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CS4040/5040 Homework # 3

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CS4040/5040
Homework # 3
(CS4040 60 pts / CS5040 70 pts)
1. (5 pts.) Draw a decision tree that achieves the smallest average number of comparisons
to sort a list of four items into the order 1, 2, 3, 4. What is the height of your tree?
What is the average number of comparisons needed for your tree assuming all orders
are equally likely? What are the best case and worst cases for your tree? Does your
tree model any of the standard sorts we have studied?
2. (5 pts.) Give a detailed description of an algorithm (pseudo code) that will sort n
integers in the range 1 to n
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in O(n) time. Be sure to justify why your algorithm takes
O(n) time.
3. (6 pts.) For each of the following standard sorts, explain exactly why it is either a
stable sort, or an unstable sort. Clearly state any assumptions you make.
(a) quicksort
(b) bubble sort
(c) insertion sort
(d) counting sort
(e) heap sort
(f) bucket sort
(g) shell sort
4. (4 pts.) Show the steps in sorting the following English words using radix sort. Start
with them in this exact relative order. Note, all these words are actual words in the
scrabble dictionary!
THIG PLOD BACK EVIL SCAG EDGE YEGG PLOT DOGE ACTS PLOW AWAY
RANG BABE ACRE WING BABY PLUG ACTA BABU PLOP FILL ACNE DOPE
BABA
5. (10 pts.) You are given an array of integers, each with an unknown number of digits.
You are also told the total number of digits of all the integers in the array is n.
Provide an algorithm that will sort the array in O(n) time no matter how the digits
are distributed among the elements in the array. (e.g. there might be one element
with n digits, or n/2 elements with 2 digits, or the elements might be of all different
lengths, etc. Be sure to justify in detail the run time of your algorithm.
6. (10 pts.) Problem 7-4, page 188.
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Wednesday, September 29, 2021
7. (10 pts.) To find the top wage earner in the country (i.e. the one with the highest
reported IRS income) requires O(n) time (use the algorithm that finds the maximum).
Suppose instead we want to find the set of people that are in the top k% by reported
income (not necessarily in sorted order). Describe an algorithm (psuedocode) to solve
this problem in the smallest possible asymptotic time complexity. Argue that your
algorithm is optimal. Analyze the space and time complexity of your algorithm in
detail.
8. (10 pts.) In the Select algorithm the input elements are divided into groups of 5
elements. Be sure to justify your answers to the questions below.
(a) Does the algorithm run in linear time if the elements are divided into groups of 3
elements?
(b) Does the algorithm run in linear time if the elements are divided into groups of 7
elements?
(c) CS5040 only: What is the optimal number of elements to divide the input into
if we care only about the detailed time complexity of the algorithm. Note: I am
NOT talking about the big-O complexity here, but rather the exact number of
operations needed to calculate the answer. Show a detailed analysis for why you
think it is the optimum number of elements.
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