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hw3(4) - CS 373U Combinatorial Algorithms Spring 2004...

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CS 373U: Combinatorial Algorithms, Spring 2004 Homework 3 Due Friday, March 12, 2004 at noon Name: Net ID: Alias: Name: Net ID: Alias: Name: Net ID: Alias: For each numbered problem, if you use more than one page, staple all those pages together. Please do not staple your entire homework together. This will allow us to more easily distribute the problems to the graders. Remember to print the name and NetID of every member of your group, as well as the assignment and problem numbers, on every page you submit. You do not need to turn in this cover page. This homework is challenging. You might want to start early. # 1 2 3 4 5 6 * Total Score Grader

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CS 373U Homework 3 (due March 12, 2004) Spring 2004 1. Let S be a set of n points in the plane. A point p in S is called Pareto-optimal if no other point in S is both above and to the right of p . (a) Describe and analyze a deterministic algorithm that computes the Pareto-optimal points in S in O ( n log n ) time. (b) Suppose each point in S is chosen independently and uniformly at random from the unit square [0 , 1] × [0 , 1]. What is the exact expected number of Pareto-optimal points in S ? 2. Suppose we have an oracle Random ( k ) that returns an integer chosen independently and uniformly at random from the set { 1 , . . . , k } , where k is the input parameter; Random is our only source of random bits. We wish to write an efficient function RandomPermutation ( n ) that returns a permutation of the integers ( 1 , . . . , n ) chosen uniformly at random.
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