Show that log(n !)=Θ(n logn) (Hint: To show an upper bound,...

Key Concepts

Factorial Function

The factorial function, denoted n!, is the product of all positive integers from 1 to n. It is a fundamental function in combinatorics and analysis, which grows very rapidly as n increases. Understanding its growth behavior is crucial in various areas of mathematics and computer science, particularly in counting and complexity analysis.

Asymptotic Analysis (Theta Notation)

Theta notation (?) is used to describe the asymptotic tight bound of a function. It provides a way to classify functions according to their growth rates by capturing the leading-order behavior for large inputs. In the context of the problem, showing that log(n!) is ?(n log n) means that log(n!) grows on the order of n log n for large n, with both an upper and a lower bound proportional to n log n.

Logarithmic Properties

Logarithmic functions transform multiplicative relationships into additive ones, making them a powerful tool when dealing with products like the factorial function. By applying the logarithm to n!, the product becomes a sum, which is easier to analyze and compare with other functions, especially when establishing asymptotic bounds.

Comparison Techniques for Upper and Lower Bounds

Comparison techniques involve bounding the function of interest between two simpler functions whose asymptotic growth rates are well understood. For the upper bound, n! is compared to n^n, while for the lower bound, it is compared to (n/2)^(n/2). These comparisons help to establish that log(n!) grows proportionally to n log n by showing that it is sandwiched between two functions that behave as ?(n log n).

Transcript

00:01 So we would like to show that log of n factorial is big theta of n times log n.

00:14 So first, to establish an upper bound, i'm going to notice that n factorial is equal to, of course, n times n minus 1 times n minus 2 times dot dot dot times 3 times 2 times 1, which is a product of n factors, right? n down to 1.

00:41 N factors, which is less than n times n times n times n dot times n, also for n factors, since almost all of these are just n, or almost all of these are less than n, the exception of n itself.

01:04 This is equal to n to the n.

01:09 So n factorial is less than n to the n, since log is increasing.

01:15 That tells us log of n factorial is less than log of n to the n, which is equal to n times log n.

01:24 Great, so we conclude then that log of n factorial is big o of n log n.

01:34 Now, conversely, i'd like to notice that n factorial, again, n times n minus 1 times n minus 2 dot dot dot 3 times 2 times 1, is greater than n times n minus 1 minus 2 dot dot dot.

01:54 So, this is true to n over 2, or n plus 1 over 2, n minus 1 over 2.

02:05 This in the case of n is even.

02:07 This in the case of n is odd.

02:10 In general, this is going to be, and let me pull this down so i have a little bit more space.

02:18 This should be n minus 2 factors.

02:25 Sorry, not n minus 2, n over 2 factors...

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