Assignment 1

Problem 1: Show that 1^d + 2^d + 3^d + .... + n^d is O(n^{d+1}) for a constant d? What is the value of c and n_0. (Note that 3^d means 3 raised to the power d and n_0 means n subscript 0). Problem 2: Solve the recurrence relation T(n)= T(xn) + T((1-x)n) + cn in terms of x and n, where x is a constant in the range 0<X<1. Is asymptotic complexity same when x="0.5," 0.1 and 0.001? What happens to constant hidden in the O() notation. Problem 3: Analyze the recurrence relation T(n)=root(n)T(root(n)) + n, where root(n) denotes the square-root of n. (This is called as the rootish-divide-and-conquer and for example plays an important role in parallel computing.) Problem 4: Devise an algorithm that can find a node in a binary tree, such that its removal leaves none of the subtrees to contain more than (2/3n + 5) nodes. (By removal of a node, we mean the removal of the node as well as all its incident edges.) Sketch your algorithm in terms of Steps. Prove that your algorithm is correct. Analyze its complexity. Problem 5: Express the function n^3/50 + 75n^2 + 25n +7 in terms of Theta-notation. You need to compute g(n), c_1, c_2, and n_0. Problem 6: V. Pan has discovered a way to multiply two 70X70 matrices using only 143640 multiplications. Ignoring the additions, what will the asymptotic complexity of Pan's algorithm for multiplying two nXn matrices? Is it better than Strassen's? Justify your answer. ## Deliverables 1) Complete and fully-functional working program(s) in executable form as well as complete source code of all work done. 2) Deliverables must be in ready-to-run condition, as follows (depending on the nature of the deliverables): a) For web sites or other server-side deliverables intended to only ever exist in one place in the Buyer's environment--Deliverables must be installed by the Seller in ready-to-run condition in the Buyer's environment. b) For all others including desktop software or software the buyer intends to distribute: A software installation package that will install the software in ready-to-run condition on the platform(s) specified in this bid request. 3) All deliverables will be considered "work made for hire" under U.S. Copyright law. Buyer will receive exclusive and complete copyrights to all work purchased. (No GPL, GNU, 3rd party components, etc. unless all copyright ramifications are explained AND AGREED TO by the buyer on the site per the coder's Seller Legal Agreement). ## Platform Microsoft Word