CS 4120/5120 Homework 2

This is a homework assignment for the course CS 4120/5120, focusing on fundamental concepts in algorithms and data structures. The assignment includes problems on drawing recursion trees, solving recurrences using substitution and the Master Method, implementing Merge Sort and a 3-way Merge Sort, and working with heaps, including building a max-heap and performing heap sort. The problems require students to demonstrate their understanding of these concepts through both theoretical analysis and practical application.

CS 4120/5120 Spring 2017 Homework 2
_______ / 50 points
DUE MONDAY FEBRUARY 13, 2017 7:30PM
Turn in both an electronic copy on Canvas and a printed copy in lecture.
Name ______________________________________
ID ______________________________________
1. [5 pts] Draw the recursion tree (see Fig 2.5, page 38 in the book for an example) for the recurrence:
T(n) = θ(1) if n = 1
= 2T(n/2) + n2 if n > 1
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CS 4120/5120 Spring 2017 Homework 2
2. [2 pts] Use the tree from Problem 1 to guess a solution to the recurrence.
3. [5 pts] Use the substitution method to solve the recurrence based on your guess in Problem 2.
4. [5 pts] Use the Master Method to solve the recurrence in Problem 1.
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CS 4120/5120 Spring 2017 Homework 2
5. [5 pts] Use MERGE-SORT algorithm (available on Page 34 in textbook) to sort the following list:
5 8 2 1 6 6 4 2 3
Show the steps (One example is illustrated in Fig 2.4 on Page 35 in the textbook).
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CS 4120/5120 Spring 2017 Homework 2
6. [7 pts] Design a 3-way merge sort algorithm, which divides the given array into three equal parts,
recursively sorts each part, then merges the results. In the main MergeSort3(A,p,r) algorithm,
you may assume the existence of an appropriate Merge3(A,p,q1,q2,r) linear-time ((n))
algorithm. Provide the pseudocode for the main algorithm (but not for the Merge3 helper). Don't forget
to properly handle the base case!
MergeSort3(A,p,r)
7. [3 pts] Give a recurrence relation T(n) for your solution to Problem 6.
8. [3 pts] Use the Master Method to solve the recurrence you gave in Problem 7.
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CS 4120/5120 Spring 2017 Homework 2
9. [3 pts] Given the following input array [40, 10, 30, 80, 5, 60] show as a “heap” (in binary tree form).
Note that this initial “heap” may not satisfy the max-heap property.
10. [5 pts] Convert the heap in Problem 9 into a maximum-heap. Show each step of BUILD-MAX-HEAP
(including marking the node i in each step, similar to Figure 6.3, page 158).
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CS 4120/5120 Spring 2017 Homework 2
11. [7 pts] Show HEAPSORT on the final maximum-heap from Problem 9 (i.e., skip line 1 of the algorithm
that calls BUILD-MAX-HEAP). Show each step (including marking the node i in each step, similar to
Figure 6.4, page 161).
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