Euler072 - application of vector operations to reduce calculation time and refactoring numpy

[quant12345]

Describe your change:

Application of vector operations to reduce calculation time and refactoring numpy.
With limit: int = 1_000_000, the calculation of the new algorithm is 40% faster, limit: int = 10_000_000 is almost twice as fast.

code test

import datetime

import numpy as np

"""
Problem 72 Counting fractions: https://projecteuler.net/problem=72

Description:

Consider the fraction, n/d, where n and d are positive integers. If n<d and HCF(n,d)=1,
it is called a reduced proper fraction.
If we list the set of reduced proper fractions for d ≤ 8 in ascending order of size, we
get: 1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 5/7,
3/4, 4/5, 5/6, 6/7, 7/8
It can be seen that there are 21 elements in this set.
How many elements would be contained in the set of reduced proper fractions for
d ≤ 1,000,000?

Solution:

Number of numbers between 1 and n that are coprime to n is given by the Euler's Totient
function, phi(n). So, the answer is simply the sum of phi(n) for 2 <= n <= 1,000,000
Sum of phi(d), for all d|n = n. This result can be used to find phi(n) using a sieve.

Time: 1 sec
"""


def solution(limit: int = 1_000_000) -> int:
    """
    Returns an integer, the solution to the problem
    >>> solution(10)
    31
    >>> solution(100)
    3043
    >>> solution(1_000)
    304191
    """

    phi = [i - 1 for i in range(limit + 1)]

    for i in range(2, limit + 1):
        if phi[i] == i - 1:
            for j in range(2 * i, limit + 1, i):
                phi[j] -= phi[j] // i

    return sum(phi[2 : limit + 1])


def solution_new(limit: int = 1_000_000) -> int:
    """
    Returns an integer, the solution to the problem
    >>> solution(10)
    31
    >>> solution(100)
    3043
    >>> solution(1_000)
    304191
    """

    # generating an array from -1 to limit
    phi = np.arange(-1, limit)

    for i in range(2, limit + 1):
        if phi[i] == i - 1:
            ind = np.arange(2 * i, limit + 1, i)# indexes for selection
            phi[ind] -= phi[ind] // i

    return np.sum(phi[2 : limit + 1])


if __name__ == "__main__":

    now = datetime.datetime.now()
    old = solution()
    print('Returns an integer old', old)
    time_original = datetime.datetime.now() - now
    print('time_original', time_original.total_seconds())

    now = datetime.datetime.now()
    new = solution_new()
    print('Returns an integer new', new)
    time_new = datetime.datetime.now() - now
    print('time_new', time_new.total_seconds())

    print('old == new', old == new)

Output:

limit: int = 1_000_000

Returns an integer old 303963552391
time_original 1.236859
Returns an integer new 303963552391
time_new 0.816209
old == new True


limit: int = 10_000_000

Returns an integer old 30396356427241
time_original 17.871657
Returns an integer new 30396356427241
time_new 9.741272
old == new True

• Add an algorithm?
• Fix a bug or typo in an existing algorithm?
• Documentation change?

Checklist:

• I have read CONTRIBUTING.md.
• This pull request is all my own work -- I have not plagiarized.
• I know that pull requests will not be merged if they fail the automated tests.
• This PR only changes one algorithm file. To ease review, please open separate PRs for separate algorithms.
• All new Python files are placed inside an existing directory.
• All filenames are in all lowercase characters with no spaces or dashes.
• All functions and variable names follow Python naming conventions.
• All function parameters and return values are annotated with Python type hints.
• All functions have doctests that pass the automated testing.
• All new algorithms include at least one URL that points to Wikipedia or another similar explanation.
• If this pull request resolves one or more open issues then the description above includes the issue number(s) with a closing keyword: "Fixes #ISSUE-NUMBER".