1. Mergesort

2. Merging two sorted arrays To merge two sorted arrays into a third (sorted) array, repeatedly compare the two least elements and copy the smaller of the two 7 9 14 18 5 10 12 20 5 7 9 10 12 14 18 20

3. Merging parts of a single array The procedure really is no different if the two arrays are really portions of a single array first part second part 7 9 14 18 5 10 12 20 5 7 9 10 12 14 18 20

4. Sorting an array If we start with an array in which the left half is sorted and the right half is sorted, we can merge them into a single sorted array We need to copy our temporary array back up 7 9 14 18 5 10 12 20 7 9 14 18 5 10 12 20 This is a sorting technique called mergesort

5. Recursion How did the left and right halves of our array get sorted? Obviously, by mergesorting them To mergesort an array: Mergesort the left half Mergesort the right half Merge the two sorted halves Each recursive call is with a smaller portion of the array The base case: Mergesort an array of size 1

6. The actual codeprivate void recMergeSort(long[] workspace, int lowerBound, int upperBound) { if (lowerBound == upperBound) return; int mid = (lowerBound + upperBound) / 2; recMergeSort(workspace, lowerBound, mid); recMergeSort(workspace, mid + 1, upperBound); merge(workspace, lowerBound, mid + 1, upperBound);} from Lafore, p. 287

7. The merge method The merge method is too ugly to show It requires: The index of the first element in the left subarray The index of the last element in the right subarray The index of the first element in the right subarray That is, the dividing line between the two subarrays The index of the next value in the left subarray The index of the next value in the right subarray The index of the destination in the workspace But conceptually, it isnt difficult!

8. Analysis of the merge operation The basic operation is: compare two elements, move one of them to the workspace array This takes constant time We do this comparison n times Hence, the whole thing takes O(n) time Now we move the n elements back to the original array Each move takes constant time Hence moving all n elements takes O(n) time Total time: O(n) + O(n) = O(n)

9. But wait...theres more So far, weve found that it takes O(n) time to merge two sorted halves of an array How did these subarrays get sorted? At the next level down, we had four subarrays, and we merged them pairwise Since merging arrays is linear time, when we cut the array size in half, we cut the time in half But there are two arrays of size n/2 Hence, all merges combined at the next lower level take the same amount of time as a single merge at the upper level

10. Analysis II So far we have seen that it takes O(n) time to merge two subarrays of size n/2 O(n) time to merge four subarrays of size n/4 into two subarrays of size n/2 O(n) time to merge eight subarrays of size n/8 into four subarrays of size n/4 Etc. How many levels deep do we have to proceed? How many times can we divide an array of size n into two halves? log2n, of course

11. Analysis III So if our recursion goes log n levels deep... ...and we do O(n) work at each level... ...our total time is: log n * O(n)... ...or in other words, O(n log n) For large arrays, this is much better than Bubblesort, Selection sort, or Insertion sort, all of which are O(n2) Mergesort does, however, require a workspace array as large as our original array

12. Stable sort algorithms A stable sort keeps equal elements in Ann 98 Ann 98 the same order Bob 90 Joe 98 This may matter Dan 75 Bob 90 when you are sorting data Joe 98 Sam 90 according to some Pat 86 Pat 86 characteristic Sam 90 Ze 86 Example: sorting Ze 86 Dan 75 students by test scores original stably array sorted

13. Unstable sort algorithms An unstable sort Ann 98 Joe 98 may or may not keep equal Bob 90 Ann 98 elements in the Dan 75 Bob 90 same order Joe 98 Sam 90 Stability is usually Pat 86 Ze 86 not important, but Sam 90 Pat 86 sometimes it matters Ze 86 Dan 75 original unstably array sorted

14. Is mergesort stable?private void recMergeSort(long[] workspace, int lowerBound, int upperBound) { if (lowerBound == upperBound) return; int mid = (lowerBound + upperBound) / 2; recMergeSort(workspace, lowerBound, mid); recMergeSort(workspace, mid + 1, upperBound); merge(workspace, lowerBound, mid + 1, upperBound);} Is this sort stable? recMergeSort does none of the actual work of moving elements aroundthats done in merge

15. Stability of merging Stability of mergesort depends on the merge method After all, this is the only place where elements get moved 7 9 14 18 5 10 12 20 5 7 9 10 12 14 18 20 What if we encounter equal elements when merging? If we always choose the element from the left subarray, mergesort is stable

16. The End