CMSC 451 Homework 1 - Spring 2009: Algorithms and Complexity - Prof. Samir Khuller, Assignments of Computer Science

Download CMSC 451 Homework 1 - Spring 2009: Algorithms and Complexity - Prof. Samir Khuller and more Assignments Computer Science in PDF only on Docsity! Spring 2009 CMSC 451: Homework 1 Samir Khuller Feb 5 Problem 1. For each pair of expressions (A,B) below, indicate whether A is O, o, Ω, ω, or Θ of B. Note that zero, one or more of these relations may hold for a given pair; list all correct ones. A B (a) n100 2n (b) 10n 100n (c) log (n!) n log n Problem 2. Prove by induction n ∑ k=1 k(k + 1) = n(n + 1)(n + 2) 3 . Problem 3. Assume that there are n numbers (some possibly negative) in a circle, and we wish to find the maximum contiguous sum on the circle. Give an efficient algorithm for solving this problem. What is its worst case running time? Problem 4. Solve the recurrence T (n) = { T (n/5) + 2 n > 1 3 otherwise assuming n is a power of 5. Show your calculations. Problem 5. A coin is tossed n times, each time with an independent probability p of coming up heads and 1 − p of coming up tails. Let H be the number of heads occurring. What is (a) E[H], the expected number of heads? (b) V [H], the variance of H? (c) the standard deviation of H? (d) the probability that H > 2? Show your calculations.