1. CSC 2300 Data Structures & Algorithms March 20, 2007 Chapter 7. Sorting

2. Today – Sorting Heapsort –  Algorithm  Analysis  Lower bound Mergesort –  Algorithm  Worst case Quicksort –  Algorithm

3. Heapsort – Algorithm What time bound can we get if we use a heap? We want to sort N integers. We build a binary heap of N elements. How much time does buildHeap take? We then perform N deleteMin operations. The elements leave the heap smallest first, in sorted order. By recording these elements in a second array and then copying them back, we sort N elements. How much time does each deleteMin take? What is the total running time? What is a problem with Heapsort?

4. Heapsort – Space The algorithm uses an extra array. Thus, the memory requirement is doubled. What about the work to copy the second array back to the first? How do we solve the problem of space?

5. Heapsort – Saving Space After each deleteMin, the heap shrinks by 1. Thus, the cell that was last in the heap can be used to store the element that was just deleted. If we use this strategy, the array will contain the elements in what sorted order? We usually want the elements in increasing sorted order. What should we do?

6. Max Heap Use a max heap. Build the heap in linear time. Then perform N – 1 deleteMax operations.

7. Heapsort Note: the array for heapsort contains data in position 0, unlike before, where the index starts at 1.

8. Heapsort – Performance Experiments show that the performance of heapsort is extremely consistent. On average, heapsort uses only slightly fewer comparisons than suggested by the worst-case bound. Why? It appears that successive deleteMax operations destroy the randomness of the heap. Can you suggest another O(N log N) sorting scheme? What do you think about when you see O(N log N)?

9. Mergesort A recursive algorithm! O(N log N) worst-case running time. We can show that the number of comparisons is nearly optimal. The fundamental operation is to merge two sorted lists. Since the lists are sorted, the merging can be done in one pass through the input, if the output is put in a third list.

10. Mergesort – Example We will go through an example in class. Try this: A:1, 13, 24, 26, 35, 44, 52, 68 B:2, 15, 27, 28, 47, 66, 72, 75 C: Merging two sorted lists with a total of N elements takes at most N – 1 comparisons. Why only N – 1? Every comparison adds an element to C, except the last comparison, which adds at least two elements.

11. Mergesort – Routines No base case in recursive routine?

12. Merge – Routine

13. Mergesort – Analysis Assume that N is a power of 2. For N = 1, the time to mergesort is constant, which we denote by 1. Otherwise, the time to mergesort N numbers is equal to the time to do two recursive mergesorts of size N/2, plus the time to merge, which is linear. Hence T(1) = 1 T(N) = 2T(N/2) + N Does anyone remember how to solve this recurrence?

14. Mergesort – Recurrence Recurrence: T(N) = 2T(N/2) + N Divide by N: T(N)/N = T(N/2)/(N/2) + 1 What do we assume about the value of N? We get: T(N)/N = T(1)/1 + log N Thus, T(N) = N log N + N = O(N log N)

15. Quicksort – Description Fastest generic sorting algorithm in practice. Average running time is O(N log N). What is worst-case running time bound? O(N 2 ). We may make the O(N 2 ) bound very unlikely with just a little effort on choosing the pivot.

16. Quicksort – Algorithm 1. If the number of elements in S is 0 or 1, then return. 2. Pick any element v in S. This is called the pivot. 3. Partition S – {v} into two disjoint groups: S 1 = { x ε S – {v} | x ≤ v} and S 2 = { x ε S – {v} | x ≥ v}. 4. Return { quicksort(S 1 ) followed by v followed by quicksort(S 2 )}.

17. Quicksort – Example

18. Quicksort – Partition Strategy Example. Input: 8, 1, 4, 9, 6, 3, 5, 2, 7, 0. Say 6 is chosen as pivot. 8149035276 ijpivot 8149035276 ij 2149035876 ij 2149035876 ij 2145039876 ij 2145039876 jipivot 2145036879 pivoti