Show that randomized quick-sort runs in O(n log n) time with probability 1 − 1/n2 . Hint: Use the Chernoff bound that states that if we flip a coin k times, then the probability that we get fewer than k/16 heads is less than 2−k/8
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Show that randomized quick-sort runs in O(n log n) time with probability 1 − 1/n2 . Hint: Use the Chernoff bound that states that if we flip a coin k times, then the probability that we get fewer than k/16 heads is less than 2−k/8.
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- An electrician has wired n lights, all initially on, so that: 1) light 1 can always be turned on/off, and 2) for k > 1, light k cannot be turned either on or off unless light k – 1 is on and all preceding lights are off for k > 1. The question we want to explore is the following: how many moves are required to turn all n lights off? For n = 5, a solution sequence has been worked out below. Fill in the missing entries. The lights are counted from left to right, so the first bit is the first light, and so on. 11111 01111 11011 10011 00010 10010 11010Determine φ (m), for m=12,15, 26, according to the definition: Check for each positive integer n smaller m whether gcd(n,m) = 1. (You do not have to apply Euclid’s algorithm.)Prove that I(X; Y |Z) ≥ I(X; Y ) . Note: X, Y, and Z are random variables. X and Z are independent.
- Let pn(x) be the probability of selling the house to the highest bidder when there are n people, and you adopt the Look-Then-Leap algorithm by rejecting the first x people. For all positive integers x and n with x < n, the probability is equal to p(n(x))= x/n (1/x + 1/(x+1) + 1/(x+2) + … + 1/(n-1)) If n = 100, use the formula above to determine the integer x that maximizes the probability n = 100 that p100(x). For this optimal value of x, calculate the probability p100(x). Briefly discuss the significance of this result, explaining why the Optimal Stopping algorithm produces a result whose probability is far more than 1/n = 1/100 = 1%.Suppose that an algorithm has runtime 20 n3 for n <= 100. For n>100, the slowest cases for each n take time 2 n2 log n. The fastest cases for each n take time 10 n2. That is, for n>100, runtime is between 10 n2 and 2 n2 log n. Which of the following are known to be true of the worst-case runtime for the algorithm? (Check all that apply, which may be more than one in a row.) O( n2 ) O( n2 log n ) O( n3 ) None of these Ω( n2 ) Ω( n2 log n ) Ω( n3 ) None of these Θ( n2 ) Θ( n2 log n ) Θ( n3 ) None of these Which of the following are known to be true of the average-case runtime? O( n2 ) O( n2 log n ) O( n3 ) None of these Ω( n2 ) Ω( n2 log n ) Ω( n3 ) None of these Θ( n2 ) Θ( n2 log n ) Θ( n3 ) None of these [Each row must have at least one answer checked.] ComposeYou are standing in front of an infinitely long straight fence; that is, the fence extends infinitely to your left and to your right. The fence has a single gate in it but you do not know where it is. Your goal is to minimize the distance you need to walk in order to find the gate. If n (which is unknown) is the distance to the gate in yards, design and an analyze an efficient algorithm for finding the gate in terms of n.
- Heuristics Prove or disprove: If h1(n), ..., hk(n) are admissible, so is h(n) = h1(n) + ... + hk(n)Pick one million sets of 12 uniform random numbers between 0 and 1. Sum up the 12 numbers in each set. Make a histogram with these one million sums, picking some reasonable binning. You will find that the mean is (obviously?) 12 times 0.5 = 6. Perhaps more surprising, you will find that the distribution of these sums looks very much Gaussian (a "Bell Curve"). This is an example of the "Central Limit Theorem", which says that the distribution of the sum of many random variables approaches the Gaussian distribution even when the individual variables are not gaussianly distributed. mean Superimpose on the histogram an appropriately normalized Gaussian distribution of 6 and standard deviation o = 1. (Look at the solutions from the week 5 discussion session for some help, if you need it). You will find that this Gaussian works pretty well. Not for credit but for thinking: why o = 1 in this case? (An explanation will come once the solutions are posted).Running Time Analysis: Give the tightest possible upper bound for the worst case running time for each of the following in terms of N. You MUST choose your answer from the following (not given in any particular order), each of which could be reused (could be the answer for more than one of a) – f)): O(N2), O(N3 log N), O(N log N), O(N), O(N2 log N), O(N5), O(2N), O(N3), O(log N), O(1), O(N4), O(NN), O(N6)
- In one pass and space O(n log n), the above algorithm constructs a (1/6)-approximate weighted matching M.give proofA bottle factory produces bottles of equal mass. During a production, the weight of one of the bottles is set incorrectly. The factory scale will be used to find this bottle. Design a decrease-andconquer algorithm which finds the that bottle. Analyze the worst-case, best-case and average-case complexities of your algorithm. Explain your algorithm.Please, answer the whole question. Suppose you toss n biased coins independently. Given positive integers n and k, along with a set of non-negative real numbers p1,..., pn in [0, 1], where pi is the probability that the ith coin comes up head, your goal is to compute the probability of obtaining exactly k heads when tossing these n biased coins. Design an O(nk)-time algorithm for this task. Explain the algorithm, write down the pseudo code and do run time analysis.