HDU 2276 Kiki & Little Kiki 2（构造矩阵）

题目链接：http://acm.hdu.edu.cn/showproblem.php?pid=2276

题意：有n盏灯，0表示不亮，1表示亮，如果 第i盏灯的左边灯是亮的，那么下一秒钟，该灯的状态要改变。问你在第t秒时，灯的状态时什么样的。

思路：由题意a1=(a1+an)%2;  a2=(a1+a2)%2;  a3=(a2+a3)%2.......an=(an-1+an)%2，对于第一组数据，可构造如下矩阵a

1 1 0 0 0 0 0
0 1 1 0 0 0 0
0 0 1 1 0 0 0
0 0 0 1 1 0 0
0 0 0 0 1 1 0
0 0 0 0 0 1 1
1 0 0 0 0 0 1

输入矩阵为b=0 1 0 1 1 1 1，可得b*a^n就是n秒后的状态。

#include <iostream>

#include <cstdio>

#include <string.h>

#include <algorithm>

#include <cmath>

#include <vector>

#include <queue>

#include <set>

#include <string>

#include <map>



#define max(x,y) ((x)>(y)?(x):(y))

#define min(x,y) ((x)<(y)?(x):(y))

#define abs(x) ((x)>=0?(x):-(x))

#define i64 long long

#define u32 unsigned int

#define u64 unsigned long long

#define clr(x,y) memset(x,y,sizeof(x))

#define CLR(x) x.clear()

#define ph(x) push(x)

#define pb(x) push_back(x)

#define Len(x) x.length()

#define SZ(x) x.size()

#define PI acos(-1.0)

#define sqr(x) ((x)*(x))



#define FOR0(i,x) for(i=0;i<x;i++)

#define FOR1(i,x) for(i=1;i<=x;i++)

#define FOR(i,a,b) for(i=a;i<=b;i++)

#define DOW0(i,x) for(i=x;i>=0;i--)

#define DOW1(i,x) for(i=x;i>=1;i--)

#define DOW(i,a,b) for(i=a;i>=b;i--)

using namespace std;





void RD(int &x){scanf("%d",&x);}

void RD(i64 &x){scanf("%lld",&x);}

void RD(u64 &x){scanf("%llu",&x);}

void RD(u32 &x){scanf("%u",&x);}

void RD(double &x){scanf("%lf",&x);}

void RD(int &x,int &y){scanf("%d%d",&x,&y);}

void RD(i64 &x,i64 &y){scanf("%lld%lld",&x,&y);}

void RD(u64 &x,u64 &y){scanf("%llu%llu",&x,&y);}

void RD(u32 &x,u32 &y){scanf("%u%u",&x,&y);}

void RD(double &x,double &y){scanf("%lf%lf",&x,&y);}

void RD(int &x,int &y,int &z){scanf("%d%d%d",&x,&y,&z);}

void RD(i64 &x,i64 &y,i64 &z){scanf("%lld%lld%lld",&x,&y,&z);}

void RD(u64 &x,u64 &y,u64 &z){scanf("%llu%llu%llu",&x,&y,&z);}

void RD(u32 &x,u32 &y,u32 &z){scanf("%u%u%u",&x,&y,&z);}

void RD(double &x,double &y,double &z){scanf("%lf%lf%lf",&x,&y,&z);}

void RD(char &x){x=getchar();}

void RD(char *s){scanf("%s",s);}

void RD(string &s){cin>>s;}



void PR(int x) {printf("%d\n",x);}

void PR(i64 x) {printf("%I64d\n",x);}

void PR(u64 x) {printf("%llu\n",x);}

void PR(u32 x) {printf("%u\n",x);}

void PR(double x) {printf("%lf\n",x);}

void PR(char x) {printf("%c\n",x);}

void PR(char *x) {printf("%s\n",x);}

void PR(string x) {cout<<x<<endl;}



const int mod=2;

const int INF=2000000000;

const int N=105;

int n;



class Matrix

{

public:

    int a[N][N];



    void init(int x)

    {

        int i;

        if(!x) clr(a,0);

        else

        {

            clr(a,0);

            FOR0(i,N) a[i][i]=1;

        }

    }



    void read(int n)

    {

        int i,j;

        FOR1(i,n) FOR1(j,n)

        {

            RD(a[i][j]);

            a[i][j]%=mod;

        }

    }



    Matrix operator+(Matrix b)

    {

        int i,j;

        Matrix c;

        FOR0(i,n) FOR0(j,n) c.a[i][j]=(a[i][j]+b.a[i][j])%mod;

        return c;

    }



    Matrix operator+(int x)

    {

        int i;

        Matrix c=*this;

        FOR0(i,n) (c.a[i][i]+=x)%=mod;

        return c;

    }





    Matrix operator*(Matrix b)

    {

        int i,j,k;

        Matrix p;

        p.init(0);

        FOR0(i,n) FOR0(j,n) FOR0(k,n)

        {

            (p.a[i][j]+=a[i][k]*b.a[k][j]%mod)%=mod;

        }

        return p;

    }



    Matrix power(int t)

    {

        Matrix ans,p=*this;

        ans.init(1);

        while(t)

        {

            if(t&1) ans=ans*p;

            p=p*p;

            t>>=1;

        }

        return ans;

    }



    void print()

    {

        int i,j;

        FOR0(i,n)

        {

            FOR0(j,n) printf("%d ",a[i][j]);

            puts("");

        }

    }

};



Matrix a;

string s;

int m;







int main()

{

    while(scanf("%d",&m)!=-1)

    {

        RD(s);

        n=Len(s);

        a.init(0);

        int i,j,k;

        FOR0(i,n)

        {

            if(i==n-1) a.a[i][i]=a.a[i][0]=1;

            else a.a[i][i]=a.a[i][i+1]=1;

        }

        a=a.power(m);

        FOR0(i,n)

        {

            k=0;

            FOR0(j,n) (k+=a.a[j][i]*(s[j]-'0'))&=1;

            printf("%d",k);

        }

        puts("");

    }

    return 0;

}