Consider the following algorithm. def long_prod(1st: list, t: int) -> int: "' Return the maximum length of any slice of lst whose product is at most t. Preconditions: t > 0; lst is non-empty; every element of lst is positive. # max length found so far for i in range (1, len(lst) + 1): m = 0 # Loop 1 j = i - 1 # product of lst[j+1:i] 1st[j] <= t: p = 1 while j >= 0 and p # Loop 2 * p = p * 1st[j] - 1 i + 1 if i m: m = i return m (a) Find, with proof, an input family for which the running time of long-prod is O(n4/3). Show your work.
Consider the following algorithm. def long_prod(1st: list, t: int) -> int: "' Return the maximum length of any slice of lst whose product is at most t. Preconditions: t > 0; lst is non-empty; every element of lst is positive. # max length found so far for i in range (1, len(lst) + 1): m = 0 # Loop 1 j = i - 1 # product of lst[j+1:i] 1st[j] <= t: p = 1 while j >= 0 and p # Loop 2 * p = p * 1st[j] - 1 i + 1 if i m: m = i return m (a) Find, with proof, an input family for which the running time of long-prod is O(n4/3). Show your work.
Computer Networking: A Top-Down Approach (7th Edition)
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Chapter1: Computer Networks And The Internet
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![Consider the following algorithm.
def long_prod(1st: list, t: int) -> int:
''' Return the maximum length of any slice of lst whose product is at most t.
Preconditions: t > 0; lst is non-emp ty; every element of lst is positive.
3
4
# max length found so far
for i in range (1, len(lst) + 1):
m = 0
# Loop 1
7
= i - 1
# product of lst[j+1:i]
while j >= 0 and p * lst[j] <= t:
1st [j]
8
p = 1
9
# Loop 2
10
р*
j
j
1
11
j + 1
12
%3D
13
if i
j > m:
14
m = i
15
return m
(а)
Find, with proof, an input family for which the running time of long-prod is O(n/3).
Show your work.
(b)
Find, with proof, an upper bound on the worst-case running time of long-prod. Show
your work. For full marks, your upper bound must match the lower bound from the next part.
(c)
Find, with proof, a lower bound on the worst-case running time of long-prod. Show
your work. For full marks, your lower bound must match the upper bound from the previous part.
(d)
Recall that the best-case running time of an algorithm is defined as follows:
BC(n) = min{RT(x) | x E In}
where RT(x) is the running time of the algorithm on input x, and In is the set of all inputs of size
n. (As discussed in lecture, this is similar to the definition of worst-case running time, but with min
in place of max.)
Find, with proof, a tight bound on the best-case running time of long-prod. Your analysis should
consist of two separate proofs for matching upper and lower bounds on the best-case running time.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0a416aa-90b7-478f-8c7c-e07ad14a269e%2Fa084d113-9f89-4a8d-a2a6-6f488e33e91f%2Ftdn5fnm_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the following algorithm.
def long_prod(1st: list, t: int) -> int:
''' Return the maximum length of any slice of lst whose product is at most t.
Preconditions: t > 0; lst is non-emp ty; every element of lst is positive.
3
4
# max length found so far
for i in range (1, len(lst) + 1):
m = 0
# Loop 1
7
= i - 1
# product of lst[j+1:i]
while j >= 0 and p * lst[j] <= t:
1st [j]
8
p = 1
9
# Loop 2
10
р*
j
j
1
11
j + 1
12
%3D
13
if i
j > m:
14
m = i
15
return m
(а)
Find, with proof, an input family for which the running time of long-prod is O(n/3).
Show your work.
(b)
Find, with proof, an upper bound on the worst-case running time of long-prod. Show
your work. For full marks, your upper bound must match the lower bound from the next part.
(c)
Find, with proof, a lower bound on the worst-case running time of long-prod. Show
your work. For full marks, your lower bound must match the upper bound from the previous part.
(d)
Recall that the best-case running time of an algorithm is defined as follows:
BC(n) = min{RT(x) | x E In}
where RT(x) is the running time of the algorithm on input x, and In is the set of all inputs of size
n. (As discussed in lecture, this is similar to the definition of worst-case running time, but with min
in place of max.)
Find, with proof, a tight bound on the best-case running time of long-prod. Your analysis should
consist of two separate proofs for matching upper and lower bounds on the best-case running time.
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