def recursive(a, low, high, x):
    '''递归写法'''
    middle = int((low + high) / 2)
    if high >= low:
        if a[middle] == x:
            return middle
        elif a[middle] > x:
            return recursive(a, low, middle - 1, x)
        else:
            return recursive(a, middle + 1, high, x)
    if x < a[middle]:
        return middle
    if x > a[middle]:
        return middle + 1


def whileloop(a, x):
    '''while 循环写法'''
    low = 0
    high = len(a) - 1
    while low <= high:
        middle = int((low + high) / 2)
        if a[middle] < x:
            low = middle + 1
        elif a[middle] > x:
            high = middle - 1
        else:
            return middle
    if x < a[middle]:
        return middle
    if x > a[middle]:
        return middle + 1


def forloop(a, x):
    '''for 循环写法'''
    for idx, item in enumerate(a):
        if item >= x:
            return idx


def builtin_bisect(a, x):
    '''使用内置的 bisect'''
    return bisect.bisect_left(a, x)

共进行 3 次测试，各次 a 的长度为 $10^3$、$10^6$ 和 $10^9$，x 为长度的 0.67，测试 CPU 环境为 Intel(R) Xeon(R) CPU @ 2.00GHz，Python 版本为 Python 3.6.8，就是 colab 的环境：

recursive_times = []
whileloop_times = []
forloop_times = []
builtin_times = []
lengths = [int(1e3), int(1e6), int(1e9)]
for i in lengths:
    a = range(i)
    x = 0.67 * i
    print(i)
    start = time.time()
    _ = recursive(a, 0, i - 1, x)
    cost = time.time() - start
    recursive_times.append(cost)

    start = time.time()
    _ = whileloop(a, x)
    cost = time.time() - start
    whileloop_times.append(cost)

    start = time.time()
    _ = forloop(a, x)
    cost = time.time() - start
    forloop_times.append(cost)

    start = time.time()
    _ = builtin_bisect(a, x)
    cost = time.time() - start
    builtin_times.append(cost)

fig = plt.figure(figsize=(15, 8))
plt.plot(range(3), [i * 1000 for i in recursive_times], '-o', label="recursive")
plt.plot(range(3), [i * 1000 for i in whileloop_times], '-o', label="whileloop")
plt.plot(range(3), [i * 1000 for i in forloop_times], '-o', label="forloop")
plt.plot(range(3), [i * 1000 for i in builtin_times], '-o', label="builtin_bisect")
plt.xticks([0, 1, 2], [3, 6, 9])
plt.xlabel("length/10^")
plt.ylabel("COST/ms")
plt.legend()
plt.grid(True)

测试结果：

测试结果

可以看到随着数组长度的扩大，for 循环方法消耗的时间也非常非常快的增长，而其他方法波动很小。具体而言，数组长度变为原来的 1000 倍，for 循环方法耗时也变为原来的 1000 倍左右，而其他方法的耗时变为原来的 1-3 倍左右。

暂且除去 for 循环，如果我们细看在 $10^3$、$10^6$ 和 $10^9$ 处的图，可以看到剩余三个方法之间的差距，递归耗时均最高：

实际上 bisect 内部实现用的就是 while 循环的方法，代码很短，我直接贴过来（吐槽下官方代码竟然没有很好的格式化）：

def bisect_left(a, x, lo=0, hi=None):
    """Return the index where to insert item x in list a, assuming a is sorted.
    The return value i is such that all e in a[:i] have e < x, and all e in
    a[i:] have e >= x.  So if x already appears in the list, a.insert(x) will
    insert just before the leftmost x already there.
    Optional args lo (default 0) and hi (default len(a)) bound the
    slice of a to be searched.
    """

    if lo < 0:
        raise ValueError('lo must be non-negative')
    if hi is None:
        hi = len(a)
    while lo < hi:
        mid = (lo+hi)//2
        if a[mid] < x: lo = mid+1
        else: hi = mid
    return lo

详细的测试时间数据如下：

recursive_times=[1.5735626220703125e-05, 2.5272369384765625e-05, 4.553794860839844e-05]
whileloop_times=[6.67572021484375e-06, 1.1205673217773438e-05, 1.8358230590820312e-05]
forloop_times=[4.982948303222656e-05, 0.058376312255859375, 58.462812423706055]
builtin_times=[6.4373016357421875e-06, 1.9311904907226562e-05, 2.4557113647460938e-05]

到此时我们回答了第一个问题，二分查找真的很快！ 在数组长度较小时，差距还不是那么的明显。但是随着数组长度扩大，差距简直指数级扩大，差距甚至几百万倍。

至于第二个问题，我认为的答案是尽可能使用内置库，这些都是经过了很多优化，速度和稳定性都有保证，写起来还简单粗暴，何乐而不为呢？

写完感觉一篇废话 😂

二分查找真的很快吗

目录

代码

Reference

END

Alan Lee