280, CSci Spring 2009, Exam 1
This take-home exam is due Friday, February 13, at 4pm. You should submit your solution on paper, either handwritten or typed. E-mailed solutions will not be accepted. Answer only four of the following ve problems. If you answer all problems, I will grade only the rst four. Each problem is worth 20 points, so your solutions will be graded out of 80 points. The quality of writing and mathematical argument will be a central component in how your solutions are graded. When I return your graded solution, I will tell you what you should do to be eligible for a nal 20 points. If your initial submission has no signicant aws, the nal 20 points will require no additional work. You should not obtain help from anybody or anything, except your instructor, your written class notes, the textbook (Dasgupta, Papadimitriou, & Vazirani, available on paper or on-line), and the course Web site. 1. Using induction, and without appealing to the Master Theorem, prove that T (n) = n log2 n + n is a correct closed-form solution to the below recurrence, assuming that n is a power of 2. T (n) = 1 2T
n 2
if n 1 + n if n > 1
2. Suppose we have two arrays of n numbers, both known to be in increasing order. We want to nd the median of the numbers in both arrays. For instance, if the arrays are 1, 2, 4 and 3, 5, 6 , the median is 3.5: If you merge the arrays together, the middle two numbers are 3 and 4, which average to 3.5. Describe an algorithm for this problem that takes O(log n) time. Argue that your algorithm is correct and that it takes the required amount of time. 3. Given an array a of numbers, we want to nd the largest sum of any subsegment a[i:j] of the array. For example, given the array 1, 4, 2, 3, 5 , the largest sum is 5, achieved in this case by summing the middle three numbers. Design a divide-and-conquer algorithm this for problem, and argue that it works correctly and that it takes O(n log n) time. (There is an O(n) algorithm for this problem based on dynamic programming. However, your solution here should explicitly use the divide-and-conquer technique.) 4. Suppose we have a dag on n vertices in which there are never multiple paths from one vertex to another. This property is true of the dag at right, but if there were an edge from A to D, it would not be, since there would be two different paths from A to E (via B and via D).
A B C D E F
Design an algorithm that computes the total number of valid topological sort orderings for such a dag. For the example dag, the output should be 6, since there are 6 valid orderings: ABDECF, ADBECF, DABECF, ABDEFC, ADBEFC, DABEFC. You should argue that your algorithm is correct and that it is takes O(n2 ) or less time. (You can assume that multiplication and addition takes O(1) time. That assumption isnt technically correct, because the numbers being added or multiplied can themselves have O(n) digits.) 5. Youre planning a marathon road trip, and you want to devel...