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final-f2010-sol

# final-f2010-sol - Introduction to Algorithms Massachusetts...

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Introduction to Algorithms December 14, 2010 Massachusetts Institute of Technology 6.006 Fall 2010 Professors Konstantinos Daskalakis and Patrick Jaillet Final Exam Solutions Final Exam Solutions Problem 1. What is Your Name? [2 points] (2 parts) (a) [1 point] Flip back to the cover page. Write your name there. (b) [1 point] Flip back to the cover page. Circle your recitation section.

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6.006 Final Exam Solutions Name 2 Problem 2. Storing Partial Maxima [30 points] (1 part) 6.006 student, Mike Velli, wants to build a website where the user can input a time interval in history, and the website will return the most exciting sports event that occurred during this interval. Formally, suppose that Mike has a chronologically sorted list of n sports events with associated integer “excitement factors” e 1 , . . . , e n . You can assume for simplicity that n is a power of 2 . A user’s query will consist of a pair ( i, j ) with 1 i < j n , and the site is supposed to return max( e i , e i +1 , . . . , e j ) . Mike wishes to minimize the amount of computation per query, since there will be a lot of traffic to the website. If he precomputes and stores max( e i , . . . , e j ) for every possible input ( i, j ) , he can respond to user queries quickly, but he needs storage Ω( n 2 ) which is too much. In order to reduce storage requirements, Mike is willing to allow a small amount of computation per query. He wants to store a cleverer selection of precomputed values than just max( e i , . . . , e j ) for every ( i, j ) , so that for any user query, the server can retrieve two precomputed values and take the maximum of the two to return the final answer. Show that now only O ( n log n ) values need to be precomputed. Solution: We are given the list e 1 , . . . , e n . For each 1 i n/ 2 , store max( e i , e i +1 , . . . , e n/ 2 ) , and for each n/ 2 < j n , store max( e n/ 2+1 , . . . , e j ) . Recurse separately on the two lists e 1 , . . . , e n/ 2 and e n/ 2+1 , . . . , e n . Stop the recursion when the list size becomes 1 . If the user’s query is ( i, j ) with i n/ 2 and j > n/ 2 , then we can return max(max( e i , e i +1 , . . . , e n/ 2 ) , max( e n/ 2+1 , . . . , e j )) . If both i, j n/ 2 or i, j > n/ 2 , then the answer is found recursively. Let S ( n ) be the number of values stored for a list of length n . By construction, S ( n ) = O ( n ) + 2 S ( n/ 2) , and therefore, S ( n ) = O ( n log n ) .
6.006 Final Exam Solutions Name 3 Problem 3. Longest Simple Cycle [30 points] (2 parts) Given an unweighted, directed graph G = ( V, E ) , a path h v 1 , v 2 , ..., v n i is a set of vertices such that for all 0 < i < n , there is an edge from v i to v i +1 . A cycle is a path such that there is also an edge from v n to v 1 . A simple path is a path with no repeated vertices and, similarly, a simple cycle is a cycle with no repeated vertices. In this question we consider two problems: L ONGEST S IMPLE P ATH : Given a graph G = ( V, E ) and two vertices u, v V , find a simple path of maximum length from u to v or output NONE if no path exists.

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final-f2010-sol - Introduction to Algorithms Massachusetts...

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