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13 Searching - 1 SORTING AND ASYMPTOTIC COMPLEXITY Lecture...

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Unformatted text preview: 9/30/2010 1 SORTING AND ASYMPTOTIC COMPLEXITY Lecture 13 CS2110 – Fall 2009 InsertionSort 2 //sort a, an array of int for (int i = 1; i < a.length; i++) { int temp = a[i]; int k; for (k = i; 0 < k && temp < a[k–1]; k––) a[k] = a[k–1]; ¡ Many people sort cards this way ¡ Invariant: everything to left of i is already sorted ¡ Works especially well when input is nearly 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 a[k] = temp; } SelectionSort 3 ¡ To sort an array of size n: ¡ Examine a[0] to a[n–1] ; find the smallest one and swap it with a[0] This is the other common way for people to sort cards Runtime ¡ 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] ¢ Worst-case O(n 2 ) ¢ Best-case O(n 2 ) ¢ Expected-case O(n 2 ) Divide & Conquer? 4 ¡ It often pays to ¡ Break the problem into smaller subproblems, ¡ Solve the subproblems separately, and then ¡ Assemble a final solution ¡ 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? MergeSort 5 ¡ 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 Merging Sorted Arrays A and B 6 ¡ 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 (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 (...
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13 Searching - 1 SORTING AND ASYMPTOTIC COMPLEXITY Lecture...

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