十个利用矩阵乘法解决的经典题目

    好像目前还没有这方面题目的总结。这几天连续看到四个问这类题目的人,今天在这里简单写一下。这里我们不介绍其它有关矩阵的知识,只介绍矩阵乘法和相关性质。
    不要以为数学中的矩阵也是黑色屏幕上不断变化的绿色字符。在数学中,一个矩阵说穿了就是一个二维数组。一个n行m列的矩阵可以乘以一个m行p列的矩阵,得到的结果是一个n行p列的矩阵,其中的第i行第j列位置上的数等于前一个矩阵第i行上的m个数与后一个矩阵第j列上的m个数对应相乘后所有m个乘积的和。比如,下面的算式表示一个2行2列的矩阵乘以2行3列的矩阵,其结果是一个2行3列的矩阵。其中,结果的那个4等于2*2+0*1:
    
    下面的算式则是一个1 x 3的矩阵乘以3 x 2的矩阵,得到一个1 x 2的矩阵:
    

    矩阵乘法的两个重要性质:一,矩阵乘法不满足交换律;二,矩阵乘法满足结合律。为什么矩阵乘法不满足交换律呢?废话,交换过来后两个矩阵有可能根本不能相乘。为什么它又满足结合律呢?仔细想想你会发现这也是废话。假设你有三个矩阵A、B、C,那么(AB)C和A(BC)的结果的第i行第j列上的数都等于所有A(ik)*B(kl)*C(lj)的和(枚举所有的k和l)。

经典题目1 给定n个点,m个操作,构造O(m+n)的算法输出m个操作后各点的位置。操作有平移、缩放、翻转和旋转
    这里的操作是对所有点同时进行的。其中翻转是以坐标轴为对称轴进行翻转(两种情况),旋转则以原点为中心。如果对每个点分别进行模拟,那么m个操作总共耗时O(mn)。利用矩阵乘法可以在O(m)的时间里把所有操作合并为一个矩阵,然后每个点与该矩阵相乘即可直接得出最终该点的位置,总共耗时O(m+n)。假设初始时某个点的坐标为x和y,下面5个矩阵可以分别对其进行平移、旋转、翻转和旋转操作。预先把所有m个操作所对应的矩阵全部乘起来,再乘以(x,y,1),即可一步得出最终点的位置。
    

经典题目2 给定矩阵A,请快速计算出A^n(n个A相乘)的结果,输出的每个数都mod p。
    由于矩阵乘法具有结合律,因此A^4 = A * A * A * A = (A*A) * (A*A) = A^2 * A^2。我们可以得到这样的结论:当n为偶数时,A^n = A^(n/2) * A^(n/2);当n为奇数时,A^n = A^(n/2) * A^(n/2) * A (其中n/2取整)。这就告诉我们,计算A^n也可以使用二分快速求幂的方法。例如,为了算出A^25的值,我们只需要递归地计算出A^12、A^6、A^3的值即可。根据这里的一些结果,我们可以在计算过程中不断取模,避免高精度运算。

经典题目3 POJ3233 (感谢rmq)
    题目大意:给定矩阵A,求A + A^2 + A^3 + … + A^k的结果(两个矩阵相加就是对应位置分别相加)。输出的数据mod m。k<=10^9。
    这道题两次二分,相当经典。首先我们知道,A^i可以二分求出。然后我们需要对整个题目的数据规模k进行二分。比如,当k=6时,有:
    A + A^2 + A^3 + A^4 + A^5 + A^6 =(A + A^2 + A^3) + A^3*(A + A^2 + A^3)
    应用这个式子后,规模k减小了一半。我们二分求出A^3后再递归地计算A + A^2 + A^3,即可得到原问题的答案。

经典题目4 VOJ1049
    题目大意:顺次给出m个置换,反复使用这m个置换对初始序列进行操作,问k次置换后的序列。m<=10, k<2^31。
    首先将这m个置换“合并”起来(算出这m个置换的乘积),然后接下来我们需要执行这个置换k/m次(取整,若有余数则剩下几步模拟即可)。注意任意一个置换都可以表示成矩阵的形式。例如,将1 2 3 4置换为3 1 2 4,相当于下面的矩阵乘法:
    
    置换k/m次就相当于在前面乘以k/m个这样的矩阵。我们可以二分计算出该矩阵的k/m次方,再乘以初始序列即可。做出来了别忙着高兴,得意之时就是你灭亡之日,别忘了最后可能还有几个置换需要模拟。

经典题目5 《算法艺术与信息学竞赛》207页(2.1代数方法和模型,[例题5]细菌,版次不同可能页码有偏差)
    大家自己去看看吧,书上讲得很详细。解题方法和上一题类似,都是用矩阵来表示操作,然后二分求最终状态。

经典题目6 给定n和p,求第n个Fibonacci数mod p的值,n不超过2^31
    根据前面的一些思路,现在我们需要构造一个2 x 2的矩阵,使得它乘以(a,b)得到的结果是(b,a+b)。每多乘一次这个矩阵,这两个数就会多迭代一次。那么,我们把这个2 x 2的矩阵自乘n次,再乘以(0,1)就可以得到第n个Fibonacci数了。不用多想,这个2 x 2的矩阵很容易构造出来:
    

经典题目7 VOJ1067
    我们可以用上面的方法二分求出任何一个线性递推式的第n项,其对应矩阵的构造方法为:在右上角的(n-1)*(n-1)的小矩阵中的主对角线上填1,矩阵第n行填对应的系数,其它地方都填0。例如,我们可以用下面的矩阵乘法来二分计算f(n) = 4f(n-1) – 3f(n-2) + 2f(n-4)的第k项:
    
    利用矩阵乘法求解线性递推关系的题目我能编出一卡车来。这里给出的例题是系数全为1的情况。

经典题目8 给定一个有向图,问从A点恰好走k步(允许重复经过边)到达B点的方案数mod p的值
    把给定的图转为邻接矩阵,即A(i,j)=1当且仅当存在一条边i->j。令C=A*A,那么C(i,j)=ΣA(i,k)*A(k,j),实际上就等于从点i到点j恰好经过2条边的路径数(枚举k为中转点)。类似地,C*A的第i行第j列就表示从i到j经过3条边的路径数。同理,如果要求经过k步的路径数,我们只需要二分求出A^k即可。

经典题目9 用1 x 2的多米诺骨牌填满M x N的矩形有多少种方案,M<=5,N<2^31,输出答案mod p的结果
    
    我们以M=3为例进行讲解。假设我们把这个矩形横着放在电脑屏幕上,从右往左一列一列地进行填充。其中前n-2列已经填满了,第n-1列参差不齐。现在我们要做的事情是把第n-1列也填满,将状态转

209 条评论

  • windywinter

    重载双目运算符尽量放到类定义的外面,以避免不同编译器的区别。

    回复:谢谢指点;这方面的东西我还不太熟,过几天来研究

  • windywinter

    另外,matrix67大牛为什么不写一个矩阵快速乘法呢?这个一直弄不明白怎么做。

    回复:是不是这个?
    http://www.comp.nus.edu.sg/~xujia/mirror/algorithm.myrice.com/algorithm/commonalg/misc/strassen/strassen.htm

  • yiyi

    第6题的那个用矩阵乘法求第n个Fibonacci数的算法很感兴趣,可看的还是不大明白,能说的再详细一点吗?

    回复:usera.imagecave.com/matrix67/fib_using_matrix.GIF

  • lsylsy2

    矩阵=matrix

  • Kimm

    经典题目9 能否再解释下
    看不是很懂
    左边的状态是怎么转移到右边的
    111又是怎么转移到011的?

  • hey

    给个建议, 规范 C++ 书写.
    #include <cstdio>, #define, scanf, printf 之类的东西不该出现哦.

  • 晗熙

    第九题  超经典
    能否写个pascal???

  • rmq

    啊,太荣幸了,大牛居然贴我blog上的一个小题目 ~~
    太开心了

    回复:您牛的blog我每天都上,都是经典题目啊

  • shinefine

    楼主能把下面这个数列用矩阵的方式求出第n项的值吗?
    数列用递归方式定义:
    Q(1)= Q(2) =1
    Q(N)=Q(N-Q(N-1))+Q(N-Q(N-2)) (N>2)

  • Tsuji

    楼上的数列好象在VIJOS上看到过

  • Excessuz

    其实类的双目运算符好像可以在类里面定义成友元函数……和在类外定义是等效的……这样看起来可能会比较舒服……

  • Melissa

    Found your site in google, and it has a lot of usefull information. Thanx.h

  • cloudygoose

    最后一题能再讲得详细的吗?完全不理解。

    以AT结尾、以AA结尾、以GG结尾、以?A结尾、以?G结尾、以?C结尾和以??结尾

    是怎么得出的?

  • Vector·Sun

    最后一题真的不好理解……
    若能把所有状态之间的关系(即那个矩阵)列出来就好了

  • andyhua__88

    请问第4个题目中是对每个置换转换成n*n的矩阵,然后对每个矩阵相乘吗?如果是这样,当n=100时,时间复杂度也会达到n^3,最少也会是7^(logn),不会超时吗?

  • topsky

    题目3
    构造一个矩阵
    A I
    O I
    自乘后
    A^2 I+A
    O I
    乘3次为
    A^3 I+A+A^2
    O I
    这样应该会更快吧,直接二分一下。

  • 第九题能不能给我一个Pascal程序,我看不懂C++

  • Earthson

    对于题目2和3,只是1维情况的推广而已,对于2,可以用逐次平方计算(其实和m67给的递归方式差不了多少)
    对于3,我能给出更好的方法,和问题2其实是类似的,基于二进制模式的递推(对应的递归是非对称的拆分),能在O(log(n))的时间给出解
    http://www.earthson.net/archives/280
    我笔拙,写得可能不是很好,但这不能掩盖这个算法的出色构造
    我还给出了更一般的二进制可递推序列的特性要求和算法
    http://www.earthson.net/archives/491

  • 小渡

    几个月前就看这篇了,今天重新学习,转载了一下并膜拜楼主一下

  • Jacob

    很详细表示很好理解~~

  • Sue0610

    第九题真的很orz您的代码啊~
    开始不明白const int validSet[]={0,3,6,12,15,24,27,30};的含义,琢磨明白之后只有orz的份了~

  • Eureka

    对于第三题这样子的可以这样做:这里的A就视为一个常量吧,k作为参数,那么这个式子就可以表示为f(k),可以知道f(k)=f(k-1)*A+A,这个东西就可以化归到题目7中去了。如果这里面的A是矩阵的话,该用1的地方用单位矩阵就好了

  • 那个

    //Complex(1);
    Rot(x,thet){x*(sin(thet)+i*cos(thet));}
    >> Rot(x, thet) #
    x=Rot(x,2);
    >> x*sin^2(2)-x*cos^2(2) + 2*x*sin(2)*cos(2)*i #
    x=Rot(x,2);
    >> x*sin^3(2)-3*x*sin(2)*cos^2(2) + (3*x*sin^2(2)*cos(2)-x*cos^3(2))*i #
    Mov(x,a,b){x+a+b*i;}
    >> Mov(x, a, b) #
    Fx(x){-Conjugate(x);}
    >> Fx(x) #
    Fy(x){Conjugate(x);}
    >> Fy(x) #
    Mult(x,n){x*n;}
    >> Mult(x, n) #
    x=Rot(x,-2);
    >> x*sin^3(2)*sin(-2)-3*x*sin^2(2)*cos(2)*cos(-2)-3*x*sin(2)*cos^2(2)*sin(-2)+x*cos^3(2)*cos(-2) + (x*sin^3(2)*cos(-2)+3*x*sin^2(2)*cos(2)*sin(-2)-3*x*sin(2)*cos^2(2)*cos(-2)-x*cos^3(2)*sin(-2))*i #
    x=Mov(x,3,4);
    >> x*sin^3(2)*sin(-2)-3*x*sin^2(2)*cos(2)*cos(-2)-3*x*sin(2)*cos^2(2)*sin(-2)+x*cos^3(2)*cos(-2)+3 + (x*sin^3(2)*cos(-2)+3*x*sin^2(2)*cos(2)*sin(-2)-3*x*sin(2)*cos^2(2)*cos(-2)-x*cos^3(2)*sin(-2)+4)*i #
    Float(x);
    >> 计算结果显示浮点数 #
    Subst(x, x, 1);
    >> sin^3(2)*sin(-2)-3*sin^2(2)*cos(2)*cos(-2)-3*sin(2)*cos^2(2)*sin(-2)+cos^3(2)*cos(-2)+3 + (sin^3(2)*cos(-2)+3*sin^2(2)*cos(2)*sin(-2)-3*sin(2)*cos^2(2)*cos(-2)-cos^3(2)*sin(-2)+4)*i #

  • 4

    for(i=0;i<n;i++)b[n]=a[a[n]];

  • cervelo

    好像可以在类里面定义成友元函数……和在类外定义是等效的……这样看起来可能会比较舒服……

  • nopy

    最后少了一段T.T 应该是博客升级更新的时候丢了部分数据。。
    百度了下补上

    我们以M=3为例进行讲解。假设我们把这个矩形横着放在电脑屏幕上,从右往左一列一列地进行填充。其中前n-2列已经填满了,第n-1列参差不齐。现在我 们要做的事情是把第n-1列也填满,将状态转移到第n列上去。由于第n-1列的状态不一样(有8种不同的状态),因此我们需要分情况进行讨论。在图中,我 把转移前8种不同的状态放在左边,转移后8种不同的状态放在右边,左边的某种状态可以转移到右边的某种状态就在它们之间连一根线。注意为了保证方案不重 复,状态转移时我们不允许在第n-1列竖着放一个多米诺骨牌(例如左边第2种状态不能转移到右边第4种状态),否则这将与另一种转移前的状态重复。把这8 种状态的转移关系画成一个有向图,那么问题就变成了这样:从状态111出发,恰好经过n步回到这个状态有多少种方案。比如,n=2时有3种方 案,111->011->111、111->110->111和111->000->111,这与用多米诺骨牌覆盖 3×2矩形的方案一一对应。这样这个题目就转化为了我们前面的例题8

  • 浣熊

    还可以用矩阵来解决soduko

  • RR

    T4可以做到O(m)(所以说m<=10真是太小了啊开到10^8吧(雾))

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