ps7 - Massachusetts Institute of Technology Handout 13...

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Unformatted text preview: Massachusetts Institute of Technology Handout 13 6.854J/18.415J: Advanced Algorithms Wednesday, October 21, 2009 David Karger Problem Set 7 Due: Wednesday, October 28, 2009. Collaboration policy: collaboration is strongly encouraged . However, remember that 1. You must write up your own solutions, independently. 2. You must record the name of every collaborator. 3. You must actually participate in solving all the problems. This is difficult in very large groups, so you should keep your collaboration groups limited to 3 or 4 people in a given week. 4. No bibles. This includes solutions posted to problems in previous years. NONCOLLABORATIVE Problem 1. You are given a collection of n points in some metric space (i.e., the distances between the points satisfy the triangle inequality). Consider the problem of dividing the points into k clusters so as to minimize the maximum diameter of (distance between any two points in) a cluster. (a) Suppose the optimum diameter d is known. Devise a greedy 2-approximation algorithm. Hint: consider any point and all points within distance d of it. (b) Consider the algorithm that ( k times) chooses as a center the point at maxi- mum distance from all previously chosen centers, then assigns each point to the nearest center. By relating this algorithm to the previous algorithm, show that you get a 2-approximation. Problem 2. Consider the problem of scheduling, on one machine, a collection of jobs with given processing times p j , due dates d j , and lateness penalties ( weights ) w j paid for jobs that miss their due dates, so as to minimize the total lateness penalty. (If we let U j denote the indicator variable for job j completing after its due date, then our problem is 1 || w j U j .) (a) Argue that any feasible subset of jobs (that can all together be completed by their due dates) might as well be scheduled in order of increasing deadline (so it...
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ps7 - Massachusetts Institute of Technology Handout 13...

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