hbs5 - penalty for that day is (200-x ) 2 . You want to...

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CS 473: Algorithms, Fall 2010 HBS 5 Problem 1. [Largest Contiguous Sum] Suppose you are given an array A [1 ...n ] of numbers, which may be positive, negative, or zero. De- scribe an algorithm that finds the largest of elements in a contiguous subarray A [ i...j ]. For example, if the array contains the numbers ( - 6 , 12 , - 7 , 0 , 14 , - 7 , 5), then the largest sum of any contiguous subarray is 19 = 12 - 7 + 0 + 14. Problem 2. [Hotel Stops] You are going on a long trip. You start on the road at mile post 0. Along the way, there are n hotels, at mile posts a 1 < a 2 < · · · < a n , where each a i is measured from the starting point. The only places you are allowed to stop are at these hotels, but you can choose which of the hotels you stop at. You must stop at the final hotel (at distance a n ), which is your destination. You’d ideally like to travel 200 miles a day, but this many not be possible (depending on the spacing of the hotels). If you travel x miles during a day, the
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Unformatted text preview: penalty for that day is (200-x ) 2 . You want to plan your trip so as to minimize the total penaltythat is, the sum over all travel days, of the daily penalties. Give an ecient algorithm that determines the optimal sequence of hotels at which to stop. Problem 3. [Longest Common Subsequence] Let A [1 ...m ] and B [1 ...n ] be two arbitrary arrays. A common subsequence of A and B is another sequence that is a subsequence of both A and B . Describe an ecient algorithm to compute the length of the longest common subsequence of A and B . A subsequence is anything obtained from a sequence by extracting a subset of elements, but keep-ing them in the same order; the elemtns of the subsequence need not be contiguous in the original sequence. For example, the strings C , DAMN , and YAIOAI , and DYNAMICPROGRAMMING are all subsequences of the sequence DYNAMICPROGRAMMING . 1...
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