11-Sorting - 6/15/2011 InsertionSort CS/ENGRD 2110...

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6/15/2011 1 CS/ENGRD 2110 Object-Oriented Programming and Data Structures Spring 2011 Thorsten Joachims Lecture 11: Sorting InsertionSort Many people sort cards this way Invariant: everything to left of i is already sorted Worst-case is O(n 2 ) Consider reverse-sorted input Best-case is O(n) Consider sorted input Expected case is O(n 2 ) Expected number of inversions is n(n 1)/4 2 //sort a[], an array of int for (int i = 1; i < a.length; i++) { int temp = a[i]; int k; 1]; k –– ) a[k] = a[k 1]; a[k] = temp; } SelectionSort To sort an array of size n: Examine a[0] to a[n 1]; find the smallest one and swap it with a[0] Examine a[1] to a[n 1]; find the smallest one and swap it with a[1] In general, in step i, examine a[i] to a[n 1]; find the smallest one and swap it with a[i] This is the other common way for people to sort cards Runtime Worst-case O(n 2 ) Best-case O(n 2 ) Expected-case O(n 2 ) 3 It often pays to Break the problem into smaller subproblems, Solve the subproblems separately, and then Assemble a final solution This technique is called divide-and-conquer Caveat: It won’t help unless the partitioning and assembly processes are inexpensive Can we apply this approach to sorting? 4 MergeSort Quintessential divide-and-conquer algorithm Divide array into equal parts, sort each part, then merge Questions: Q1: How do we divide array into two equal parts? A1: Find middle index: a.length/2 Q2: How do we sort the parts? A2: call MergeSort recursively! Q3: How do we merge the sorted subarrays? A3: We have to write some (easy) code 5 Merging Sorted Arrays A and B Create an array C of size = size of A + size of B Keep three indices: i into A j into B k into C Initialize all three indices to 0 (start of each array) Compare element A[i] with B[j], and move the smaller element into C[k] Increment i or j, whichever one we took, and k When either A or B becomes empty, copy remaining elements from the other array (B or A, respectively) into C 6
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11-Sorting - 6/15/2011 InsertionSort CS/ENGRD 2110...

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