Advanced Algorithms Practice Midterm - West Virginia University - Prof. Krishnamurthy Subr, Exams of Discrete Mathematics

Download Advanced Algorithms Practice Midterm - West Virginia University - Prof. Krishnamurthy Subr and more Exams Discrete Mathematics in PDF only on Docsity! Advanced Analysis of Algorithms - Practice Midterm L. Kovalchick LCSEE, West Virginia University, Morgantown, WV {lynn@csee.wvu.edu} Please attempt as many problems as you can in class. The practice midterm will not be graded, i.e. there are no points. The solutions are posted at: http://www.csee.wvu.edu/˜ksmani/courses/fa03/algos/algos.html 1 Problems 1. Show that n ∑ i=1 i3 = [ n · (n + 1) 2 ]2. 2. Consider the recurrence relation: T (n) = 1, if n = 1 = T (n− 1) + 2n, otherwise Show that T (n) = 2n+1 − 3. 3. Give an algorithm which will return the third largest element of a heap with heap− size ≥ 3, in O(1) time. Assume that all elements are unique. 4. Describe an algorithm that, given n integers in the range from 1 to k, preprocesses its input and then answers any query about how many of the n integers fall into a range from [a · · · b] in O(1) time. Your algorithm should use O(n + k) preprocessing time. 5. Suppose a biker is about to go on a ride on a bike trail carrying a single knapsack. Suppose further that she knows the maximum total weight W that she can carry, and she has a set S of n different useful items that she can potentially take with her. Let us assume that each item j has an integer weight wj and a benefit value bj , which is a value that the biker assigns to item j. Her problem is to optimize the value of the items that she places into her knapsack, without going over the weight limit W (i.e., maximize ∑ j∈T bj subject to ∑ j∈T wj ≤W ). Give a recursive definition for the profit value of the optimal solution to this problem. 1