis, highlighting the best, worst, and average case scenarios for Insertion Sort.

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Understanding Insertion Sort: Analysis and Applications

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  1. Algorithm Design and Analysis (ADA) 242-535, Semester 1 2013-2014 • Objective • asymptotic analysis of insertion sort 3. 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:8 2 4 9 3 6 Output:2 3 4 6 8 9

  4. 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 INSERTION-SORT (A, n) ⊳ A[1 . . n] forj ← 2ton dokey ← A[ j] i ← j – 1 while i > 0 and A[i] > key doA[i+1] ← A[i] i ← i – 1 A[i+1] = key “pseudocode” 1 i j n A: key sorted

  8. Example of Insertion Sort 8 2 4 9 3 6

  9. Example of Insertion Sort 8 2 4 9 3 6

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

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

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

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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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; • 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 for1-9 block1-9 while4-7 2 3 8 block4-7 5 6

  21. this is the hard part – assume the worse case wherethe loop has to move the most elements) • 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

  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

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