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Main points of this past exam are: Random Permutation, N Speakers, Last-Minute Preparation, Every Speaker, Random Permutation, Random Re-Ordering, Chernoff Bounds, Probability, Chebyshev Bound, Perfect Matching
Typology: Exams
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This is a closed-book exam with 6 questions. You have 3 hours. All questions are worth equal points, so be sure to budget 30 minutes per question. You are allowed to use the 4 sides of notes you brought with you. No other notes are allowed. No calculators please. Write all your answers in this booklet. Good luck!
SID Number
NOTE: Please print your name and SID CLEARLY. We can’t be responsible for scores for exams without a legible name.
(a) Suppose you made a random permutation of the N talks on Monday to get the order for Tuesday. What is the probability that no speaker on Tuesday follows the speaker that they followed on Monday? Assume N is large.
(b) Suppose now that each speaker also requires some time after their talk to collect their materials, so that they always miss the talk of the speaker after them (as well as the one before). What is the probability that a random permutation of the Monday talks leads to no conflicts? i.e. No two talks that were consecutive (in either order) on Monday should be consecutive on Tuesday?
(c) Assuming a random re-ordering of the talks, what is the expected number of talks that end up in the same time slot on Monday and Tuesday? A talk is in the same time slot if it is the k th^ talk in the Monday ordering and then the k th^ talk in the Tuesday ordering.
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edges are added to G at random with replacement. That is, we do the following step
2
times: choose vertices u and v 6 = u at random from the vertices of G, and add edge fu; v g (with replacement) to G.
(a) What is the expected number of edges in G?
(b) Now suppose we choose a random perfect matching in G (a set of N = 2 edges with no common vertices), and contract those edges giving a new graph G^0. The result will be a graph with N = 2 vertices. Each of the new vertices represents a contraction of two of the original vertices. What is the expected number of edges in G^0?
(c) Now suppose we repeat the contraction operation above k times, giving a new graph G^00. After k such contractions, the graph will have N = 2 k^ vertices, and each new vertex will represent 2 k^ vertices of G. How large should k be so that there is a high probability that G^00 is a complete graph? Hint: think about the coupon collectors problem.
(a) What is (n), the order of the multiplicative group Z n?
(b) What are the possible orders of subgroups of Z n?
(c) Since n is a power of an odd prime, Z n is a cyclic group, and it has at least one gener- ator. What fraction of the elements in Z n are generators?
(b) Now suppose we want to do the same thing but with Y remaining anonymous. That is, X asks Y to cash a check and to satisfy the properties above, but you cannot use Y’s public RSA key, or anything that identifies Y. You should still prove to the bank that the person with the bits (Y) is the same person the check was written for and that they agreed to accept it.
