Faster Sorting Methods

1. Faster Sorting Methods Chapter 9

2. Chapter Contents • Merge Sort • Merging Arrays • Recursive Merge Sort • The Efficiency of Merge Sort • Merge Sort in the Java Class Library • Quick Sort • The Efficiency of Quick Sort • Creating the Partition • Quick Sort in the Java Class Library • Radix Sort • The Efficiency of Quick Sort • Comparing the Algorithms

3. Merge Sort • Divide an array into halves • Sort the two halves • Merge them into one sorted array • Referred to as a divide and conquer algorithm • This is often part of a recursive algorithm • However recursion is not a requirement

4. Merge Sort Merging two sorted arrays into one sorted array.

5. Merge Sort The major steps in a merge sort.

6. Merge Sort Algorithm mergeSort(a, first, last) // Sorts the array elements a[first] through a[last] recursively. if (first < last) { mid = (first + last)/2 mergeSort(a, first, mid) mergeSort(a, mid+1, last) Merge the sorted halves a[first..mid] and a[mid+1..last] }

7. Merge Sort The effect of the recursive calls and the merges during a merge sort.

8. Merge Sort • Recursive Division step: • Assume that n is a power of 2, so we can divide n by 2 evenly. • Initial call divide into two subarrays of n/2 elements each. • Second call divide into four subarrays of n/2^2 element each • Third call divide array into eight subarrays of n/2^3 elements each. • If n= 8 = 2^3, it takes three level of recursive call to obtain subarrays of one element each. • So total k level of recursive calls result in k levels of merges. K = log2n

9. Merge Sort • Merge step: • Need at most n-1 comparisons among the n elements in the two subarrays; n moves to temporary array; n moves back to original array; In total, each merge requires at most 3n-1 • 2, 6 4,8 • 2,4,6,8 • 1)2<4, copy 2 to new array • 2) 6>4, copy 4 to new array • 3) 6<8, copy 6 to new array • 4) copy 8 to new array • If 6 is replaced with 3? How many comparisons? • in last slide, when n=8, number of merges at each level: • 3n -1 = 3n – 2^0: third merge level • (3n/2 -1)*2 = 3n-2^1: second merge level • (3n/2^2 -1 )*2^2 = 3n-4 = 3n – 2^2: first merge level • Merge at each level is O(n), and total number of level k is log2n, the complexity of mergeSort is O(n logn)

10. Merge Sort • Efficiency of the merge sort • Merge sort is O(n log n) in all cases • It's need for a temporary array is a disadvantage • Merge sort in the Java Class Library • The class Arrays has sort routines that uses the merge sort for arrays of objects public static void sort(Object[] a); public static void sort(Object[] a, int first, int last);

11. Quick Sort • Divides the array into two pieces • Not necessarily halves of the array • An element of the array is selected as the pivot • Elements are rearranged so that: • The pivot is in its final position in sorted array • Elements in positions before pivot are less than the pivot • Elements after the pivot are greater than the pivot

12. Quick Sort Algorithm quickSort(a, first, last) // Sorts the array elements a[first] through a[last] recursively. if (first < last) { Choose a pivot Partition the array about the pivot pivotIndex = index of pivot quickSort(a, first, pivotIndex-1) // sort SmallerquickSort(a, pivotIndex+1, last) // sort Larger}

13. Quick Sort A partition of an array during a quick sort.

14. Quick Sort • Quick sort is O(n log n) in the average case • O(n2) in the worst case • Worst case can be avoided by careful choice of the pivot

15. Quick Sort A partition strategy for quick sort … continued→

16. Quick Sort A partition strategy for quick sort.

17. Quick Sort Median-of-three pivot selection: (a) the original array; (b) the array with its first, middle, and last elements sorted

18. Quick Sort (a) The array with its first, middle, and last elements sorted; (b) the array after positioning the pivot and just before partitioning.

19. Quick Sort • Quick sort rearranges the elements in an array during partitioning process • After each step in the process • One element (the pivot) is placed in its correct sorted position • The elements in each of the two sub arrays • Remain in their respective subarrays • The class Arrays in the Java Class Library uses quick sort for arrays of primitive types

20. Radix Sort • Does not compare objects • Treats array elements as if they were strings of the same length • Groups elements by a specified digit or character of the string • Elements placed into "buckets" which match the digit (character) • Originated with card sorters when computers used 80 column punched cards

21. Radix Sort (a) Original array and buckets after first distribution; (b) reordered array and buckets after second distribution … continued →

22. Radix Sort (c) reordered array and buckets after third distribution; (d) sorted array

23. Radix Sort • Pseudo code Algorithm radixSort(a, first, last, maxDigits)// Sorts the array of positive decimal integers a[first..last] into ascending order;// maxDigits is the number of digits in the longest integer. for (i = 1 to maxDigits){ Clear bucket[0], bucket[1], . . . , bucket[9]for (index = first to last) { digit = ith digit from the right of a[index]Place a[index] at end of bucket[digit] }Place contents of bucket[0], bucket[1], . . . , bucket[9] into the array a} Radix sort is O(d*n) =O(n) but can only be used for certain kinds of data

24. Comparing the Algorithms The time efficiency of various algorithms in Big Oh notation

25. Comparing the Algorithms A comparison of growth-rate functions as n increases.