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hw4 - 5-3 1-2 and k = 2 then the median element is 1 and...

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CS 473: Algorithms, Fall 2010 HW 4 (due Tuesday, September 28th) This homework contains four problems. Read the instructions for submitting homework on the course webpage . In particular, make sure that you write the solutions for the problems on separate sheets of paper; the sheets for each problem should be stapled together. Write your name and netid on each sheet. Collaboration Policy: For this home work, Problems 1-3 can be worked in groups of up to 3 students each. Problem 0 should be answered in Compass as part of the assessment HW2-Online and should be done individually. 1. (30 pts) You are given an array A of n distinct integers, and an integer k such that 1 k n . The square distance between a pair of integers x, y is defined as the quantity ( x - y ) 2 . Your goal is to design an O ( n ) time algorithm to find k elements in A with the smallest square distance to the median (i.e. the element of rank b n/ 2 c in A ). For instance, if
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Unformatted text preview: , 5 ,-3 , 1 ,-2] and k = 2, then the median element is 1, and the 2 elements in A with the smallest square distance to the median are { 1 ,-2 } . If k = 3, then you can output either { 1 ,-2 ,-3 } or { 1 ,-2 , 5 } . Hint: Use the linear time Selection algorithm. 2. (30 pts) Describe an O ( n log n ) time algorithm to compute the longest decreasing subsequence of a given sequence of n numbers. ( Hint : use an appropriate data structure to speed up the dynamic programming algorithm.) 3. (30 pts) Let T be rooted tree with integer lengths on its edges (they can be positive, negative or zero). Describe an algorithm that ﬁnds among all paths from a node in T to one of its descendants the one with the smallest length. The instance of the problem given below has the smallest path E-F-H of length-9. A B C D E F G H I J 1 2-5-4-5 2-2-7 3 1...
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