1. Sorting Lower Bounds n Beating Them

2. Recap Divide and Conquer –Know how to break a problem into smaller problems, such that –Given a solution to the smaller problems we can easily find a solution to the bigger problem

3. Recap Divide and Conquer –Divide and solve the smaller pieces recursively –Solve the big problem using the solutions to the smaller problems.

4. Recap The Quicksort algorithm: –chooses an element, called pivot. –Splits the input into three groups: smaller than the pivot equal to pivot larger than pivot. – Recursively sort the smaller and the larger groups independently

5. How fast can we sort? Can we do better than O(n lg n)? Well… it depends? It depends on the model of computation!

6. Comparison sorting Only comparisons are used to determine the order of elements Example –Selection Sort –Insertion sort –Merge Sort –Quick Sort –Heapsort

7. Lower Bounds How do we prove that any algorithm will need cnlogn time in the worst case? First Assumption: All these algorithms are allowed to only compare elements in the input. (Say only < ?)

8. I choose a permutation of {1,2,3,…N} Comparison Sort Game I ask you yes/no questions. Time is the number of questions asked in the worst case. I answer. I determine the permuation.

9. Decision Trees Another way of looking at the game Example: 3 element sort (a 1, a 2, a 3 ) One yes/no question? (Is x < y?) a 1 ? a 2 “Less than” leads to left branch “Greater than” leads to right branch

10. Decision Trees Example: 3 element sort (a 1, a 2, a 3 ) a 1 ? a 2 a 2 ? a 3

11. Decision Trees Example: 3 element sort (a 1, a 2, a 3 ) a 2 ? a3a 1 ? a 2 a 1, a 2, a 3

12. Decision Trees Example: 3 element sort (a 1, a 2, a 3 ) a 2 ? a 3 a 1 ? a 2 a 1 ? a 3 a 2 ? a 3 a 1 ? a 3 a 1, a 2, a 3 a 3, a 1, a 2 a 1, a 3, a 2 a 2, a 1, a 3 a 2, a 3, a 1 a 3, a 2, a 1

13. Decision Trees Each leaf contains a permutation indicating that permutation of order has been established A decision tree can model a comparison sort

14. Decision Trees It is the tree of all possible instruction traces Running time = length of path Worst case running time = length of the longest path (height of the tree)

15. Bounding log N! N! = 1 × 2 × 3 × … × N / 2 × … × N N factors each at most N. N / 2 factors each at least N / 2.  N N N / 2  N/2N/2 N / 2 log( N / 2 )  log(N!)  N log(N) =  (N log(N)).

16. Lower bound for comparison sorting Theorem: Any decision tree that sorts n elements has height  (n lg n) Proof: There are n! leaves A height h binary tree has no more than 2 h nodes and it should be able to resolve all n! permutations.

17. Optimal Comparison Sorting Algorithms Merge sort Quick sort Heapsort