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Naïve Sorting Algorithms Bubble Sort Insertion Sort 11/4/2010 Naïve Sorting Algorithms There are three standard algorithms Bubble or exchange sort Naïve Sorts Insertion Sort Selection Sort Naïve sorting algorithms Bubble Sort Bubble sort: scan for flips, until all are fixed Big Idea: Compare adjacent entries and swap when A[i] > A[i+1] 3 2 1 6 5 4 Example: 2 3 1 6 5 4 2 1 3 6 5 4 2 1 3 5 6 4 2 1 3 5 4 6 Etc... Naïve Sorting Insertion Sort for i=1 to n-1 Big Idea: Assume the list [0,1…,i-1] is sorted. Insert A[i] { for j=0 to n-i-1 if (A[j] > A[j+1]) element to right place. swap(A[j], A[j+1]); Example: if (no swaps) break; } What is the complexity of the algorithm, if All keys are equal? Keys are sorted in reverse order? Keys are sorted? keys are randomly distributed? 1 11/4/2010 Insertion sort Insertion sort Sorted subarray • Algorithm 105 47 13 99 30 222 for i = 1 to n-1 do 47 105 13 99 30 222 insert a[i] in the proper place 13 47 105 99 30 222 in a[0:i-1] • Correctness 13 47 99 105 30 222 •Consider the Loop Invariant: after i steps, the sub-array A[0:i] is 13 30 47 99 105 222 sorted •Show that loop invariant holds at the beginning and is true at the end 105 47 13 99 30 222 of the loop •More on correctness of algorithms later Selection Sort Complexity of Naïve Sorts Big Idea: Find the max of the list and exchange with Analysis the last element Bubble Example: Insertion Selection Divide-and-conquer Merge Sort 2 11/4/2010 Merging Two Sorted Arrays All the work in merge sort is done at the merge step. Example Quick Sort 1 13 24 26 2 15 27 38 Quicksort idea Quicksort Quicksort was invented in 1960 by Tony Hoare. Choose a pivot. Quicksort has O(N 2) worst-case performance, and on average O(N log N). Rearrange so that More importantly, it is the fastest known comparison- pivot is in the “right” based sorting algorithm in practice . spot. Recurse on each half and conquer! Quicksort algorithm Quicksort105 47algorithm13 17 30 222 5 19 If array A has 1 (or 0) elements, then done. Choose a pivot element x from A. 19 5 17 13 47 30 222 105 Divide A-{x} into two arrays: B = {y ∈A | y≤x} 13 47 ∈ ≥ C = {y A | y x} 5 17 30 222 105 Result is B+{x}+C. Recurse on arrays B and C 105 222 3 11/4/2010 Place the pivot algorithm Example Assume we have an array of size n Show how to place the pivot (choose middle element) Pick a pivot x in the right place Place the pivot x in the last place of the array 34 24 56 17 19 45 90 23 36 Have two pointers i = 0, j = n-2 while ( i <= j) { while (A[j] > pivot) j--; while (A[i] < pivot) i--; swap (A[i], A[j]); } swap (A[i], pivot); Naïve sorting algorithms Bubble sort: scan for flips, until all are fixed 3 2 1 6 5 4 2 3 1 6 5 4 Ananda Gunawardena 2 1 3 6 5 4 2 1 3 5 6 4 2 1 3 5 4 6 Etc... Insertion sort Naïve Sorting Sorted subarray for i=1 to n-1 105 47 13 99 30 222 { for j=0 to n-i-1 if (A[j].compareTo(A[j+1])>0) 47 105 13 99 30 222 swap(A[j], A[j+1]); if (no swaps) break; 13 47 105 99 30 222 } What happens if 13 47 99 105 30 222 All keys are equal? Keys are sorted in reverse order? 13 30 47 99 105 222 Keys are sorted? keys are randomly distributed? 105 47 13 99 30 222 Exercise: Count the number of operations in bubble sort and find a Big O analysis for bubble sort 4 11/4/2010 Insertion sort How fast is insertion sort? • Algorithm To insert a[i] into a[0:i-1], slide all elements larger for i = 1 to n-1 do than a[i] to the right. insert a[i] in the proper place tmp = a[i]; in a[0:i-1] for (j = i; j>0 && a[j-1]>tmp; j--) • Correctness a[j] = a[j-1]; •Note: after i steps, the sub-array A[0:i] is sorted a[j] = tmp; # of slides = O(#inversions) very fast if array is nearly sorted to begin with Selection sort Sorting Comparison • Algorithm for i = n-1 to 1 do Discuss the pros and cons of each of the naïve sorting Find the largest entry in the algorithms in the subarray A[0:i] Swap with A[i] What is the runtime complexity of selection sort? Quick Fastest algorithm Sort in practice Algorithm Find a pivot Move all elements smaller than pivot to left Advanced Sorting Move all elements bigger than pivot to right Recursively sort each half O(n log n) algorithm 5 11/4/2010 Merge Sort Heap Sort Divide the array into two equal halves Build a max heap Divide each half recursively until each array is of size 1 Delete Max (attach to end of array) until heap is empty Merge two (sorted) arrays of size 1 Resulting array is sorted Complete the process recursively Complexity Bucket sort In addition to comparing pairs of elements, we require these additional restrictions: all elements are non-negative integers Bucket Sort all elements are less than a predetermined maximum value Elements are usually keys paired with other data Bucket sort Bucket sort characteristics 1 3 3 1 2 Runs in O(N) time. Easy to implement each bucket as a linked list. Is stable : If two elements (A,B) are equal with respect to sorting, and they appear in the input in order (A,B), then they remain in the same order in the output. 1 2 3 6 11/4/2010 Radix sort If your integers are in a larger range then do bucket sort on each digit Radix Sort Start by sorting with the low-order digit using a STABLE bucket sort. Then, do the next-lowest,and so on Radix sort Radix sort characteristics Example: Each sorting step can be performed via bucket sort, and is thus O(N). 2 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 5 1 0 1 1 0 0 1 0 1 0 1 0 2 If the numbers are all b bits long, then there are b 1 0 0 1 1 1 0 0 0 1 0 1 1 3 sorting steps. 7 1 1 1 1 0 1 0 1 0 1 0 0 4 3 0 1 1 0 0 1 1 1 0 1 0 1 5 Hence, radix sort is O(bN). 4 1 0 0 1 1 1 1 1 1 1 1 0 6 6 1 1 0 0 1 1 0 1 1 1 1 1 7 Each sorting step must be stable. What Radix sort about can be used non-binary? for decimal numbers and alphanumeric strings. 0 3 2 0 3 1 0 1 5 0 1 5 2 2 4 0 3 2 0 1 6 0 1 6 0 1 6 2 5 2 1 2 3 0 3 1 0 1 5 1 2 3 2 2 4 0 3 2 0 3 1 2 2 4 0 3 1 1 2 3 1 6 9 0 1 5 0 3 2 1 6 9 1 2 3 0 1 6 2 5 2 2 2 4 2 5 2 1 6 9 1 6 9 2 5 2 7.
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