1. Solve the following recurrences by giving tight Θ-notation bounds in terms of n for sufficiently large n. Assume that T(·) represents the running time of an algorithm, i.e. T(n) is a positive and non-decreasing function of n. For each part below, briefly describe the steps along with the final answer.
(a) T(n) = 4T(n/2) + n2logn
(b) T(n) = 8T(n/6) + nlogn
▪ √
(c) T(n) = 6000 T(n/2) + n 6000
(d) T(n) = 10T(n/2) + 2n
(e) T(n) = 2T(√n) + log2n
2. Solve Kleinberg and Tardos, Chapter 5, Exercise 3.
3. Solve Kleinberg and Tardos, Chapter 5, Exercise 5.
4. Assume that you have a blackbox that can multiply two integers. Describe an algorithm that when given an n-bit positive integer a and an integer x, computes xa with at most O(n) calls to the blackbox.
5. Consider two strings a and b and we are interested in a special type of similarity called the “J-similarity”. Two strings a and b are considered J-similar to each other in one of the following two cases: Case 1) a is equal to b, or Case 2) If we divide a into two substrings a1 and a2 of the same length, and divide b in the same way, then one of following holds: (a) a1 is J-similar to b1, and a2 is J-similar to b2 or (b) a2 is J-similar to b1, and a1 is J-similar to b2. Caution: the second case is not applied to strings of odd length.
Prove that only strings having the same length can be J-similar to each other. Further, design an algorithm to determine if two strings are J-similar within O(n logn) time (where n is the length of strings).
6. Given an array of n distinct integers sorted in ascending order, we are interested in finding out if there is a Fixed Point in the array. Fixed Point in an array is an index i such that arr[i] is equal to i. Note that integers in the array can be negative.
Example: Input: arr[] = -10, -5, 0, 3, 7 Output: 3, since arr[3] is 3
a) Present an algorithm that returns a Fixed Point if there are any present in the array, else returns -1. Your algorithm should run in O(log n) in the worst case.
b) Use the Master Method to verify that your solutions to part a) runs in O(log n) time.
c) Let’s say you have found a Fixed Point P. Provide an algorithm that determines whether P is a unique Fixed Point. Your algorithm should run in O(1) in the worst case.
1 of 1 Professor: Dr. Shahriar Shamsian
