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Recursive Complexity
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Let's now take a look at a recursive function. A recursive function is a function that calls itself. Can we analyze its complexity? The following function, written in Python, evaluates the factorial of a given number. The factorial of a positive integer number is found by multiplying it with all the previous positive integers together. For example, the factorial of 5 is 5 * 4 * 3 * 2 * 1. We denote that '5!' and pronounce it 'five factorial'
2.
if
n
=
=
1
:
4.
return
n
*
factorial( n
-
1
)
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Let us analyze the complexity of this function. This function doesn't have any loops in it, but its complexity isn't constant either. What we need to do to find out its complexity is again to go about counting instructions. Clearly, if we pass some n to this function, it will execute itself n times. If you're unsure about that, run it 'by hand' now for n = 5 to validate that it actually works. For example, for n = 5, it will execute 5 times, as it will keep decreasing n by 1 in each call. We can see therefore that this function is then Θ( n ).
If you're unsure about this fact, remember that you can always find the exact complexity by counting instructions. If you wish, you can now try to count the actual instructions performed by this function to find a function f( n ) and see that it's indeed linear (recall that linear means Θ( n )).
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See Figure 5 for a diagram to help you understand the recursions performed when factorial( 5 ) is called.This should clear up why this function is of linear complexity.
