Algorithms Networking - Homework Assignment 1 | CMSC 858, Assignments of Computer Science

Download Algorithms Networking - Homework Assignment 1 | CMSC 858 and more Assignments Computer Science in PDF only on Docsity! CMSC 858S: Algorithms in Networking Fall 2004 Homework Assignment #1 (will be graded) Due date: Beginning of class on September 30, 2004 Note: As with all homework assignments for this class, this needs to be done along with your group-mates. Please submit one final, polished set of answers for your group, and do not include unnecessary material. (For instance, don’t include your preliminary work/calculations/intuition that led to the final answer.) Partial credit will be given where appropriate: if you think you have some ideas that deserve partial credit, please itemize them concisely, so that the grading process can be accurate. The problem marked (*) may be more difficult than the others. Also, the Chernoff-bound approach of bounding: (i) Pr[X ≥ a] by Pr[etX ≥ eta], and (ii) Pr[X ≤ a] by Pr[e−tX ≥ e−ta] (and then applying Markov, simplifying, and choosing the optimal positive t), is quite general: the case where X is a sum of bounded and independent random variables seen in class, is just a special case. Problems 2 and 3 deal with such generalizations. 1. We have a random variable X whose distribution is some D; our aim is to estimate E[X] using “not too many” samples from D. Suppose we can draw independent random samples X1, X2, . . . from D; we want to draw some t such samples, and output the estimate Y = (X1 + X2 + · · · + Xt)/t for E[X]. Our main goal is that with probability at least 1 − δ, the absolute value of the difference between our estimate and E[X], should be at most  ·E[X]. We now explore how to solve this problem. Suppose the value ` = ⌈√ Var[X]/E[X] ⌉ is given, but that we do not know anything more about D. (P1) Show that O(`2/(2δ)) samples suffice. (We will see how to improve this in (P3).) (P2) Suppose we only need a “weak estimate” Z such that: with probability at least 0.6, the absolute value of the difference between Z and E[X], is at most  · E[X]. Show that O(`2/2) samples suffice to compute such a weak estimate. (P3) Show that by judiciously combining O(log(1/δ)) independently-computed weak esti- mates, we can solve our main problem: i.e., we can get an estimate such that with probability at least 1 − δ, the absolute value of the difference between our estimate and E[X], is at most  ·E[X]. (Thus, O(`2 log(1/δ)/2) samples suffice. A hint: the mean may not be the correct choice for the “judicious combination”!) 2. Suppose X is a Poisson random variable with mean λ; i.e., X takes values in the non-negative integers, with Pr[X = i] = e−λ · λi/i!. Given δ > 0, use a Chernoff-type approach to bound Pr[X ≥ λ(1 + δ)]. Does your result resemble any bound seen in class? 3 (*). We are given a black-box that outputs numbers independently and uniformly at random from the set {1, 2, . . . , 2n}. We use this to construct a random permutation π of {1, 2, . . . , n} as follows. We will construct a one-to-one function f : {1, 2, . . . , n} → {1, 2, . . . , 2n}, and then output the numbers in {1, 2, . . . , n} in the order of their f(·) values. To construct f , we proceed as follows, for j = 1, 2, . . . , n: choose f(j) by repeatedly obtaining numbers from the black-box, and setting f(j) to be the first number found such that f(j) 6= f(i) for all i < j. (Note that we ensure that f is one-to-one.) Let X denote the number of calls to the black-box. 1