GATE CSE 2012 | Question: 18

Question

Let $W(n) $ and $A(n)$ denote respectively, the worst case and average case running time of an algorithm executed on an input of size $n$.  Which of the following is ALWAYS TRUE?

• A. $A(n) = \Omega (W(n))$
• B. $A(n) = \Theta (W(n))$
• C. $A(n) = \text{O} (W(n))$
• D. $A(n) = \text{o} (W(n))$

Comments

As arjun sir explained dats the reason c is the answer

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Answer will be C

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Average case of quick sort is $O(nlogn)$ and worst case is $O(n^2)$.
$$Best Case \leq AvgCase \leq WorstCase$$
Hence, (C) is the correct option.

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is this always holds true?

BC<=Avg Case<=WC

Answer 1

Worst case complexity can never be lower than the average case complexity, but it can be higher. So, $(C)$ is the answer.
$A(n) = O(W(n))$.

Comments

Thanks for this statement. Really helped in clearing my concept.

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I am not able to interpret the options properly. 

A(n)=O(W(n))

Says, W(n) is acting as Upper Bound for A(n), right? 

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@kirtipurohit Yes, W(n) is acting as an upper bound for A(n) because the time for Average case can never exceed the time for Worst case. Also note that W(n) > A(n) and NOT  W(n) > A(n) because for some algorithms (for eg in mergesort) the time for Worst case and the time for Average case can be same ,i.e, W(n) = A(n). So option D is false and option C is the correct answer.

Answer 2

Let F(n)=A(n) , G(n)=W(n)

Rule :- For Omega :-   F(n)>=cG(n)

Option A -  A(n)=Ω(W(n))   it means A(n)>=cW(n) , which is false becoz Worst case time  can not be less than avg case time.

Rule :-  For Theta :-     c1G(n) <= F(n) <= c2G(n)

Option B -  A(n)=Θ(W(n))  it means c1W(n)<=A(n)<=c2W(n) , Which can not be always true. (i.e it is false due to Bold markd line.. )

Rule :- For Big Oh :- F(n)<=cG(n)

option C-  A(n)=O(W(n)) it means A(n)<=cW(n) , which is always true........

options D:- I dont know..

Comments

I am not able to interpret the options properly. 

A(n)=O(W(n))

Says, W(n) is acting as Upper Bound for A(n), right? 

Answer 3

if f(n) ≤ Cg(n)

then f(n) = O(g(n))

 

BestCase ≤ AvgCase≤WorstCase

so AvgCase ≤ WorstCase

AvgCase = O(WorstCase) will be always true

 

=> A(n)=O(W(n))

Answer 4

The average case time can be lesser than or even equal to the worst case.
So, A(n) would be upper bounded by W(n) and it will not be strict upper bound as it can even be same (e.g. Bubble Sort and merge sort).
A(n)=O(W(n))
Note: Option A is wrong because A(n) is not equal to Ω(w(n)) .

 

Hence it is option C

Comments

I am not able to interpret the options properly. 

A(n)=O(W(n))

Says, W(n) is acting as Upper Bound for A(n), right? 

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@kirtipurohit

Take an example of quick sort

worst case W(n) = n$^2$ (when elements are same or are in ascending or descending order)

average case A(n) = nlogn

nlogn is always <= n$^2$

nlogn = O(n$^2$)

Hence A(n) = O(W(n))

 

 

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Thanks, it helped. So my statement is ultimately correct, right? 

Says, W(n) is acting as Upper Bound for A(n)

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No ,

Worst case doesn’t have anything to do with upper bound.

 

reference :https://news.ycombinator.com/item?id=2322051