Question 6: Professor Holmes has come up with a new sorting algorithm. He calls it Trinary Sort and "claims" that it is asymptotically faster than Merge Sort, despite the fact both the algorithms operate using similar logic. But, unlike Merge Sort, Trinary Sort splits the input array into (roughly) 3 equal parts at each step of the recursion as long as the array is splittable (i.e., has at least 3 elements). Trinary Sort's merge subroutine, similar in principle to the one used by Merge Sort, takes 3 sorted subarrays and merges them to produce a single sorted array. Given all of this, answer the following questions. (a) In Merge Sort, the merge subroutine makes n-1 comparisons to merge 2 arrays of size n/2, which takes (n) time. How many comparisons will the merge subroutine of Trinary Sort make to merge 3 arrays of size n/3? What would be the (...) bound on the running time for this subroutine? (b) What is the (...) bound on the running time of the Trinary Sort algorithm? Come up with the answer using the Recursion Tree Method from class. (c) What is the recurrence relation for the Trinary Sort algorithm? Also explain it in plain English what each term in the recurrence relation stands for. (d) What is the (...) bound on the running time of the Trinary Sort algorithm? This time come up with the answer through Recurrence Relation Expansion of the recurrence relation you came up with in part (c). (e) Is Professor Holmes right to claim that Trinary Sort is asymptotically faster than Merge Sort? Why/why not? Reason through your answer using a mathematical proof. (Hint: Use the limit method demonstrated in class as well as the recitation on Asymptotic Analysis & Recurrence Relations)

Database System Concepts
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Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
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Question 6:
Professor Holmes has come up with a new sorting algorithm. He calls it Trinary Sort and
"claims" that it is asymptotically faster than Merge Sort, despite the fact both the algorithms
operate using similar logic. But, unlike Merge Sort, Trinary Sort splits the input array into
(roughly) 3 equal parts at each step of the recursion as long as the array is splittable (i.e., has
at least 3 elements). Trinary Sort's merge subroutine, similar in principle to the one used by
Merge Sort, takes 3 sorted subarrays and merges them to produce a single sorted array. Given
all of this, answer the following questions.
(a)
In Merge Sort, the merge subroutine makes n-1 comparisons to merge 2 arrays
of size n/2, which takes (n) time. How many comparisons will the merge subroutine of
Trinary Sort make to merge 3 arrays of size n/3? What would be the (...) bound on
the running time for this subroutine?
(b)
What is the (...) bound on the running time of the Trinary Sort algorithm?
Come up with the answer using the Recursion Tree Method from class.
(c)
What is the recurrence relation for the Trinary Sort algorithm? Also explain it
in plain English what each term in the recurrence relation stands for.
(d)
What is the (...) bound on the running time of the Trinary Sort algorithm?
This time come up with the answer through Recurrence Relation Expansion of the
recurrence relation you came up with in part (c).
(e)
Is Professor Holmes right to claim that Trinary Sort is asymptotically faster
than Merge Sort? Why/why not? Reason through your answer using a mathematical
proof.
(Hint: Use the limit method demonstrated in class as well as the recitation on Asymptotic
Analysis & Recurrence Relations)
Transcribed Image Text:Question 6: Professor Holmes has come up with a new sorting algorithm. He calls it Trinary Sort and "claims" that it is asymptotically faster than Merge Sort, despite the fact both the algorithms operate using similar logic. But, unlike Merge Sort, Trinary Sort splits the input array into (roughly) 3 equal parts at each step of the recursion as long as the array is splittable (i.e., has at least 3 elements). Trinary Sort's merge subroutine, similar in principle to the one used by Merge Sort, takes 3 sorted subarrays and merges them to produce a single sorted array. Given all of this, answer the following questions. (a) In Merge Sort, the merge subroutine makes n-1 comparisons to merge 2 arrays of size n/2, which takes (n) time. How many comparisons will the merge subroutine of Trinary Sort make to merge 3 arrays of size n/3? What would be the (...) bound on the running time for this subroutine? (b) What is the (...) bound on the running time of the Trinary Sort algorithm? Come up with the answer using the Recursion Tree Method from class. (c) What is the recurrence relation for the Trinary Sort algorithm? Also explain it in plain English what each term in the recurrence relation stands for. (d) What is the (...) bound on the running time of the Trinary Sort algorithm? This time come up with the answer through Recurrence Relation Expansion of the recurrence relation you came up with in part (c). (e) Is Professor Holmes right to claim that Trinary Sort is asymptotically faster than Merge Sort? Why/why not? Reason through your answer using a mathematical proof. (Hint: Use the limit method demonstrated in class as well as the recitation on Asymptotic Analysis & Recurrence Relations)
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