Python：寻找除数的性能

Question

对这些功能进行基准测试：

def divisors_optimized(number):
    square_root = int(math.sqrt(number))

    for divisor in range(1, square_root):
        if number % divisor == 0:
            yield divisor
            yield number / divisor

    if square_root ** 2 == number:
        yield square_root

def number_of_divisors_optimized(number):
    count = 0
    square_root = int(math.sqrt(number))

    for divisor in range(1, square_root):
        if number % divisor == 0:
            count += 2

    if square_root ** 2 == number:
        count += 1

    return count

你可以看到两者的基本结构是相同的。

基准代码：

number = 9999999
for i in range(10):
    print(f"iteration {i}:")
    start = time.time()
    result = list(utils.divisors_optimized(number))
    end = time.time()
    print(f'len(divisors_optimized) took {end - start} seconds and found {len(result)} divisors.')

    start = time.time()
    result = utils.number_of_divisors_optimized(number)
    end = time.time()
    print(f'number_of_divisors_optimized took {end - start} seconds and found {result} divisors.')

    print()

输出：

iteration 0:
len(divisors_optimized) took 0.00019598007202148438 seconds and found 12 divisors.
number_of_divisors_optimized took 0.0001919269561767578 seconds and found 12 divisors.

iteration 1:
len(divisors_optimized) took 0.00019121170043945312 seconds and found 12 divisors.
number_of_divisors_optimized took 0.00020599365234375 seconds and found 12 divisors.

iteration 2:
len(divisors_optimized) took 0.000179290771484375 seconds and found 12 divisors.
number_of_divisors_optimized took 0.00019049644470214844 seconds and found 12 divisors.

iteration 3:
len(divisors_optimized) took 0.00019025802612304688 seconds and found 12 divisors.
number_of_divisors_optimized took 0.00020170211791992188 seconds and found 12 divisors.

iteration 4:
len(divisors_optimized) took 0.0001785755157470703 seconds and found 12 divisors.
number_of_divisors_optimized took 0.00017905235290527344 seconds and found 12 divisors.

iteration 5:
len(divisors_optimized) took 0.00022721290588378906 seconds and found 12 divisors.
number_of_divisors_optimized took 0.00020170211791992188 seconds and found 12 divisors.

iteration 6:
len(divisors_optimized) took 0.0001919269561767578 seconds and found 12 divisors.
number_of_divisors_optimized took 0.00018930435180664062 seconds and found 12 divisors.

iteration 7:
len(divisors_optimized) took 0.00017881393432617188 seconds and found 12 divisors.
number_of_divisors_optimized took 0.00017905235290527344 seconds and found 12 divisors.

iteration 8:
len(divisors_optimized) took 0.00017976760864257812 seconds and found 12 divisors.
number_of_divisors_optimized took 0.0001785755157470703 seconds and found 12 divisors.

iteration 9:
len(divisors_optimized) took 0.00024819374084472656 seconds and found 12 divisors.
number_of_divisors_optimized took 0.00020766258239746094 seconds and found 12 divisors.

您可以看到执行时间非常接近，每次都有利于。

有人可以向我解释一下，为什么从生成器中创建列表并检索其长度与迭代时计数一样快？我的意思是，内存分配（list()）不应该比分配贵得多吗？

我正在使用 Python 3.6.3。

Answer 1

你测试的东西远远多于你生产的东西。 int 与 list 在“找到的因子”情况下的生成器操作的成本与完成的总工作量相比相形见绌。您正在执行 3000 多个试验部门；十二个yields 与十二个添加是对这类工作的小改动。将加法/yields 替换为 pass（什么都不做），您会发现它仍然运行（大致）相同的时间：

def ignore_divisors_optimized(number):
    square_root = int(math.sqrt(number))

    for divisor in range(1, square_root):
        if number % divisor == 0:
            pass

    if square_root ** 2 == number:
        pass

以及使用ipython 的%timeit 魔术进行微基准测试：

>>> %timeit -r5 number_of_divisors_optimized(9999999)
266 µs ± 1.85 µs per loop (mean ± std. dev. of 5 runs, 1000 loops each)
>>> %timeit -r5 list(divisors_optimized(9999999))
267 µs ± 1.29 µs per loop (mean ± std. dev. of 5 runs, 1000 loops each)
>>> %timeit -r5 ignore_divisors_optimized(9999999)
267 µs ± 1.43 µs per loop (mean ± std. dev. of 5 runs, 1000 loops each)

number_of_divisors 快了一微秒这一事实无关紧要（重复测试的抖动高于一微秒）；它们的速度基本相同，因为 >99% 的工作是循环和测试，而不是测试通过时所做的工作。

这是 90/10 优化规则的一个示例：大约 90% 的时间花在 10% 的代码上（在这种情况下，是试用部门本身）； 10% 用于其他 90% 的代码。您正在优化运行时间为 10% 的 90% 代码的一小部分，但这并没有帮助，因为绝大多数时间都花在了 if number % divisor == 0: 行上。如果您删除该测试以支持只循环 range 什么都不做，则运行时间在我的本地微基准测试中下降到 ~78 µs，这意味着该测试占用了将近 200 µs 的运行时间，是所有其余部分的两倍多代码放在一起需要。

如果您想对此进行优化，您需要查看加速试验分割线本身的方法（这基本上意味着不同的 Python 解释器或使用 Cython 将其编译为 C），或者运行该线的方法更少的时间（例如，预先计算可能的素因子直到某个界限，因此对于任何给定的输入，您可以避免测试非素因子，然后从已知的素因子及其多重性中产生/计算非素因子的数量）。