演算法提高 遞推求值 (C程式碼只有50分, 附上答案C++程式碼,看起來比較吃力)
阿新 • • 發佈:2019-02-11
問題描述
已知遞推公式:
F(n, 1)=F(n-1, 2) + 2F(n-3, 1) + 5,
F(n, 2)=F(n-1, 1) + 3F(n-3, 1) + 2F(n-3, 2) + 3.
初始值為:F(1, 1)=2, F(1, 2)=3, F(2, 1)=1, F(2, 2)=4, F(3, 1)=6, F(3, 2)=5。
輸入n,輸出F(n, 1)和F(n, 2),由於答案可能很大,你只需要輸出答案除以99999999的餘數。 輸入格式 輸入第一行包含一個整數n。 輸出格式 輸出兩行,第一行為F(n, 1)除以99999999的餘數,第二行為F(n, 2)除以99999999的餘數。 樣例輸入 4 樣例輸出 14
21 資料規模和約定 1<=n<=10^18。
F(n, 1)=F(n-1, 2) + 2F(n-3, 1) + 5,
F(n, 2)=F(n-1, 1) + 3F(n-3, 1) + 2F(n-3, 2) + 3.
初始值為:F(1, 1)=2, F(1, 2)=3, F(2, 1)=1, F(2, 2)=4, F(3, 1)=6, F(3, 2)=5。
輸入n,輸出F(n, 1)和F(n, 2),由於答案可能很大,你只需要輸出答案除以99999999的餘數。 輸入格式 輸入第一行包含一個整數n。 輸出格式 輸出兩行,第一行為F(n, 1)除以99999999的餘數,第二行為F(n, 2)除以99999999的餘數。 樣例輸入 4 樣例輸出 14
21 資料規模和約定 1<=n<=10^18。
#include <stdio.h> #define BIG 99999999 int iskonw[4][3]={ {0}, {0,2,3}, {0,1,4}, {0,6,5} }; int jieguo[99999][3];//動態規劃 /*int fun(int n,int num){ //粗糙的動態規劃,題目系統裡只得了40分 int temp=0; if(n-3>=1){ if(jieguo[n][num]!=0) return jieguo[n][num]%BIG; if(num==1) temp=fun(n-1,2)+2*fun(n-3,1)+5; else if(num==2) temp=fun(n-1,1)+3*fun(n-3,1)+2*fun(n-3,2)+3; jieguo[n][num]=temp; } else return iskonw[n][num]; return temp%BIG; } */ int fun2(long n,int num,int tag){ long temp=0; long F1[4],F2[4]; long i,j; F1[1]=6; F2[1]=5; F1[2]=1; F2[2]=4; F1[3]=2; F2[3]=3; if(n>=4) for(i=4;i<=n;i++){ F1[0]=(F2[1]+2*F1[3]+5)%BIG; F2[0]=(F1[1]+3*F1[3]+2*F2[3]+3)%BIG; for(j=3;j>=1;j--){ F1[j]=F1[j-1]; F2[j]=F2[j-1]; } } else return iskonw[n][num]; if(tag==1) return F1[0]; return F2[0]; } int main() { long n=99134193; scanf("%d",&n); printf("%d\n",fun2(n,1,1)); printf("%d\n",fun2(n,2,2)); /**/ //14 21 return 0; }
#include<iostream> #include<cstring> #define MOD 99999999 using namespace std; typedef long long** Mat; void mul_mat(Mat a, Mat b, Mat &c, int R_number_a, int C_number_a, int C_number_b) { int i, j, k; Mat C = new long long*[R_number_a]; for (int i = 0;i<R_number_a;i++) { C[i] = new long long[C_number_b]; for (int j = 0;j<C_number_b;j++) C[i][j] = 0ll; } for ( i = 0;i<R_number_a;i++) for ( k = 0;k < C_number_a;k++) for ( j = 0;j<C_number_b;j++) C[i][j] += a[i][k] * b[k][j]%MOD; delete c; c = C; } Mat pow_mat(Mat region, int R_number, int C_number, long long n) { Mat ans = new long long*[R_number]; Mat t = new long long*[R_number]; for (int i = 0;i<R_number;i++) { ans[i] = new long long[C_number]; for (int j = 0;j<C_number;j++) ans[i][j] = (long long)(i == j); } for (int i = 0;i<R_number;i++) { t[i] = new long long[C_number]; for (int j = 0;j<C_number;j++) t[i][j] = region[i][j]; } while (n) { if (n & 1) mul_mat(ans, t, ans, R_number, C_number, C_number); mul_mat(t, t, t, R_number, C_number, C_number); n = n >> 1; } return ans; } int main() { Mat region = new long long*[7], ans; long long n, a[7] = {5ll,6ll,4ll,1ll,3ll,2ll,1ll}; cin >> n; region[0] = new long long[7]; region[1] = new long long[7]; for (int i = 0;i < 7;i++) region[i] = new long long[7],memset(region[i],0,sizeof(long long)*7); region[0][1]= region[1][0]= region[2][0]= region[3][1]= region[4][2]= region[5][3]= region[6][6]=1ll; region[0][4] = 2ll; region[0][5] = 3ll; region[0][6] = 3ll; region[1][5] = 2ll; region[1][6] = 5ll; if (n < 4) { switch (n) { case 1:cout << 2 << endl << 3 << endl;break; case 2:cout << 1 << endl << 4<< endl;break; case 3:cout << 6 << endl << 5 << endl;break; } } else { ans = pow_mat(region, 7, 7, n - 3); long long fn1 = 0ll, fn2 = 0ll; for (int i = 0;i < 7;i++) fn1 += ans[1][i] * a[i], fn2+= ans[0][i] * a[i]; cout << fn1%MOD << endl << fn2%MOD << endl; } return 0; }