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# HW5 - (b Give an algorithm for sorting a list of n keys...

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Fall Semester 2008 CS300 Algorithms Homework #5 (due 11/7) 1. Assume that a list of distinct keys is sorted in the decreasing order. You want to sort them in increasing order by Heapsort. (a) How many comparisons of keys are done in the heap construction phase if there are 10 keys? (b) How many are done if there are n keys? Show how you drive your answer. (c) Is this list in the decreasing order a best case for the heap construction? Why or why not? 2. Throughout most of sorting methods, we have assumed that the keys in the list to be sorted were distinct. Often, there are duplicate keys. Such duplication could make sorting easier, but algorithms that were designed for distinct (or mostly distinct) keys may not take advantage of the duplication. Let us consider the extreme case where there are only two possible key values, 0 and 1. (a) What is the order of the number of key comparisons done by InsertionSort in the worst case? (Describe a worst-case input.)
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Unformatted text preview: (b) Give an algorithm for sorting a list of n keys whose member is either 0 or 1 in O(n). (You are allowed to read the list only ONCE.) 3. We are given an n × n array A . A =       a 11 a 12 ··· a 1 n a 21 a 22 ··· a 2 n . . . . . . a n 1 a n 2 ··· a nn       The elements in each row are sorted, i.e., a i,j ≤ a i,j +1 , for j = 1 , 2 , ··· ,n-1 for each row i , where 1 ≤ i ≤ n . The elements in each column are also sorted, i.e., a i,j ≤ a i +1 ,j , for i = 1 , 2 , ··· ,n-1 for each column j , where 1 ≤ j ≤ n . Propose a method to ﬁnd the element a ij in A that is equal to a query value x . In other words, your method should report ( i,j ) if x = a ij for some 1 ≤ i,j ≤ n , and (0 , 0), if x is not equal to any element in A . The number of comparisons should be less than or equal to 2 n . 1...
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