CS 350 Final Algorithms and Complexity. It is recommended that you read through the exam before you begin. Answer all questions in the space provided.

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1 It is recommended that you read through the exam before you begin. Answer all questions in the space provided. Name: Answer whether the following statements are true or false and briefly explain your answer in the space provided. 1. [TRUE / FALSE] Any algorithm with a pair of nested loops has a complexity of at least Θ(n 2 ). [5 pts] 2. [TRUE / FALSE] The dynamic programming approach to the {0,1} Knapsack problem is more [5 pts] feasible for large instances than the Integer Programming (branch and bound) approach. CS 350 Final Page 1 of 10

2 3. [TRUE / FALSE] A brute-force solution is always less efficient than a divide and conquer [5 pts] solution to the same problem. 4. [TRUE / FALSE] All bounded optimization problems are reducible to decision problems in [5 pts] polynomial time. 5. [TRUE / FALSE] If P NP, then NP=EXPTIME. [5 pts] CS 350 Final Page 2 of 10

3 6. Consider problem 1 from homework 4: [10 pts] Given an unsorted array, A of all integers in the range 0...n except for one integer, denoted the missing number, find the missing number. Prove (without giving an algorithm) that this problem is in P. 7. Prove that the general Integer Linear Programming (ILP) problem is NP-Complete. (Hint: [10 pts] reduce from one of the problems we ve seen a lot of.) CS 350 Final Page 3 of 10

4 8. (a) Briefly explain the difference between Memoization and Tabulation with regards to dynamic [4 pts] programming. (b) When would memoization result in a faster run time than tabulation? [3 pts] (c) When would tabulation result in a faster running time? [3 pts] 9. (a) What is the best case space complexity of Breadth-first Search? (You do not need to prove [5 pts] this, just briefly describe your reasoning.) (b) What is the worst case space complexity of Depth-first Search? (You do not need to prove this, just briefly describe your reasoning.) [5 pts] CS 350 Final Page 4 of 10

5 10. Majority Element An array A[1...n] is said to have a majority element if more than half of its entries are the same. Given an array, design an algorithm to tell whether the input has a majority element, and, if so, to find that element. To compare elements you have access to a constant time transitive and reflexive oracle function, same(a, b) which returns true if a and b are in the same equivalence class and false otherwise. You may not compare elements of the input in any other way. [Sanjoy Dasgupta, Christos H. Papadimitriou, and Umesh Vazirani Algorithms (1 ed.). McGraw-Hill, Inc., New York, NY, USA.] (a) What is the complexity of the brute force solution? (Hint: there is no way of knowing how many distinct elements there are before you ve finished comparing them.) [5 pts] (b) Consider the following divide and conquer algorithm: majority(a): if A = 1: return A[1] l <- majority A[1:n/2] r <- majority A[n/2:n] count elements of A equal to l, if > n/2, return l count elements of A equal to r, if > n/2, return r What is this algorithm s runtime? Prove its correctness by induction. [5 pts] CS 350 Final Page 5 of 10

6 (c) There is a more efficient divide and conquer approach. Consider the following subroutine: Pair the elements of A to get n 2 pairs. Look at each pair: if the two elements are different, discard both of them; if they are the same, keep just one. Show that after this procedure there are at most n 2 elements left, and that they have a majority element if A does. Describe an algorithm that uses this subroutine to find a majority element. What is its runtime? [10 pts] CS 350 Final Page 6 of 10

7 It is recommended that you read through the exam before you begin. Choose just one of the following questions. Circle your chosen question and write your solution on the following pages. Be sure to specify the parameters you are using to measure the input size and identify the basic operation. Name: 1. Write an algorithm that implements the word wrap operation. It should take a string and an [40 pts] integer representing the screen width. The output should be a list of strings, each representing a line of text. Make sure your algorithm doesn t split any strings and correctly handles new line characters that are encountered. Find the time and space complexity of your algorithm. 2. Suppose you are investing. You want to buy low and sell high to maximize your profit. Thanks [40 pts] to the magic of time travel you have acquired an array of future prices, but can only perform two trades, one buy and one sell. Your array of prices is in chronological order and your buy order must precede your sell order. Design a O(n log n) divide and conquer algorithm for finding the maximum profit you can make. Find the space complexity of your algorithm. 3. Design an algorithm to implement the paint fill function that one might see on many image [40 pts] editing programs. That is, given a screen (represented by a two dimensional array of colors), a point, and a new color, fill in the surrounding area until the color changes from the original color. Find the time and space complexity of your algorithm. 4. Given a string, write an algorithm to find the longest substring that is a palindrome. Find the [40 pts] time and space complexity of your algorithm. Example: suppose the input is: sampleracecartest, you should return racecar. (Note: The optimal solution for this problem with use dynamic programming and use Θ(n 2 ) time and space. Partial credit will be awarded for less efficient solutions, with a max of 35 points being available.) CS 350 Final Page 7 of 10

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