What is the computational complexity of `itertools.combinations` in python?

I would say it is θ[r (n choose r)], the n choose r part is the number of times the generator has to yield and also the number of times the outer while iterates.

In each iteration at least the output tuple of length r needs to be generated, which gives the additional factor r. The other inner loops will be O(r) per outer iteration as well.

This is assuming that the tuple generation is actually O(r) and that the list get/set are indeed O(1) at least on average given the particular access pattern in the algorithm. If this is not the case, then still Ω[r (n choose r)] though.

As usual in this kind of analysis I assumed all integer operations to be O(1) even if their size is not bounded.

I had this same question too (For itertools.permutations) and had a hard time tracing the complexities. This led me to visualize the code using matplotlib.pyplot;

The code snippet is shown below

result=[]
import matplotlib.pyplot as plt
import math
x=list(range(1,11))
def permutations(iterable, r=None): 
    count=0
    global result
    pool = tuple(iterable)
    n = len(pool)
    r = n if r is None else r
    if r > n:
        return
    indices = list(range(n))
    cycles = list(range(n, n-r, -1))
    yield tuple(pool[i] for i in indices[:r])
    while n:
        for i in reversed(range(r)):
            count+=1
            cycles[i] -= 1
            if cycles[i] == 0:
                indices[i:] = indices[i+1:] + indices[i:i+1]
                cycles[i] = n - i
            else:
                j = cycles[i]
                indices[i], indices[-j] = indices[-j], indices[i]
                yield tuple(pool[i] for i in indices[:r])
                break
        else:
            resulte.append(count)
            return
for j in x:
    for i in permutations(range(j)):
        continue

x=list(range(1,11))
plt.plot(x,result)

Time Complexity graph for itertools.permutation

From the graph, it is observed that the time complexity is O(n!) where n=Input Size