首页 > 算法优化 > Euler Project之Retraction【算法优化向】

Euler Project之Retraction【算法优化向】

2013年12月8日 发表评论 阅读评论
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pe_banner_light好吧,其实我是在新博客这边练习敲公式来着。。。

题目来源还是是Euler Project,之前官方连续公布了三道类似的题目,都是基于一种叫做Retraction数的东西,它的定义是:

对于所有整数\(\),定义一族函数\(\)\(\)均为整数,且满足0<\(\)<\(\),\(\),\(\)
如果对所有\(\)均有:

\(\)

那么,我们管\(\)叫做一个Retraction;
我们需要计算的是\(\),定义为包含\(\)的Retraction的个数。

虽然这么说,但是计算\(\)并不是题目的最终结果,只是之前公布的三道题目全都是跟这个\(\)有关的,所以我这里就来算一下怎么算\(\)最快,好吧,不一定是最快的,但是我觉得还是挺快的。。

如果真正运用到题目的话,倒是不会用到我这个计算\(\)的东西,因为配合题目的别的条件,可以更快的算出题目真正要求的东西。不过我就是要算\(\),谁也别拦我在这里写公式!!以前在blog.com被不能写数学公式的烦躁感压抑太久了!!【万一哪一天Latex插件失败了,我这个页面将会显示一大堆乱码。。。想想就好可怕。。】

首先,我们先用Python写一个最简单(思维量上)的版本,就是把题目翻译一下的那个版本:

def f(n,a,b,x):
    return mod(a*x+b,n)

def IsRetraction(a,b,n):
    for x in range(n):
        if f(n,a,b,f(n,a,b,x)) != mod(f(n,a,b,x),n):
            return False
    return True

def R(n):
    counter = 0;
    for a in range(1,n):
        for b in range(0,n):
            if IsRetraction(a,b,n):
                counter += 1;
    return counter;

有兴趣的同学可以自己去跑一下这段代码,慢的不可理喻!!因为随着n的增大,计算复杂度是\(\)

所以我们可以对题目的条件进行一下推导:

\(\)\(\)

\(\)\(\)

\(\)\(\)-----------------------------※

\(\)\(\)

代入\(\),其中引文\(\)<\(\),所以有\(\),所以:
\(\)

所以我们得到第一个结论:\(\)

上面的结论成立必然会有在\(\)的时候成立(因为我们就是把\(\)代进去推出来的嘛~),所以我们可以以此为“导火索”,采用数学归纳法!!

也就是说,我们只要可以从\(\)成立推导出\(\)成立的条件,因为\(\)已经成立了,所以必然有对于一切\(\)成立!!

上面的※式中,我们有\(\),【\(\)是整数,下面不再累述】然后我们把\(\)代进去:

\(\)-----------------------------※2

同余计算里面有以下结论:
\(\)\(\)

\(\)\(\)

所以之前的※2式子就变成:\(\)

因为上面红色式子对\(\)成立的时候必然有:\(\)

所以代入之后又:\(\)

\(\)\(\)

这个时候\(\)不能带进去,所以两边先对\(\)取模:

\(\)

这个时候\(\)是可以带进里面去的,所以就变成:

\(\)

\(\)

\(\)

之前我推到这里简直震惊了!!因为我们要判断\(\)是不是Retraction,只要判断下面这两个式子是否成立就可以了!!

\(\)

于是乎,代码就变成这个样子了:

def R_version2(n):
    counter = 0;
    for a in range(1,n):
        if mod(a*a-a,n) != 0:
            continue
        for b in range(0,n):
            if mod(a*b,n) == 0:
                counter += 1;
    return counter;

代码瞬间就变成只有一个函数了~
我在我的破机子上跑,计算R(1000),最开始那个代码要20.1578052518秒,现在只要0.0142358875219,这个已经算时间计算误差了,基本就是秒出了。
但是,其实还可以优化的!!

之前的思路是对\(\)从1扫到n-1,然后判断是否满足\(\),满足,再把\(\)从0扫到\(\),判断是否满足\(\),如果满足,那么这一组\(\)就是一个Retraction。

但是我们再看看第二个条件:假设我们已知了\(\)满足\(\),应该可以直接算出有多少个\(\)满足\(\)的对吧!!没错,就是a和n的最大公约数!!!

可能有人想不通,下面我详细解释一下:

原因是这样子的,我们要计算满足\(\)\(\),也就是要寻找满足\(\)\(\),也就是说,\(\)最小为0,再其次的话,第二小的\(\)里面一定要包含\(\)的质约数与\(\)的质约数的差集,比如\(\)\(\),那么最小的\(\)为0,\(\),\(\),所以第二小的\(\)的约数是集合{2,2,2,5}减去集合{2,2,2,2}【这里说集合好像不是很对,因为集合不能有重复的元素的~嘛~别在意啦。。】,也就是5啦!而且,\(\)取5,\(\)就是去2,再然后每个\(\)就不断地增加就是了,所以得到的\(\)就是0,5,10,15,20,25,30,35,八个。

那么最小的非零\(\)怎么算,就是用\(\)的最小公约数除以\(\),也就是\(\),最大的\(\)就是\(\),说白了就是\(\)的取值是:

\(\)

显然,因为公差是\dfrac{LCM(a,n)}{a},所以最后得到的\(\)的个数就是\(\),小学奥数告诉我们,两个数最小公倍数乘最大公约数就是两个数的乘积,所以\(\)!!

证明完毕!!Q.E.D!!

最后代码就会变成:

def R_version2(n):
    counter = 0;
    for a in range(1,n):
        if mod(a*a-a,n) != 0:
            continue
        counter += gcd(a,n);
    return counter;

简洁吧,这都是数学的功劳!!现在的时间复杂度就是\(\)了,如果想要再加速,那么就要想怎么快速的计算出满足\(\)的a来了。。。有人有好的思路么?
嗯,好吧,先写到这里吧!敲了半天公式,一本满足~


【完】

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  1. Nima
    2014年7月10日02:37 | #1

    时间复杂度还是太大,lz最后算出吗?

    • 2014年7月10日13:01 | #2

      写这篇东西就是为了敲公式,也没怎么细想,最后也忘了这道题了。。你有啥好想法么?

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