这段代码工作正常，但我的直觉告诉我它比它需要的要慢答案

【问题标题】：This code works fine but my intuition is telling me that it's way slower than it need's to be这段代码工作正常，但我的直觉告诉我它比它需要的要慢
【发布时间】：2021-04-16 16:14:07
【问题描述】：

此代码的目的是求解 Collatz 序列。如果您不熟悉，这是顺序：

取一个数N

如果 N 是奇数： N = 3N + 1

如果 N 是偶数： N = N/2

迭代直到 N = 1

10 --> 5 --> 16 --> --> 8 --> 4 --> 2 --> 1

我的基本代码假设一旦你找到一个值，它的顺序总是相同的。因此，一旦您解决了 10 个问题，您就可以从您的列表中“筛选”它，类似于素数。

顺便说一下，这里的值是找到 1,000,000 以下的最长序列，理由是所有 500,000 以下的数字在 1,000,000 以下都有 2 的倍数，因此它们永远不会有最长的序列。

代码：

list_of_tries = [*range(500000, 1000000)]
set_of_past_tries = set()
past_str = []
for x in list_of_tries:
  new_str = []
  if x not in set_of_past_tries:
    x == 1
  if x != 1:
    new_str.append(x)
  while x != 1:
    if x % 2 == 0:
      x /= 2
      new_str.append(x)
      set_of_past_tries.add(x)
    else:
      x *= 3
      x += 1
      new_str.append(x)
      set_of_past_tries.add(x)
  if len(new_str) > len(past_str):
    past_str = new_str

我希望它存储每个序列，所以一旦它发现 X = 10 例如它会停止并打印它已经知道的序列，但我不知道如何快速做到这一点。

最终，代码可以正常工作，只是速度很慢。任何见解将不胜感激！

【问题讨论】：

指令x == 1 在代码开头没有任何用处。是有意的吗？此外，x /= 2 将 x 的类型更改为浮点数（不再是 int）

标签： python list set

【解决方案1】：

您对记忆的假设是不正确的。您不能从 500,000 开始，因为如果您查看 999,998，您将除以 2 并得到 499,999，您将没有缓存值。您是正确的，低于 500,000 的值永远不会是最长序列的答案，但是，您需要知道低于 500,000 的序列才能优化序列生成。

您确实希望将目前看到的系列存储在dict 中，并键入给定的N 值，并且您需要从开头 1 开始以充分利用缓存。

this algorithm 的修改版本可用于根据您的要求获取“每个序列”。

def longest_collatz_sequence(upper_bound):
    assert upper_bound >= 2  # upper_bound needs to be longer than prepopulated cache
    upper_bound += 1
    cache = {n: [] for n in range(1, upper_bound)}

    cache[1] = []
    cache[2] = [1]

    for n in range(3, upper_bound):
        sequence = []
        current = n
        while n > 1:
            if n < current:
                sequence.extend(cache[n])
                cache[current] = sequence
                break
            if n % 2 == 0:
                n = n // 2
            else:
                n = 3 * n + 1
            sequence.append(n)
    return cache


if __name__ == '__main__':
    d = longest_collatz_sequence(1_000_000)
    longest_series_start, longest_series = max(d.items(), key=lambda item: len(item[1]))
    series = [longest_series_start, *longest_series]
    print(f"Longest Series Where N = {longest_series_start}.")
    print(f"Longest Series is {series}")
    print(f"Longest Series is {len(series)} steps.")

然后您可以打印出返回的字典以查看 1 到 upper_bound 范围内的每个可能的 Collatz 序列。

您可以使用具有指定键的max 函数找到最长的系列，您可以通过将存储在键中的N 值与存储在值中的其余系列连接起来来构建最长的系列。

以上程序输出：

Longest Series Where N = 837799.
Longest Series is [837799, 2513398, 1256699, 3770098, 1885049, 5655148, 2827574, 1413787, 4241362, 2120681, 6362044, 3181022, 1590511, 4771534, 2385767, 7157302, 3578651, 10735954, 5367977, 16103932, 8051966, 4025983, 12077950, 6038975, 18116926, 9058463, 27175390, 13587695, 40763086, 20381543, 61144630, 30572315, 91716946, 45858473, 137575420, 68787710, 34393855, 103181566, 51590783, 154772350, 77386175, 232158526, 116079263, 348237790, 174118895, 522356686, 261178343, 783535030, 391767515, 1175302546, 587651273, 1762953820, 881476910, 440738455, 1322215366, 661107683, 1983323050, 991661525, 2974984576, 1487492288, 743746144, 371873072, 185936536, 92968268, 46484134, 23242067, 69726202, 34863101, 104589304, 52294652, 26147326, 13073663, 39220990, 19610495, 58831486, 29415743, 88247230, 44123615, 132370846, 66185423, 198556270, 99278135, 297834406, 148917203, 446751610, 223375805, 670127416, 335063708, 167531854, 83765927, 251297782, 125648891, 376946674, 188473337, 565420012, 282710006, 141355003, 424065010, 212032505, 636097516, 318048758, 159024379, 477073138, 238536569, 715609708, 357804854, 178902427, 536707282, 268353641, 805060924, 402530462, 201265231, 603795694, 301897847, 905693542, 452846771, 1358540314, 679270157, 2037810472, 1018905236, 509452618, 254726309, 764178928, 382089464, 191044732, 95522366, 47761183, 143283550, 71641775, 214925326, 107462663, 322387990, 161193995, 483581986, 241790993, 725372980, 362686490, 181343245, 544029736, 272014868, 136007434, 68003717, 204011152, 102005576, 51002788, 25501394, 12750697, 38252092, 19126046, 9563023, 28689070, 14344535, 43033606, 21516803, 64550410, 32275205, 96825616, 48412808, 24206404, 12103202, 6051601, 18154804, 9077402, 4538701, 13616104, 6808052, 3404026, 1702013, 5106040, 2553020, 1276510, 638255, 1914766, 957383, 2872150, 1436075, 4308226, 2154113, 6462340, 3231170, 1615585, 4846756, 2423378, 1211689, 3635068, 1817534, 908767, 2726302, 1363151, 4089454, 2044727, 6134182, 3067091, 9201274, 4600637, 13801912, 6900956, 3450478, 1725239, 5175718, 2587859, 7763578, 3881789, 11645368, 5822684, 2911342, 1455671, 4367014, 2183507, 6550522, 3275261, 9825784, 4912892, 2456446, 1228223, 3684670, 1842335, 5527006, 2763503, 8290510, 4145255, 12435766, 6217883, 18653650, 9326825, 27980476, 13990238, 6995119, 20985358, 10492679, 31478038, 15739019, 47217058, 23608529, 70825588, 35412794, 17706397, 53119192, 26559596, 13279798, 6639899, 19919698, 9959849, 29879548, 14939774, 7469887, 22409662, 11204831, 33614494, 16807247, 50421742, 25210871, 75632614, 37816307, 113448922, 56724461, 170173384, 85086692, 42543346, 21271673, 63815020, 31907510, 15953755, 47861266, 23930633, 71791900, 35895950, 17947975, 53843926, 26921963, 80765890, 40382945, 121148836, 60574418, 30287209, 90861628, 45430814, 22715407, 68146222, 34073111, 102219334, 51109667, 153329002, 76664501, 229993504, 114996752, 57498376, 28749188, 14374594, 7187297, 21561892, 10780946, 5390473, 16171420, 8085710, 4042855, 12128566, 6064283, 18192850, 9096425, 27289276, 13644638, 6822319, 20466958, 10233479, 30700438, 15350219, 46050658, 23025329, 69075988, 34537994, 17268997, 51806992, 25903496, 12951748, 6475874, 3237937, 9713812, 4856906, 2428453, 7285360, 3642680, 1821340, 910670, 455335, 1366006, 683003, 2049010, 1024505, 3073516, 1536758, 768379, 2305138, 1152569, 3457708, 1728854, 864427, 2593282, 1296641, 3889924, 1944962, 972481, 2917444, 1458722, 729361, 2188084, 1094042, 547021, 1641064, 820532, 410266, 205133, 615400, 307700, 153850, 76925, 230776, 115388, 57694, 28847, 86542, 43271, 129814, 64907, 194722, 97361, 292084, 146042, 73021, 219064, 109532, 54766, 27383, 82150, 41075, 123226, 61613, 184840, 92420, 46210, 23105, 69316, 34658, 17329, 51988, 25994, 12997, 38992, 19496, 9748, 4874, 2437, 7312, 3656, 1828, 914, 457, 1372, 686, 343, 1030, 515, 1546, 773, 2320, 1160, 580, 290, 145, 436, 218, 109, 328, 164, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1]
Longest Series is 525 steps.

我在这个实现和你的实现上运行了timeit。

Original Implementation: 348.300429 
Modified Implementation: 46.332222100000024

注意：此修改仅在您确实需要范围内的每个序列时才有意义。如果你只想要最长的序列，你应该使用未修改的算法获取837799并使用CodeReview上this answer的生成器生成序列。

def longest_collatz_sequence(upper_bound):
    assert upper_bound >= 2  # upper_bound needs to be longer than prepopulated cache
    upper_bound += 1
    cache = {n: 0 for n in range(1, upper_bound)}

    cache[1] = 1
    cache[2] = 2

    for n in range(3, upper_bound):
        counter = 0
        current = n
        while n > 1:
            if n < current:
                cache[current] = cache[n] + counter
                break
            if n % 2 == 0:
                n = n / 2
                counter += 1
            else:
                n = 3 * n + 1
                counter += 1
    return cache


def collatz_gen(starting_num):
    yield starting_num

    num = starting_num
    while num > 1:
        num = (num // 2) if num % 2 == 0 else (3 * num + 1)
        yield num


if __name__ == '__main__':
    d = longest_collatz_sequence(1_000_000)
    longest_series_start = list(d.values()).index(max(d.values())) + 1
    series = list(collatz_gen(longest_series_start))
    print(f"Longest Series Where N = {longest_series_start}.")
    print(f"Longest Series is {series}")
    print(f"Longest Series is {len(series)} steps.")

timeit 在24.5141495 测量了这个实现，包括生成最长的序列。

【讨论】：