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Unformatted text preview: CS4102, Spring 2010. Homework 3: Solutions Textbook, p. 253. Question 13. This depends on which partition algorithm is used. If we use Lomuto’s partition algorithm (p. 245 in the book), then the if
statement fails for each iteration of the loop, and the value of h is never incremented before it is returned. Thus it remains equal to the index of the first item in the sub
list being partitioned. This leads to the worst
case situation for quicksort. The number of comparisons will be: Problem Rec2: See solution posted with Homework 2 solutions. Problem Sort
LB
1: n n −1 W ( n ) = ∑ ( k − 1) = ∑ k =
k =2 k =1 n ( n − 1) ∈ Θ( n 2 ) 2 € The decision tree argument says this about the worst
case number of comparisons for a sorting algorithm that compares keys: W ( n ) ≥ Ⱥlg n!Ⱥ ≥ Ⱥn lg n − 1.443nȺ For n=5, n! is 120. Since lg 128 is 7, W(5) ≥ 7. So the lower
bound for n=5 is 7. Problem Invers
1: A simple modification to merge accomplishes this task. Referring to the pseudo
code to the right, the change is made in the final else. When the first
item in the remaining part of B (the right
half of the list) is less
than the first
item in what’s remaining in A (the left
half of the list), then that item from B is out
of
order with respect to all the remaining items in A. So keep a running sum where you add the number of remaining items in A at the start of that else
clause. € Merge(A, B, C) // merging A and B into C if (A is empty) rest of C = rest of B else if (B is empty) rest of C = rest of A else if (first of A <= first of B) first of C = first of A merge (rest of A, B, rest of C) else // first of B < first of A first of C = first of B merge (A, rest of B, rest of C) Example: n is 8 and you’re merging the first 4 with the last 4 for this list: [1, 3, 6, 9, 0, 2, 7, 10] Right way the 0 will be placed into C. But it is an inversion with the 4 items: 1, 3, 6, 9. Later, after 1 from A is put into C, you need to put 2 into C. That means it’s an inversion with 3, 6 and 9. Later 7 is put into C before 9, so that’s another inversion. There are thus 4+3+1=9 inversions in that list. Problem QS
Med3
1: You need at least 3 comparisons to find the median of three values. (Draw a decision tree to convince yourself of this.) You could code it with nested if’s, like the decision tree. If you have the median of three for the partition element, you can’t have the worst
case where you have sub
lists of size 0 and n
1 to sort. But, if the median could be 2nd largest or 2nd smallest, which means you’d have sub
lists of size 1 and n
2 to sort. So the recurrence would be: W(n) = (n
1) + 3 + W(1) + W(n
2) = W(n
2) + n + 2 ...
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This note was uploaded on 03/21/2010 for the course CS 4102 taught by Professor Horton during the Spring '10 term at UVA.
 Spring '10
 HORTON
 Algorithms

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