We have a Standard Six-Sided Die - IT Assignment Help

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Problem 1 (3+4+3 = 10 points) We have a standard six-sided die. Let X be the number of times that a 6 occurs over n throws of the die. Let p be the probability of the event X ∼ n/4. Compute the best upper bounds on p that you can obtain using Markov’s inequality. Chebyshev’s inequality, and Chernoff bounds. Note: You will only get full marks for stating relevant theorems and showing the full calculation. Problem 2 (5+5 = 10 points) A family of hash functions, H, from U to {0, 1, · · · m} is called an `-strongly universal family of hash functions if the following holds. Let a1, a2, · · · a` be any ` distinct elements in U and α1, α2, · · · α` be elements from {0, 1, · · · m}, not necessarily distinct. Then, P[(h(a1) = α1) ∩ (h(a2) = α2) ∩ · · ·(h(a`) = α`)] = 1 m` For each of the following, say True or False with reasons.

Note:You get 4 points for the reason and 1 point for saying True or False correctly. a. The following hash family from U = {a, b} to {0, 1} is 2-strongly universal. (m = 2). a b h1 0 0 h2 1 0 b. The following hash family from U = {a, b, c} to {0, 1} is 3-strongly universal. a b c h1 0 0 0 h2 1 0 1 h3 0 1 1 h4 1 1 0 Problem 3 (20 points) Consider the following algorithm to generate a uniformly random permutation of natural numbers 1, 2, · · · n. Start with the sorted array. At each iteration, pick A[1] (that is the first element) and insert it at any of the n positions chosen uniformly at random. If it is inserted at A[i], then the index of all numbers from i down to 2 is decremented by 1. Repeat this process till n − 1 becomes the first element and then the above operation is done one final time.

• (8 points) Prove that on termination, the algorithm indeed produces a uniformly random permutation (Hint : Try to prove that at any iteration, if j is the position of number n − 1, then the subarray A[j, j + 1, · · · n] is a uniformly random permutation of the numbers contained in the subarray. What happens after the final iteration? ) 1

• (12 points)What is the expected number of operations performed by this algorithm (Hint : Use number n − 1 as a tracker !) ?

Problem 4 (12+8 = 20 points) Recall that Reservoir sampling is a family of randomized algorithms for randomly choosing a sample of k items from a stream σ of m items, where m is either very large or unknown until the list is traversed. Suppose we used reservoir sampling on a stream σ1 of length m1 and stored a k-sample (with replacement) S1. We also used reservoir sampling on another stream σ2 of length m2 and stored a k-sample (with replacement) S2. a. (12 points) Show how you can use reservoir sampling to obtain a k-sample (with replacement) S for the concatenated stream σ = σ1 · σ2 from S1 and S2 in O(k) space. You have to give a formal description for your algorithm (8 points) and prove that your algorithm does indeed randomly sample (uniformly) from the concatenated stream (4 points). b. (8 points) Generalize for the concatenation of h streams σ1, σ2, . . . , σh of lengths m1, m2, . . . , mh and associated k-samples S1, S2, . . . , Sh in O(hk) space. You have to give a formal description for your algorithm (4 points) and prove that your algorithm does indeed randomly sample (uniformly) from the concatenated stream (4 points). Problem 5 (20 points) Recall the norm-preserving version of the JL Lemma done in the lectures. Given a set S of n vectors in d-dimensional space, there exists a random k×d matrix Π where k = O(log n/ε2 ), such that for any x ∈ S (1 − ε)||x||2 2 ≤ ||Πx||2 2 ≤ (1 + ε)||x||2 2 , with probability at least 1 − 1/n3 . Remember that each entry of the matrix Π is an independent Gaussian. In this problem, we are going to consider a different random matrix as follows. For each i ∈ {1 · · · n} we pick a uniformly random number hi ∈ {1, · · · k}. We then set Πhi,i = ±1 for each i ∈ {1, · · · n} (the sign is chosen uniformly at random from {−1, 1}), and all other entries of Π are set to 0. Show that for this Π, ||Πx||2 2 is an unbiased estimator of ||x||2 2 .

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