1. COSC 1030 Lecture 11 Sorting

2. Topics Sorting Themes – Low bound of sorting Priority queue methods – Selection sort – Heap sort Divide-and-conquer methods – Merge sort – Quick sort Insertion based methods Address-calculation methods

3. Sorting Themes Why sort is an important theme of CS – Performance analysis – Low boundary of problems – Model to implementation Comparison-based sorting – Insertion – Priority Queue – Divide-and-conquer Address-calculation sorting

4. Low Bound of Sorting Low bound – of a problem P – A complexity class O(P), for any solution S to P, the complexity is of O(S) in the worst case, then O(S)  O(P) – There is a solution S such that O(S) = O(P) To sorting, the low bound is O(n log n) How do we find the low bound?

5. Decision Tree In-node represents comparison Leaf represents an order Extended binary tree – For n elements, there are n! orders Every path determines an order Minimal external path length   lg (n!)  lg(n!) > ½ n lg n

6. Priority Queue Sorting Insert n elements into a priority queue Remove elements from the priority queue one by one Performance? – Dependent on priority queue implementation List impl. – remove is of O(n)  SelectionSort Heap impl. – remove is of O(lg n)  HeapSort – O(n lg n) if heap is used Do we need additional space?

7. void selectionSort(KeyType[] A) { KeyType key; int n = A.length; int i = n –1; // initially Q is empty while(i > 0) { // PQ has more element int j = i; // j point to last key of PQ for(int k = 0; k < I; k++) { // find max in PQ if(A[k].compareTo(A[j]) > 0) j = k; } temp = A[i]; A[i] = A[j]; A[j] = temp; // swap i--; // move boundary down }

8. void heapSort(KeyType[] A) { KeyType temp; int n = A.length; for(int j = n / 2; j>1; j--) { // heapify all sub trees siftUp(A, j, n); // except root tree }// only non-leaf sub tree need to heapify for(int i = n; i >1; i--) { // reheapify starting at root siftUp(A, 1, i); // to the last leaf temp = A[1]; A[1] = A[i]; A[i] = temp; // swap }

9. void siftUp(KeyType[] A, int i, int n) { // O(lg (n –i)) // i the index of root, n the last leaf index KeyType rootKey = A[i]; int j = 2*i; // point to left child of i boolean notFinished = (j <= n); // if j is in the range, not finished while(notFinished) { // if right child of i exists and it’s bigger than left, j point to right if(j 0) j ++; if(A[j].compareTo(rootKey) <= 0) { notFinished = false; // no more keys sift up } else { A[i] = A[j]; // sift A[j] up one level i = j; // i point to larger child j = 2*i; // j point to the new left child of i notFinshed = (j <= n); } // end of while A[i] = rootKey; }

10. Divide-and-conquer void sort(KeyType[] A, int m, int n) { if(n-m > 1) { // if more than 1 element in the range // divide A[m:n] into subarrays: // A[m:i] and A[i+1:n] sort(A, m, i); sort(A, i, n); // combine two sorted array // to yield the sorted original array }

11. Merge Sort void mergeSort(KeyType[] A, int m, int n) { if(n - m > 1) { // if more than 1 element in the range // divide A[m:n] into subarrays: // A[m:i] and A[i+1:n] int i = (m + n)/2; // I point in the middle mergeSort(A, m, i); mergeSort(A, I+1, n); merge(A, m, i, n); }

12. Merge void merge(KeyType[] A, int m, int i, int n) { // O(n – m +1) KeyType[] temp = new KeyType[n – m + 1]; int k =0; int j = i + 1; int l = m; while(l <= i && j <= n) { if(A[l] <= A[j]) { // copy smaller one into temp temp[k++] = A[l++]; } else { temp[k++] = A[j++]; } while(l <= i) temp[k++] = A[l++]; // copy remain part into temp while(j <= n) temp[k++] = A[j++]; // only one will be executed for(j = 0; j < k; j++) A[m+j] = temp[j]; // move back }

13. Quick Sort Void quickSort(KeyType[] A, int m, int n) { if(m < n) { int p = partition(A, m, n); quickSort(A, m, p); quickSort(A, p+1, n); }

14. int partition(KeyType[] A, int i, int j) { // O( j – i + 1) KeyType pivot, temp; int k, middle, p; middle = (m +n) /2; pivot = A[middle]; A[middle] = A[i]; A[i] = pivot; p = i; // place pivot in the A[i], p is the pivot position for(k = i+1; k <= j; k++) { if(A[k].compareTo(pivot) < 0) { // swap the smaller one to left sub array temp = A[++p]; A[p] = A[k]; A[k] = temp; } // loop invariant (A[I+1:p] = pivot) } temp = A[i]; A[i] = A[p]; A[p] = temp; return p; }

15. Quick Sort Performance Average: O(n lg n) Worst case: – if every time partition makes one sub array empty, the other size –1 – O(n ^ 2) Why Quick Sort get the name? – Two times faster than HeapSort in average – Only one loop in partition – Coefficient less than other sort methods

16. Insertion Sort Assume A’s left part already sorted up to k. Insert a[k+1] into A[0:k] by: – Extract A[k+1] and save it in temp – Compare temp to A[j], j = 0..k, to find the right position j – Shift A[j:k] one position right – Insert temp into A[j] O( )?

17. Address Calculation Sort Proxmap sort – O(n) – Indexing – maps key domain into [0, n) key   (key – min(KEY))/(max(KEY) – min(KEY)) *n  – Counting – number of keys in each slot – Positioning – shift position of each slot Position(k)  Count(k-1) – Inserting – each key into its sub array using insertSort

18. Address Calculation Sort Radix sort – O(n) – Sort a deck of cards – Each card is represented by where n  [2, 3, …, 10, J, Q, K, A] and s  [Diamond, Club, Heart, Spear] – Separate cards by their shape – Put them back in order of shape – Separate them by number while maintain original order in each deck – Put them back in order of number

19. Performance Comparison HeapSortMergeSortQuickSortTreeSortInsertSort AverageO(n log n) O(n^2) Worst Case O(n log n) O(n^2) Additional Space O(1)O(n)O(1)O(n)O(1)