Python Time Complexity (run-time)

Question

def f2(L):
    sum = 0
    i = 1
    while i < len(L):
        sum = sum + L[i]
        i = i * 2
    return sum

Let n be the size of the list L passed to this function. Which of the following most accurately describes how the runtime of this function grow as n grows?

(a) It grows linearly, like n does. (b) It grows quadratically, like n^2 does.

(c) It grows less than linearly. (d) It grows more than quadratically.

I don't understand how you figure out the relationship between the runtime of the function and the growth of n. Can someone please explain this to me?

Answer 1

ok, since this is homework:

this is the code:

def f2(L):
    sum = 0
    i = 1
    while i < len(L):
        sum = sum + L[i]
        i = i * 2
    return sum

it is obviously dependant on len(L).

So lets see for each line, what it costs:

sum = 0
i = 1
# [...]
return sum

those are obviously constant time, independant of L. In the loop we have:

    sum = sum + L[i] # time to lookup L[i] (`timelookup(L)`) plus time to add to the sum (obviously constant time)
    i = i * 2 # obviously constant time

and how many times is the loop executed? it's obvously dependant on the size of L. Lets call that loops(L)

so we got an overall complexity of

loops(L) * (timelookup(L) + const)

Being the nice guy I am, I'll tell you that list lookup is constant in python, so it boils down to

O(loops(L)) (constant factors ignored, as big-O convention implies)

And how often do you loop, based on the len() of L?

(a) as often as there are items in the list (b) quadratically as often as there are items in the list?

(c) less often as there are items in the list (d) more often than (b) ?

Answer 2

I am not a computer science major and I don't claim to have a strong grasp of this kind of theory, but I thought it might be relevant for someone from my perspective to try and contribute an answer.

Your function will always take time to execute, and if it is operating on a list argument of varying length, then the time it takes to run that function will be relative to how many elements are in that list.

Lets assume it takes 1 unit of time to process a list of length == 1. What the question is asking, is the relationship between the size of the list getting bigger vs the increase in time for this function to execute.

This link breaks down some basics of Big O notation: http://rob-bell.net/2009/06/a-beginners-guide-to-big-o-notation/

If it were O(1) complexity (which is not actually one of your A-D options) then it would mean the complexity never grows regardless of the size of L. Obviously in your example it is doing a while loop dependent on growing a counter i in relation to the length of L. I would focus on the fact that i is being multiplied, to indicate the relationship between how long it will take to get through that while loop vs the length of L. Basically, try to compare how many loops the while loop will need to perform at various values of len(L), and then that will determine your complexity. 1 unit of time can be 1 iteration through the while loop.

Hopefully I have made some form of contribution here, with my own lack of expertise on the subject.

Update
To clarify based on the comment from ch3ka, if you were doing more than what you currently have inside your with loop, then you would also have to consider the added complexity for each loop. But because your list lookup L[i] is constant complexity, as is the math that follows it, we can ignore those in terms of the complexity.