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 = log 2 n

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 log 2 n, the complexity of mergeSort is O(n log n )

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 Smaller quickSort(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(n 2 ) 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.