Sahami Mehran CS103B
Handout #16 February 2, 2009
Problem Set #3 Due: 11:00am on Wednesday, February 11th
Note: in the problems below in which you are asked for a big-Oh running time, we are looking for a tight big-Oh bound (analogous to the big-Theta () notation mentioned in class and in Handout #9), but you only need to show the upper bound (big-Oh). You do not need to show the lower bound. 1. In Handout #11, we concluded that the recurrence relation for the running time of the recursive Binary search algorithm is (using constants x and y here): T(1) = x T(n) = T(n/2) + y Determine the big-Oh running time for this recurrence relations using: a. Repeated substitution. Use induction to prove that your closed form formula for T(n) is correct. b. Theorem 1 (Master Theorem) from Handout #10. Show how you got your result by explicitly stating the values of a, b, c, and d from the Theorem in terms of this recurrence, and how you computed the final big-Oh as a result. 2. Consider the number of permutations of a set with n elements. a. Find a recurrence relation for the number of permutations of a set with n elements. Make sure to explain how you came up with this recurrence. b. Use the recurrence you found in part (a) to find the number of permutations of a set with n elements using repeated substitution. (Note: you do not need to provide an inductive proof showing your formula is correct). 3. In this problem we consider a recurrence relation on bit strings (similar to the problem we did in class and Example 5 in Handout #10). a. Find a recurrence relation for the number of bit strings of length n that contain the substring 01. Make sure to explain how you came up with this recurrence. b. What are the initial conditions for the recurrence relation? c. How many bit strings of length 7 contain the substring 01?
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4. Consider the recurrence relation: T(1) = a T(n) = 9T(n/3) + bn Determine the big-Oh running time for this recurrence relations using repeated substitution. Use induction to prove that your closed form formula for T(n) is correct. Note: this is one of the more involved problems on this problem set. You might want to check the correctness of your solution using the Master Theorem. 5. Consider the recurrence relation: T(1) = 1 T(n) = (1/2)T(n-1) + n Determine the big-Oh running time for this recurrence relations using repeated substitution. Use induction to prove that your form closed formula for T(n) is correct. Note: this is one of the more involved problems on this problem set. 6. Say that we came up with a new sorting algorithm that we name SuperQuickSort. SuperQuickSort works just like regular QuickSort, except that rather than picking a single pivot element and creating two sub-arrays, it picks (k 1) pivot elements ordered in increasing order, and generates k sub-arrays L1, L2, , Lk that contain the array elements with values between each respective pair of pivots. Hence, it makes k recursive calls to SuperQuickSort with each of these sub-arrays. Prove that SuperQuickSort still runs in time O(n log n) for any integer k >= 2, when each of the k sub-arrays have roughly equal numbers of elements in them. 7. In class, we discussed the Towers of Hanoi problem and explained that there is a deterministic O(2n) algorithm for solving this problem, where n is the number of disks. If we had a non-deterministic algorithm to solve this problem, would we be able to obtain a better running time? In other words, explain (informally) if Towers of Hanoi is really a problem in NP or in the Exponential complexity class. Make sure to explain the role of non-determinism in your answer.
8. A monomial is a conjunction of literals, where each literal is either some boolean variable or its negation. For example, (x1 & x2 & ~x3) is a monomial. Consider the problem of determining whether arbitrary disjunctions of two monomials are satisfiable (i.e., is there some assignment to the variables in the logical formula such that it is true). Explain informally whether this problem is in P or NP. In other words, is non-determinism required to solve such a restricted satisfiability problem in polynomial time? If the problem is in P, provide a polynomial time algorithm to solve the problem. If the problem is in NP, briefly explain why non-determinism is required to solve this problem in polynomial time.
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9. Find the length of the longest common subsequence of the lists abacaabacab and bacababa using the dynamic programming algorithm discussed in class and in Handout #15. Make sure to also show the table that you generated in applying the algorithm. 10. Use the table you constructed in Problem #9 to find all paths for the longest common subsequences of the lists abacaabacab and bacababa. Explain the path (possibly multiple) taken through the table for each LCS.