What Are the Big-O Notations for n^(n-1) and (n-1)^n?

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In summary, the big-O notations for both functions, n^(n-1) and (n-1)^n, would be O(nn). However, it is incorrect to say that O(n^(n-1)) = n^(n-1) or that O((n-1)^n) = (n-1)^n. The notation "O()" is used to represent an asymptotic upper bound, not a function itself. It is also unlikely that there are any upper bounds simpler than nn-1 and tighter than nn for these functions.
  • #1
peterlam
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Hi!

For the following functions, what are their big-O notation?

1. n^(n-1)
2. (n-1)^n

Should their big-O notations be the same as the original functions? i.e.

1. O(n^(n-1)) = n^(n-1)?
2. O((n-1)^n) = (n-1)^n?

Please help!
Many thanks!
 
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  • #2
If you look at the definition of big-O notation, it is an asymptotic inequality. Roughly speaking, f(x) = O(g(x)) whenever g grows as fast or faster than f.

In this case, g(x) = nn grows faster than both. So it would be both convenient and correct to say that both are O(nn).

Without more context, it is impossible to say whether this is "good enough" for your purposes.
 
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  • #3
Thanks! I can understand that O(n^(n-1)) = n^n. But can I say O(n^(n-1)) = n^(n-1)? I am trying to find a tighter asymptotic upper bound.

Similar to the second case.

Thanks!
 
  • #4
peterlam said:
Thanks! I can understand that O(n^(n-1)) = n^n. But can I say O(n^(n-1)) = n^(n-1)? I am trying to find a tighter asymptotic upper bound.

I have a quibble with the notation you use above. It is correct to say that nn-1 = O(nn). This is not a real equality. It's just a notation. One might loosely read "=O(g)" as "is of order g". It is incorrect to say that O(nn-1) = nn. The latter notation suggests that O() is a function which returns a single function as its value.

I can't think of any upper bounds that are both simpler than nn-1 and tighter than nn.
 
  • #5


Hi there!

The big-O notation for these functions would be different. Let's break it down:

1. n^(n-1) - This function has a complexity of O(n^n), as n^(n-1) can be written as n * n * n * ... * n (n-1 times). This means that as n grows, the complexity also grows exponentially.

2. (n-1)^n - This function has a complexity of O((n-1)^n), as (n-1)^n can be written as (n-1) * (n-1) * (n-1) * ... * (n-1) (n times). This means that as n grows, the complexity also grows exponentially.

Therefore, the big-O notation for both of these functions would not be the same as the original functions. It is important to note that big-O notation is used to describe the worst-case scenario complexity of a function, so it does not necessarily have to be the exact same as the original function.

I hope this helps! Let me know if you have any other questions. Happy researching!
 

FAQ: What Are the Big-O Notations for n^(n-1) and (n-1)^n?

What is Big-O Notation and why is it important?

Big-O Notation is a mathematical concept used in computer science to describe the time complexity of an algorithm. It measures how the runtime of an algorithm grows as the input size increases. It is important because it allows us to compare the efficiency of different algorithms and choose the most efficient one for a given problem.

How is Big-O Notation calculated?

Big-O Notation is calculated by considering the dominant term or the term with the highest degree in the runtime of an algorithm. The constant coefficients and lower degree terms are ignored. For example, in the expression n^(n-1), the dominant term is n^(n-1) and the Big-O Notation would be O(n^(n-1)).

What does n^(n-1) represent in Big-O Notation?

n^(n-1) represents an algorithm with an exponential time complexity. This means that the runtime of the algorithm increases significantly as the input size increases. It is typically considered to be a very inefficient algorithm and should be avoided whenever possible.

How does (n-1)^n differ from n^(n-1) in terms of Big-O Notation?

Although both (n-1)^n and n^(n-1) represent exponential time complexity, there is a slight difference in their growth rates. (n-1)^n has a slightly lower growth rate compared to n^(n-1), which means it is a slightly more efficient algorithm. However, both algorithms should still be avoided if possible due to their exponential time complexity.

Can Big-O Notation be used to compare algorithms with different input sizes?

Yes, Big-O Notation can be used to compare algorithms with different input sizes. It is specifically designed to measure the growth rate of an algorithm as the input size increases, which allows for easy comparison between different algorithms. However, it is important to note that Big-O Notation only considers the worst-case scenario and may not accurately reflect the actual runtime of an algorithm on a specific input size.

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