这段代码工作正常，但我的直觉告诉我它比它需要的要慢

如何解决这段代码工作正常，但我的直觉告诉我它比它需要的要慢

此代码的目的是求解 Collatz 序列。如果您不熟悉，这是顺序：

取一个数字 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，它的序列总是相同的。所以一旦你解决了 10 个，你就可以像质数一样从你的列表中“筛选”它。

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 以下的数字都将是 1,000 以下的 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，它就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。

在一天结束时，代码可以工作，只是速度很慢。任何见解将不胜感激！

解决方法

您对记忆化的假设是不正确的。您不能只从 500,000 开始，因为如果您查看 999,998，您将除以 2 得到 499,999，您将没有缓存值。您是正确的，低于 500,000 的值永远不会成为最长系列的答案，但是，您确实需要知道低于 500,000 的系列才能优化序列生成。

您确实想将迄今为止看到的系列存储在以给定 dict 值为键的 N 中，并且您需要从开头 1 开始以充分利用缓存。

修改后的 this algorithm 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。

您可以使用带有指定键的 max 函数找到最长的系列，您可以通过将存储在键中的 N 值与存储在价值。

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.

我在此实现和您的实现上运行了 timeit。

Original Implementation: 348.300429 
Modified Implementation: 46.332222100000024

注意：这种修改只有在您确实需要范围内的每个序列时才有意义。如果您只想要最长的序列，您应该使用未修改的算法 get 837799 并使用 CodeReview 上 this answer 的 Generator 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 测量了此实现，包括生成最长序列。