Download link:# Algorithms (Sec: 102) Homework 2: Sorting
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Problem 1. Workflow of Heapsort and Quicksort 25 points
Demonstrate HEAP-SORT and QUICK-SORT iterations for both the following arrays:
- A1 = {2, 6, 4, 3, 1, 5}, and (ii) A2 = {1, 5, 2, 3, 0, 2, 2, 1, 4, 5}.
Problem 2. Empirical Analysis of Heapsort and Quicksort 25 points
Implement HEAP-SORT (Page 170 with supporting functions in Pages 165, 167, all in CLRS) and QUICK-SORT (Page 183, CLRS) in Python, and validate its average run-time performance (similar to Problem 2 in Homework 1).
Problem 3. Modified Quicksort 25 points
Traditional quicksort routine chooses a pivot q such that A[p : q −1] ≤ A[q] ≤ A[q + 1, r]. Instead,
present an analysis when the quicksort algorithm partitions the array A[p : r] into three parts using
two pivots q1 and q2 such that A[p : q1 − 1] ≤ A[q1] = · · · = A[q2] ≤ A[q2 + 1 : r].
(Hint: Assume that the entries in A are picked from {1, · · · , m}, where m < n.)
Problem 4. Sort by Frequency 25 points
Write a program in Python that sorts all the integer entries in an input array A of size n according to the decreasing frequency of occurrence. If the frequency of two numbers is the same, then sort them in the increasing order of value. Assume that A[j] ∈ {0, 1, · · · , k} for all j = 1, · · · , n, and let k ≪ n to allow enough number of repetitions.
(Hint: You can find frequencies using COUNTING-SORT).
Example: Let A = {3, 5, 2, 1, 0, 1, 2, 3, 4, 2, 0, 3, 4, 2, 1}. Note that n = 15 and k = 5. Let f(i) denote the frequency of occurrence of a number i in A. Then, we have
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f(0) = 2, f(3) = 3, f(1) = 3, f(4) = 2, f(2) = 4, f(5) = 1.
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Then, the output should look like: B = {2, 2, 2, 2, 1, 1, 1, 3, 3, 3, 0, 0, 4, 4, 5}.
You are strongly encouraged to solve this problem.
SELECTION-SORT(A) sorts the input array A by first finding the jth smallest element in A and swapping it with the element in A[j], in the order j = 1, j = 2, · · · , j = n − 1. Write pseudocode for SELECTION-SORT, and find the best-case and worst-case running times of SELECTION-SORT in Θ-notation.
