Empirical Analysis and Visualization of Various Approaches to the Sieve of Eratosthenes
I discovered this algorithm at the end of the 10th chapter (Maps, Hash Tables, and Skip Lists)of "Data Structures & Algorithms in Python". I ended up writing three versions:
- A naive implementation with a nasty cubic runtime (amongst many other issues, later resolved).
- A suboptimal version after some thorough research, but I still wanted to get rid of the square-based multiples.
- A fast and efficient version on par with the selected answer performance-wise, but with additive multiples (more intuitive for those less-mathematically inclined).
Naive SOE
Out of all the issues, the additional space usage in the return statement takes the cake as the worst.
def naive_sieve(m: int): BA = [True] * m for i, k in zip(range(2, m + 1), range(len(BA))): if BA[k] is False: continue for j in range(2, i): if i % j == 0: BA[k] = False f = k + j while f < len(BA): BA[f] = False f += j break return [i for i,j in zip(range(2, m + 1), BA) if j is True]Suboptimal SOE
A drastic improvement, but still lacking on space usage, and an unfriendly inner-loop interval progression.
def suboptimal_sieve(m: int): BA = [True] * m for i, k in zip(range(2, m + 1), range(2, len(BA))): if BA[k] is False: continue for j in range(i**2, m, i): BA[j] = False return [i for i,j in zip(range(2, m + 1), BA[2:]) if j is True]Fast SOE
Easy to understand (note the usage of fractional exponent to be explicitly consistent with the mathematical definition of roots)and just as performant as @Pi Delport's answer.
def fast_sieve(m: int): BA = [True] * m rtm = int(m**(1/2)) + 1 for i in range(2, len(BA)): if BA[i]: yield i if i < rtm: f = i while f < len(BA): BA[f] = False f += iEmpirical Analysis
To compare all three implementations, along with the selected answer from @Pi Delport,I ran it through 45 iterations from primes in range 100, until primes in range 4500, at intervals of 100(that's the sweet spot for the visualization, because despite the consistency of the shape of the graphs,the growth of the naive implementation dwarves the visibility of the other three).You can tweak the visualization code on the GitHub gist, but here's one sample output:
