Homework Answers
a. Let T(n) be the number of minutes needed to fry n pancakes on
both sides by the
algorithm given. Then we have the following recurrence for
T(n):
T(n) = T(n − 2) + 2 for n > 2, T(1) = 2, T(2) = 2.
T(n) = T(n-2) + 2
= T(n-4) + 4
= T(n-6) + 6
.....
T(n) = T(n-(n-2))+2n = T(2) + 2n = 2 + 2n
T(n) = O(n).
solution is T(n) = n for every even n > 0 and
T(n) = n + 1 for every odd n > 0
and can be obtained either by backward substitutions or by applying the formula for the generic term of an arithmetical progression.
b)
The algorithm fails to execute the task of frying pancakes in the minimum amount of time when n is odd for all n>1.
For example:
T(3) = 4 minutes to fry 3 pancakes but there is a much quicker process for that
First, fry pancakes 1 and 2 on one side. -> Takes 1 minute
Then fry pancake 1 on the second side together with pancake 3 on its first side. -> Takes 1 minute
Finally, fry both pancakes 2 and 3 on the second side. -> takes 1 minute
Total time taken is 3 minutes but as per algorithm, it will take much more long-time i.e, 4 minutes.
c)
If n ≤ 2, fry the pancake (or the two hamburgers together if n =
2) on each side.
If n = 3, fry the pancakes in 3 minutes as indicated in the
previous answer.
If n > 3, fry two pancakes together on each side and then fry
the remaining n − 2 pancakes by the same
algorithm. The recurrence for the number of minutes needed to fry n
pancakes looks now as follows:
T(n) = T(n − 2) + 2 for n > 3, T(1) = 2, T(2) = 2, T(3) =
3.
Add Answer to:
can you plzzz explain in detial
There are n pancakes to be fried on a small...
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subarrays. Finally, you would merge one pair of four-entry
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in each pair of subarrays contain the same number of entries.
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