A Short problem

Time Limit: 2000/1000 MS (Java/Others)    Memory Limit: 32768/32768 K (Java/Others)
Total Submission(s): 1716    Accepted Submission(s): 631

Problem Description

　　According to a research, VIM users tend to have shorter fingers, compared with Emacs users.
　　Hence they prefer problems short, too. Here is a short one:
　　Given n (1 <= n <= 10¹⁸), You should solve for
g(g(g(n))) mod 10⁹ + 7
　　where
g(n) = 3g(n - 1) + g(n - 2)
g(1) = 1
g(0) = 0

Input

　　There are several test cases. For each test case there is an integer n in a single line.
　　Please process until EOF (End Of File).

Output

　　For each test case, please print a single line with a integer, the corresponding answer to this case.

Sample Input

0
1
2

Sample Output

0
1
42837

Source

2012 ACM/ICPC Asia Regional Chengdu Online

此题出得比较精妙，

分析:假设g(g(g(n)))=g(x),x可能超出范围,但是由于mod 10^9+7,所以可以求出x的循环节

求出x的循环节后,假设g(g(g(n)))=g(x)=g(g(y)),即x=g(y),y也可能非常大,但是由x的循环节可以求出y的循环节

如何求循环节点：

 1 /*采用事先处理自己可以求出来*/
 2 LL work(LL mod){
 3   LL a=0,b=1;
 4     for(LL i=2;;++i)
 5     {
 6       a=(b*3+a)%mod;
 7         a=a^b;
 8         b=a^b;
 9         a=a^b;
10         if(a == 0 && b == 1)  return i;
11  }

所以依次将mod1带入得到mod2=222222224;

然后将mod2带入得到mod3=183120;

然后就是快速矩阵了。

代码：

 1 //#define LOCAL
 2 #include<iostream>
 3 #include<cstdio>
 4 #include<cstring>
 5 #define LL __int64
 6 using namespace std;
 7 const int mod1 =1000000007;
 8 const int mod2=222222224;
 9 const int mod3=183120;
10
11 LL mat[2][2];
12 LL ans[2][2];
13 LL n;
14
15 void Matrix(LL a[][2],LL b[][2],LL mod)
16 {
17     LL cc[2][2]={0};
18     for(int i=0;i<2;i++)
19     {
20       for(int j=0;j<2;j++)
21       {
22           for(int k=0;k<2;k++)
23         {
24             cc[i][j]=(cc[i][j]+a[i][k]*b[k][j])%mod;
25         }
26       }
27     }
28     for(int i=0;i<2;i++)
29     {
30       for(int j=0;j<2;j++)
31       {
32           a[i][j]=cc[i][j];
33       }
34     }
35 }
36
37 void pow(LL w,LL mod)
38 {
39   while(w>0)
40   {
41     if(w&1) Matrix(ans,mat,mod);
42      w>>=1;
43      if(w==0)break;
44      Matrix(mat,mat,mod);
45   }
46 }
47 void input(LL w,LL mod)
48 {
49      mat[0][0]=3;
50      mat[0][1]=mat[1][0]=1;
51      mat[1][1]=0;
52      ans[0][0]=ans[1][1]=1;
53      ans[0][1]=ans[1][0]=0;
54      pow(w,mod);
55 //     printf("%I64d\n",ans[0][0]);
56 }
57 void work(int i,__int64 w)
58 {
59   if(i==4||w==0||w==1)
60   {
61       if(w==0)
62          printf("0\n");
63       else if(w==1)
64           printf("1\n");
65       else
66       printf("%I64d\n",ans[0][0]);
67       return ;
68   }
69   LL mod;
70   if(i==1)mod=mod3;
71   else if(i==2)mod=mod2;
72   else if(i==3)mod=mod1;
73   input(w-1,mod);
74   work(i+1,ans[0][0]);
75 }
76 int main()
77 {
78   #ifdef LOCAL
79    freopen("test.in","r",stdin);
80   #endif
81   while(scanf("%I64d",&n)!=EOF)
82      work(1,n);
83  return 0;
84 }