MA 252: Data Structures and Algorithms Lecture 9. Partha Sarathi Mandal. Dept. of Mathematics, IIT Guwahati

Transcription

1 MA 252: Data Structures and Algorithms Lecture 9 Partha Sarathi Mandal Dept. of Mathematics, IIT Guwahati

2 More about quicksort We are lucky if the partitioning is balance i.e., (n/2, n/2). We are unlucky if the partitioning is imbalance i.e., (1, n-1) or (n-1, 1).

3 More intuition Suppose we alternate lucky, unlucky, lucky, unlucky, lucky,. L(n)= 2U(n/2) + Θ(n) lucky U(n)= L(n 1) + Θ(n) unlucky Solving: L(n) = 2(L(n/2 1) + Θ(n/2)) + Θ(n) = 2L(n/2 1) + Θ(n) = Θ(nlg n) How can we make sure we are usually lucky?

4 Randomized quicksort IDEA: Partition around a random element. Running time is independent of the input order. No assumptions need to be made about the input distribution. No specific input elicits the worst-case behavior. The worst case is determined only by the output of a random-number generator.

5 Pseudocode for Randomized quicksort

6 Randomized quicksort analysis Let T(n) =the random variable for the running time of randomized quicksorton an input of size n, assuming random numbers are independent. For k= 0, 1,, n 1, define the indicator random variable: E[X k ] = 0.Pr{X k =0} + 1.Pr{X k =1} = Pr{X k = 1} = 1/n, since all splits are equally likely, assuming elements are distinct.

7 Analysis (continued)

8 Calculating expectation

9 Calculating expectation

10 Calculating expectation

11 Calculating expectation

12 Calculating expectation

13 Exercise (The k =0, 1terms can be absorbed in the Θ(n) ) Exercise: Prove: E[T(n)] anlg nfor constant a> 0. Choose a large enough so that anlgndominates E[T(n)] for sufficiently small n 2. Use fact:

14 Substitution method

15 Substitution method

16 Substitution method

17 Substitution method

18 Quicksort in practice Quicksortis a great general-purpose sorting algorithm. Quicksortis typically over twice as fast as merge sort. Quicksortcan benefit substantially from code tuning. Quicksortbehaves well even with caching and virtual memory.

19 How fast can we sort? All the sorting algorithms we have seen so far are comparison sorts: only use comparisons to determine the relative order of elements. E.g., insertion sort, merge sort, quicksort, heapsort. The best worst-case running time that we ve seen for comparison sorting is O(nlgn). Is O(nlgn)the best we can do? Decision tree scan help us answer this question.

20 Decision-tree example Each internal node is labeled i:jfor i, j {1, 2,, n}. The left subtreeshows subsequent comparisons if a i a j. The right subtreeshows subsequent comparisons if a i a j.

21 Decision-tree example Each internal node is labeled i:jfor i, j {1, 2,, n}. The left subtreeshows subsequent comparisons if a i a j. The right subtreeshows subsequent comparisons if a i a j.

22 Decision-tree example Each internal node is labeled i:jfor i, j {1, 2,, n}. The left subtreeshows subsequent comparisons if a i a j. The right subtreeshows subsequent comparisons if a i a j.

23 Decision-tree example Each internal node is labeled i:jfor i, j {1, 2,, n}. The left subtreeshows subsequent comparisons if a i a j. The right subtreeshows subsequent comparisons if a i a j.

24 Decision-tree example Each leaf contains a permutation π(1), π(2),, π(n) to indicate that the ordering a π(1) a π(2) a π(n) has been established.

25 Decision-tree model A decision tree can model the execution of any comparison sort: One tree for each input size n. View the algorithm as splitting whenever it compares two elements. The tree contains the comparisons along all possible instruction traces. The running time of the algorithm = the length of the path taken. Worst-case running time = height of tree.

26 Lower bound for decision-tree sorting Theorem:Any decision tree that can sort n elements must have height Ω(nlg n). Proof:The tree must contain n!leaves, since there are n!possible permutations. A height-hbinary tree has 2 h leaves. Thus, n! 2 h. h lg (n!) (lg is monotonically increasing) lg ((n/e) n ) (Stirling s formula) = nlg n nlg e = Ω(nlg n)

27 Lower bound for comparison sorting Corollary: Heapsortand merge sort are asymptotically optimal comparison sorting algorithms.