Algorithm Design and Analysis (ADA) 242-535, Semester 1 2017-2018 2.5. Insertion Sort Objective asymptotic analysis of insertion sort

Overview 1. What is Sorting? 2. Insertion Sort

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

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

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?

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

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

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

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) } }

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

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)

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