在Python中理解Euler項目的解決方案

Question

我目前正在經歷歐拉項目。 我已經開始使用JavaScript了，昨天我已經切換到Python，因為我陷入了一個似乎很復雜的問題，用Javascript解決但很容易用Python解決，所以我決定從第一個問題開始蟒蛇。

我遇到問題5，它要求我找到最小的正數，它可以被1到20的所有數字整除。

我知道如何使用紙和筆來解決它，我已經使用編程解決了問題，但是為了優化它，我在歐拉項目的論壇中越過了這個解決方案。

它看起來很優雅，與我的相比它速度相當快，很荒謬，因為解決1到1000號的相同問題需要大約2秒鍾，我需要幾分鍾。

我試圖理解它，但我仍然很難理解它到底在做什么。 這里是：

i = 1
for k in (range(1, 21)):
    if i % k > 0:
        for j in range(1, 21):
            if (i*j) % k == 0:
                i *= j
                break
print i

它最初由名為lassevk的用戶發布

Answer 1

該代碼計算從1到20 （或您使用的任何其他range ）的數字的最小公倍數 。

它從1開始，然后對於該范圍內的每個數字k ，它檢查k是否是i的因子，如果不是（即，當i % k > 0 ），它將該數字作為因子。

然而這樣做i *= k不產生最小公倍數 ，因為例如當你有i = 2和k = 6它足以乘i由3至有i % 6 == 0 ，所以內循環發現至少數字j使得i*j具有k作為因子。

到底每隔數k范圍是這樣的， i % k == 0 ，我們總是添加最小因素，以做到這一點，因此我們計算的最小公倍數 。

---

也許准確地顯示值的變化有助於理解過程：

i = 1
k = 1
i % k == 0  -> next loop

i = 1
k = 2
i % k == 1 > 0
   j = 1
   if (i*j) % k == 1 -> next inner loop
   j = 2
   if (i*j) % k == 0
      i *= j
i = 2
k = 3
i % k == 2 > 0
    j = 1
    if (i*j) % k == 2 -> next inner loop
    j = 2
    if (i*j) % k == 4 % 3 == 1 -> next inner loop
    j = 3
    if (i*j) % k == 6 % 3 == 0
        i *= j
i = 6
k = 4
i % k == 2 > 0
    j = 1
    if (i*j) % k == 6 % k == 2 -> next inner loop
    j = 2
    if (i*j) % k == 12 % k == 0
        i *= j
i = 12
k = 5
i % k == 2 > 0
    j = 1
    if (i*j) % k == 12 % k == 2 -> next inner loop
    j = 2
    if (i*j) % k == 24 % k == 4 -> next inner loop
    j = 3
    if (i*j) % k == 36 % k == 1 -> next inner loop
    j = 4
    if (i*j) % k == 48 % k == 3 -> next inner loop
    j = 5
    if (i*j) % k == 60 %k == 0
       i *= j
i = 60
...

你可以立即注意到一些事情：

• range(1, 21)可以安全地改變到range(2, 21)因為1秒從不做任何事情
• 每當k是素數時，內部循環在j=k時結束，並且將以i *= k 。 這是預期的，因為當k是素數時它沒有因子，因此沒有選擇較小的數字j ，這將使i成為k的倍數。
• 在其它casesthe數i是由數字乘以j < k使所有的因素k現在是在i 。

Answer 2

Bakuriu通過解釋lassevk的“階乘”算法回答了你的問題。 但是，有一種更簡單的方法可以做到這一點，對於較大的輸入來說要快得多。

令num為序列中的最大數字。 所以對於你的例子num = 20 。

簡單地乘以所有從2到數num在一起，並在當前的乘法器由GCD每個步驟分（最大公約數）和當前的產品。

在這段代碼中，我將產品初始化為num ，只是為了讓循環看起來更好一些。

num = 20

p = num
for i in range(2, num):
    # Compute GCD(p, i) using Euclid's algorithm
    # When the loop ends, a is the GCD
    a, b = p, i
    while b:
        a, b = b, a % b
    p *= i // a

print(p)

產量

232792560

對於較小的num值，此算法與lassevk的算法大致相同。 但是當num = 1000它快4倍，而當num = 2000它快14倍。

---

正如Bakuriu在評論中提到的， fractions模塊提供了gcd功能。 這使代碼更短，但它在我的測試中沒有提供加速。

from fractions import gcd

num = 20

p = num
for i in range(2, num):
    p *= i // gcd(p, i)

print(p)

---

下面是一些Python 2 / Python 3代碼，它們執行實際的timeit測試，比較我的算法的兩種變體。 在Python 2.6.6上，使用fractions.gcd的版本慢了大約10％，但在Python 3.6上它可能慢5到10倍！ 這兩項測試都是在運行Debian派生Linux的老式2GHz機器上進行的。

''' Test the speed of calculating the Least Common Multiple 
    via an inline implementation of Euclid's GCD algorithm
    vs the gcd function from the fractions module

    See http://stackoverflow.com/q/38074440/4014959

    Written by PM 2Ring 2016.06.28
'''

from timeit import Timer
from fractions import gcd

def lcm0(num):
    p = num
    for i in range(2, num):
        a, b = p, i
        while b:
            a, b = b, a % b
        p *= i // a
    return p

def lcm1(num, gcd=gcd):
    p = num
    for i in range(2, num):
        p *= i // gcd(p, i)
    return p

funcs = (lcm0, lcm1)

def time_test(loops, reps):
    ''' Print timing stats for all the functions '''
    for func in funcs:
        fname = func.__name__
        setup = 'from __main__ import num,' + fname
        cmd = fname + '(num)'
        t = Timer(cmd, setup)
        r = t.repeat(reps, loops)
        r.sort()
        print('{0}: {1}'.format(fname, r))

num = 5
loops = 8192
reps = 5
for _ in range(10):
    print('\nnum={0}, loops={1}'.format(num, loops))
    time_test(loops, reps)
    num *= 2
    loops //= 2

Python 2.6輸出

num=5, loops=8192
lcm0: [0.055649995803833008, 0.057304859161376953, 0.057752132415771484, 0.060063838958740234, 0.064462900161743164]
lcm1: [0.067559003829956055, 0.068048954010009766, 0.068253040313720703, 0.069074153900146484, 0.084647893905639648]

num=10, loops=4096
lcm0: [0.058645963668823242, 0.059965133666992188, 0.060016870498657227, 0.060331821441650391, 0.067235946655273438]
lcm1: [0.072937965393066406, 0.074002981185913086, 0.074270963668823242, 0.074965953826904297, 0.080986976623535156]

num=20, loops=2048
lcm0: [0.063373088836669922, 0.063961029052734375, 0.064354896545410156, 0.071543216705322266, 0.10234284400939941]
lcm1: [0.079973936080932617, 0.080717802047729492, 0.082272052764892578, 0.086506843566894531, 0.11265397071838379]

num=40, loops=1024
lcm0: [0.077324151992797852, 0.077867984771728516, 0.07857513427734375, 0.087296962738037109, 0.10289192199707031]
lcm1: [0.095077037811279297, 0.095172882080078125, 0.095523834228515625, 0.095964193344116211, 0.10543298721313477]

num=80, loops=512
lcm0: [0.09699702262878418, 0.097161054611206055, 0.09722590446472168, 0.099267005920410156, 0.10546517372131348]
lcm1: [0.1151740550994873, 0.11548399925231934, 0.11627888679504395, 0.11672496795654297, 0.12607502937316895]

num=160, loops=256
lcm0: [0.10686612129211426, 0.10825586318969727, 0.10832309722900391, 0.11523914337158203, 0.11636996269226074]
lcm1: [0.12528896331787109, 0.12630200386047363, 0.12688708305358887, 0.12690496444702148, 0.13400888442993164]

num=320, loops=128
lcm0: [0.12498903274536133, 0.12538790702819824, 0.12554287910461426, 0.12600493431091309, 0.13396120071411133]
lcm1: [0.14431190490722656, 0.14435195922851562, 0.15340209007263184, 0.15408897399902344, 0.159912109375]

num=640, loops=64
lcm0: [0.15442395210266113, 0.15479183197021484, 0.15657520294189453, 0.16451501846313477, 0.16749906539916992]
lcm1: [0.17400288581848145, 0.17454099655151367, 0.18450593948364258, 0.18503093719482422, 0.19588208198547363]

num=1280, loops=32
lcm0: [0.21137905120849609, 0.21206808090209961, 0.21211409568786621, 0.21935296058654785, 0.22051215171813965]
lcm1: [0.23439598083496094, 0.23578977584838867, 0.23717594146728516, 0.24761080741882324, 0.2488548755645752]

num=2560, loops=16
lcm0: [0.34246706962585449, 0.34283804893493652, 0.35072207450866699, 0.35794901847839355, 0.38117814064025879]
lcm1: [0.3587038516998291, 0.36004209518432617, 0.36267900466918945, 0.36284589767456055, 0.37285304069519043]

Python 3.6輸出

num=5, loops=8192
lcm0: [0.0527388129994506, 0.05321520800134749, 0.05394392299785977, 0.0540059859995381, 0.06133090399816865]
lcm1: [0.45663526299904333, 0.4585357750002004, 0.45960231899880455, 0.4768777699973725, 0.48710195899911923]

num=10, loops=4096
lcm0: [0.05494695199740818, 0.057305197002278874, 0.058495635999861406, 0.07243769099659403, 0.07494244600093225]
lcm1: [0.5807856120009092, 0.5809524680007598, 0.5971023489983054, 0.6006399979996786, 0.6021203519994742]

num=20, loops=2048
lcm0: [0.06225249999988591, 0.06330173400056083, 0.06348088900267612, 0.0639248730003601, 0.07240132099832408]
lcm1: [0.6462642230035271, 0.6486189150018618, 0.6605903060008131, 0.6669839690002846, 0.7464891349991376]

num=40, loops=1024
lcm0: [0.06812337999872398, 0.06989315700047882, 0.07142737200047122, 0.07237963000079617, 0.07640906400047243]
lcm1: [0.6938937240011, 0.7021358079982747, 0.7238045579979371, 0.7265497620028327, 0.7266306150013406]

num=80, loops=512
lcm0: [0.07672808099960093, 0.07784233300117194, 0.07959756200216361, 0.08742279999933089, 0.09116945599816972]
lcm1: [0.7249167879999732, 0.7272519250000187, 0.7329213439988962, 0.7570086350024212, 0.75942590500199]

num=160, loops=256
lcm0: [0.08417846500015003, 0.08528995099914027, 0.0856771619983192, 0.08571110499906354, 0.09348897000018042]
lcm1: [0.7382230039984279, 0.7425414600002114, 0.7439042109981528, 0.7505959240006632, 0.756812355000875]

num=320, loops=128
lcm0: [0.10246147399811889, 0.10322481399998651, 0.10324400399986189, 0.10347093499876792, 0.11325025699989055]
lcm1: [0.7649764790003246, 0.7903363080004056, 0.7931463940003596, 0.8012050910001562, 0.8284494129984523]

num=640, loops=64
lcm0: [0.13264304200129118, 0.13345745100014028, 0.13389246199949412, 0.14023518899921328, 0.15422578799916664]
lcm1: [0.8085992009982874, 0.8125102049998532, 0.8179558970005019, 0.8299506059993291, 0.9141929620018345]

num=1280, loops=32
lcm0: [0.19097876199884922, 0.19147844200051622, 0.19308012399778818, 0.19317538399991463, 0.20103917100277613]
lcm1: [0.8671656119986437, 0.8713741569990816, 0.8904907689975516, 0.9020749549999891, 0.9131527989993629]

num=2560, loops=16
lcm0: [0.3099351109995041, 0.31015214799845126, 0.3101941059976525, 0.32628724800088094, 0.3492128660000162]
lcm1: [0.9883516860027157, 0.988955139000609, 0.9965159560015309, 1.0160803129983833, 1.0170008439999947]