noip 2009 靶形数独

题目描述 Description

　　小城和小华都是热爱数学的好学生，最近，他们不约而同地迷上了数独游戏，好胜的他
们想用数独来一比高低。但普通的数独对他们来说都过于简单了，于是他们向Z 博士请教，
Z 博士拿出了他最近发明的“靶形数独”，作为这两个孩子比试的题目。
靶形数独的方格同普通数独一样，在 9 格宽×9 格高的大九宫格中有9 个3 格宽×3 格
高的小九宫格（用粗黑色线隔开的）。在这个大九宫格中，有一些数字是已知的，根据这些

　　数字，利用逻辑推理，在其他的空格上填入1 到9 的数字。每个数字在每个小九宫格内不能
重复出现，每个数字在每行、每列也不能重复出现。但靶形数独有一点和普通数独不同，即
每一个方格都有一个分值，而且如同一个靶子一样，离中心越近则分值越高。

　　上图具体的分值分布是：最里面一格（黄色区域）为 10 分，黄色区域外面的一圈（红
色区域）每个格子为9 分，再外面一圈（蓝色区域）每个格子为8 分，蓝色区域外面一圈（棕
色区域）每个格子为7 分，最外面一圈（白色区域）每个格子为6 分，如上图所示。比赛的
要求是：每个人必须完成一个给定的数独（每个给定数独可能有不同的填法），而且要争取
更高的总分数。而这个总分数即每个方格上的分值和完成这个数独时填在相应格上的数字
的乘积的总和。如图，在以下的这个已经填完数字的靶形数独游戏中，总分数为2829。游
戏规定，将以总分数的高低决出胜负。

由于求胜心切，小城找到了善于编程的你，让你帮他求出，对于给定的靶形数独，能
够得到的最高分数。

输入描述 Input Description

　　一共 9 行。每行9 个整数（每个数都在0—9 的范围内），表示一个尚未填满的数独方
格，未填的空格用“0”表示。每两个数字之间用一个空格隔开。

输出描述 Output Description

　　输出可以得到的靶形数独的最高分数。如果这个数独无解，则输出整数-1。

样例输入 Sample Input

【输入输出样例 1】

7 0 0 9 0 0 0 0 1
1 0 0 0 0 5 9 0 0
0 0 0 2 0 0 0 8 0
0 0 5 0 2 0 0 0 3
0 0 0 0 0 0 6 4 8
4 1 3 0 0 0 0 0 0
0 0 7 0 0 2 0 9 0
2 0 1 0 6 0 8 0 4
0 8 0 5 0 4 0 1 2

【输入输出样例 2】

0 0 0 7 0 2 4 5 3
9 0 0 0 0 8 0 0 0
7 4 0 0 0 5 0 1 0
1 9 5 0 8 0 0 0 0
0 7 0 0 0 0 0 2 5
0 3 0 5 7 9 1 0 8
0 0 0 6 0 1 0 0 0
0 6 0 9 0 0 0 0 1
0 0 0 0 0 0 0 0 6

样例输出 Sample Output

【输入输出样例 1】

2829

【输入输出样例 1】

2852

数据范围及提示 Data Size & Hint

【数据范围】
40%的数据，数独中非0 数的个数不少于30。
80%的数据，数独中非0 数的个数不少于26。
100%的数据，数独中非0 数的个数不少于24。

　　这道题很不正经。。。先来用个正常点的思维，对于剩下的分数和乘8比当前最优解还小，直接return掉，再加个位运算优化，然后codevs和vijos轻松A掉4000毫秒左右。然后交到教师机去测，竟然T掉。。。这可不科学。。。

Code1

  1 #include<iostream>
  2 #include<fstream>
  3 #include<sstream>
  4 #include<cstdio>
  5 #include<cstdlib>
  6 #include<cstring>
  7 #include<ctime>
  8 #include<cctype>
  9 #include<cmath>
 10 #include<algorithm>
 11 #include<stack>
 12 #include<queue>
 13 #include<set>
 14 #include<map>
 15 #include<vector>
 16 using namespace std;
 17 typedef bool boolean;
 18 #define smin(a, b)    (a) = min((a), (b))
 19 #define smax(a, b)    (a) = max((a), (b))
 20 template<typename T>
 21 inline void readInteger(T& u){
 22     char x;
 23     int aFlag = 1;
 24     while(!isdigit((x = getchar())) && x != '-');
 25     if(x == '-'){
 26         x = getchar();
 27         aFlag = -1;
 28     }
 29     for(u = x - '0'; isdigit((x = getchar())); u = (u << 3) + (u << 1) + x - '0');
 30     ungetc(x, stdin);
 31     u *= aFlag;
 32 }
 33 
 34 const int score[9][9] = {{6, 6, 6, 6, 6, 6, 6, 6, 6}, 
 35                          {6, 7, 7, 7, 7, 7, 7, 7, 6},
 36                          {6, 7, 8, 8, 8, 8, 8, 7, 6},
 37                          {6, 7, 8, 9, 9, 9, 8, 7, 6},
 38                          {6, 7, 8, 9, 10,9, 8, 7, 6},
 39                          {6, 7, 8, 9, 9, 9, 8, 7, 6},
 40                          {6, 7, 8, 8, 8, 8, 8, 7, 6},
 41                          {6, 7, 7, 7, 7, 7, 7, 7, 6},
 42                          {6, 6, 6, 6, 6, 6, 6, 6, 6}};
 43 int result = -1;
 44 short bmat[9][9];
 45 int cells[9];
 46 int lines[9];
 47 int rows[9];
 48 int sum;
 49 
 50 void search(int s, int x, int y){
 51     if(y == 9){
 52         smax(result, s);
 53         return;
 54     }
 55     if(sum * 8 + s <= result)    return;
 56     if(bmat[x][y] == 0){
 57         int ce = (x / 3) * 3 + y / 3;
 58         int seced = (rows[x] | lines[y]) | cells[ce];
 59         for(int i = 8; i >= 0; i--){
 60             if(!(seced & (1 << i))){
 61                 int mask = 1 << i;
 62                 rows[x] |= mask, lines[y] |= mask, cells[ce] |= mask, sum -= i + 1;
 63                 search(s + (i + 1) * score[x][y], ((x == 8) ? (0) : (x + 1)), 
 64                     ((x == 8) ? (y + 1) : (y)));
 65                 rows[x] ^= mask, lines[y] ^= mask, cells[ce] ^= mask, sum += i + 1;
 66             }
 67         }
 68     }else{
 69         sum -= bmat[x][y];
 70         search(s + bmat[x][y] * score[x][y], ((x == 8) ? (0) : (x + 1)), 
 71                     ((x == 8) ? (y + 1) : (y)));
 72         sum += bmat[x][y];
 73     }
 74 }
 75 
 76 inline void init(){
 77     for(int i = 0; i < 9; i++){
 78         for(int j = 0; j < 9; j++){
 79             readInteger(bmat[i][j]);
 80             if(bmat[i][j] != 0){
 81                 int mask = (1 << (bmat[i][j] - 1));
 82                 rows[i] |= mask, lines[j] |= mask, cells[(i / 3) * 3 + j / 3] |= mask;
 83             }
 84         }
 85     }
 86     sum = 405;
 87 }
 88 
 89 inline void solve(){
 90     search(0, 0, 0);
 91     printf("%d", result);
 92 }
 93 
 94 int main(){
 95         freopen("sudoku.in", "r", stdin);
 96     freopen("sudoku.out", "w", stdout);
 97     init();
 98     solve();
 99     return 0;
100 }
朴素剪枝

　　教师机跑得比较慢，但是还是要“体谅”一下。然后再换个方法剪枝，改变搜索顺序，先搜约束条件少的

Code2

  1 #include<iostream>
  2 #include<sstream>
  3 #include<cstdio>
  4 #include<cstdlib>
  5 #include<cstring>
  6 #include<ctime>
  7 #include<cctype>
  8 #include<cmath>
  9 #include<algorithm>
 10 #include<stack>
 11 #include<queue>
 12 #include<set>
 13 #include<map>
 14 #include<vector>
 15 using namespace std;
 16 typedef bool boolean;
 17 #define smin(a, b)    (a) = min((a), (b))
 18 #define smax(a, b)    (a) = max((a), (b))
 19 template<typename T>
 20 inline void readInteger(T& u){
 21     char x;
 22     int aFlag = 1;
 23     while(!isdigit((x = getchar())) && x != '-');
 24     if(x == '-'){
 25         x = getchar();
 26         aFlag = -1;
 27     }
 28     for(u = x - '0'; isdigit((x = getchar())); u = (u << 3) + (u << 1) + x - '0');
 29     ungetc(x, stdin);
 30     u *= aFlag;
 31 }
 32 
 33 typedef class Cell {
 34     public:
 35         int x, y;
 36         int mask;
 37         
 38         Cell(const int x = 0, const int y = 0, const int mask = 0):x(x), y(y), mask(mask) {           }
 39         
 40         boolean operator < (Cell b) const {
 41             return mask > b.mask;
 42         }
 43 }Cell;
 44 
 45 const int score[9][9] = {{6, 6, 6, 6, 6, 6, 6, 6, 6}, 
 46                          {6, 7, 7, 7, 7, 7, 7, 7, 6},
 47                          {6, 7, 8, 8, 8, 8, 8, 7, 6},
 48                          {6, 7, 8, 9, 9, 9, 8, 7, 6},
 49                          {6, 7, 8, 9, 10,9, 8, 7, 6},
 50                          {6, 7, 8, 9, 9, 9, 8, 7, 6},
 51                          {6, 7, 8, 8, 8, 8, 8, 7, 6},
 52                          {6, 7, 7, 7, 7, 7, 7, 7, 6},
 53                          {6, 6, 6, 6, 6, 6, 6, 6, 6}};
 54 int result = -1;
 55 short bmat[9][9];
 56 int cells[9];
 57 int lines[9];
 58 int rows[9];
 59 int sum;
 60 Cell table[81];
 61 
 62 void search(int s, int t){
 63     if(t == 81){
 64         smax(result, s);
 65         return;
 66     }
 67     if(sum * 9 + s <= result)    return;
 68     int x = table[t].x, y = table[t].y;
 69     if(bmat[x][y] == 0){
 70         int ce = (x / 3) * 3 + y / 3;
 71         int seced = (rows[x] | lines[y]) | cells[ce];
 72         for(int i = 8; i >= 0; i--){
 73             if(!(seced & (1 << i))){
 74                 int mask = 1 << i;
 75                 rows[x] |= mask, lines[y] |= mask, cells[ce] |= mask, sum -= i + 1;
 76                 search(s + (i + 1) * score[x][y], t + 1);
 77                 rows[x] ^= mask, lines[y] ^= mask, cells[ce] ^= mask, sum += i + 1;
 78             }
 79         }
 80     }else{
 81         sum -= bmat[x][y];
 82         search(s + bmat[x][y] * score[x][y], t + 1);
 83         sum += bmat[x][y];
 84     }
 85 }
 86 
 87 inline void init(){
 88     for(int i = 0; i < 9; i++){
 89         for(int j = 0; j < 9; j++){
 90             readInteger(bmat[i][j]);
 91             if(bmat[i][j] != 0){
 92                 int mask = (1 << (bmat[i][j] - 1));
 93                 rows[i] |= mask, lines[j] |= mask, cells[(i / 3) * 3 + j / 3] |= mask;
 94             }
 95         }
 96     }
 97     sum = 405;
 98 }
 99 
100 const int full = (1 << 9) - 1;
101 
102 inline void solve(){
103     for(int i = 0; i < 9; i++) {
104         for(int j = 0; j < 9; j++) {
105             if(bmat[i][j] != 0)
106                 table[i * 9 + j] = Cell(i, j, full);
107             else table[i * 9 + j] = Cell(i, j, (rows[i] | lines[j]) | cells[(i / 3) * 3 + j / 3]);
108         }
109     }
110     sort(table, table + 81);
111     search(0, 0);
112     printf("%d", result);
113 }
114 
115 int main(){
116     freopen("sudoku.in", "r", stdin);
117     freopen("sudoku.out", "w", stdout);
118     init();
119     solve();
120     return 0;
121 }
改变搜索顺序

　　于是刚刚卡我的那个点A掉了，但是多T掉了3个点，内心是崩溃的。不行再结合一下之前的搜索顺序

Code3

  1 #include<iostream>
  2 #include<sstream>
  3 #include<cstdio>
  4 #include<cstdlib>
  5 #include<cstring>
  6 #include<ctime>
  7 #include<cctype>
  8 #include<cmath>
  9 #include<algorithm>
 10 #include<stack>
 11 #include<queue>
 12 #include<set>
 13 #include<map>
 14 #include<vector>
 15 using namespace std;
 16 typedef bool boolean;
 17 #define smin(a, b)    (a) = min((a), (b))
 18 #define smax(a, b)    (a) = max((a), (b))
 19 template<typename T>
 20 inline void readInteger(T& u){
 21     char x;
 22     int aFlag = 1;
 23     while(!isdigit((x = getchar())) && x != '-');
 24     if(x == '-'){
 25         x = getchar();
 26         aFlag = -1;
 27     }
 28     for(u = x - '0'; isdigit((x = getchar())); u = (u << 3) + (u << 1) + x - '0');
 29     ungetc(x, stdin);
 30     u *= aFlag;
 31 }
 32 
 33 typedef class Cell {
 34     public:
 35         int x, y;
 36         int mask;
 37         
 38         Cell(const int x = 0, const int y = 0, const int mask = 0):x(x), y(y), mask(mask) {           }
 39         
 40         boolean operator < (Cell b) const {
 41             if(x != b.x)    return x > b.x;
 42             return mask > b.mask;
 43         }
 44 }Cell;
 45 
 46 const int score[9][9] = {{6, 6, 6, 6, 6, 6, 6, 6, 6}, 
 47                          {6, 7, 7, 7, 7, 7, 7, 7, 6},
 48                          {6, 7, 8, 8, 8, 8, 8, 7, 6},
 49                          {6, 7, 8, 9, 9, 9, 8, 7, 6},
 50                          {6, 7, 8, 9, 10,9, 8, 7, 6},
 51                          {6, 7, 8, 9, 9, 9, 8, 7, 6},
 52                          {6, 7, 8, 8, 8, 8, 8, 7, 6},
 53                          {6, 7, 7, 7, 7, 7, 7, 7, 6},
 54                          {6, 6, 6, 6, 6, 6, 6, 6, 6}};
 55 int result = -1;
 56 short bmat[9][9];
 57 int cells[9];
 58 int lines[9];
 59 int rows[9];
 60 int sum;
 61 Cell table[81];
 62 
 63 void search(int s, int t){
 64     if(t == 81){
 65         smax(result, s);
 66         return;
 67     }
 68     if(sum * 9 + s <= result)    return;
 69     int x = table[t].x, y = table[t].y;
 70     if(bmat[x][y] == 0){
 71         int ce = (x / 3) * 3 + y / 3;
 72         int seced = (rows[x] | lines[y]) | cells[ce];
 73         for(int i = 8; i >= 0; i--){
 74             if(!(seced & (1 << i))){
 75                 int mask = 1 << i;
 76                 rows[x] |= mask, lines[y] |= mask, cells[ce] |= mask, sum -= i + 1;
 77                 search(s + (i + 1) * score[x][y], t + 1);
 78                 rows[x] ^= mask, lines[y] ^= mask, cells[ce] ^= mask, sum += i + 1;
 79             }
 80         }
 81     }else{
 82         sum -= bmat[x][y];
 83         search(s + bmat[x][y] * score[x][y], t + 1);
 84         sum += bmat[x][y];
 85     }
 86 }
 87 
 88 inline void init(){
 89     for(int i = 0; i < 9; i++){
 90         for(int j = 0; j < 9; j++){
 91             readInteger(bmat[i][j]);
 92             if(bmat[i][j] != 0){
 93                 int mask = (1 << (bmat[i][j] - 1));
 94                 rows[i] |= mask, lines[j] |= mask, cells[(i / 3) * 3 + j / 3] |= mask;
 95             }
 96         }
 97     }
 98     sum = 405;
 99 }
100 
101 const int full = (1 << 9) - 1;
102 
103 inline void solve(){
104     for(int i = 0; i < 9; i++) {
105         for(int j = 0; j < 9; j++) {
106             if(bmat[i][j] != 0)
107                 table[i * 9 + j] = Cell(i, j, full);
108             else table[i * 9 + j] = Cell(i, j, (rows[i] | lines[j]) | cells[(i / 3) * 3 + j / 3]);
109         }
110     }
111     sort(table, table + 81);
112     search(0, 0);
113     printf("%d", result);
114 }
115 
116 int main(){
117     freopen("sudoku.in", "r", stdin);
118     freopen("sudoku.out", "w", stdout);
119     init();
120     solve();
121     return 0;
122 }
暗黑剪枝

　　OJ上1600ms左右，教师机上20个点终于都过了。

　　最后发现，倒着搜索，是最简单的AC方法，OJ上2100ms左右

Code4

  1 #include<iostream>
  2 #include<sstream>
  3 #include<cstdio>
  4 #include<cstdlib>
  5 #include<cstring>
  6 #include<ctime>
  7 #include<cctype>
  8 #include<cmath>
  9 #include<algorithm>
 10 #include<stack>
 11 #include<queue>
 12 #include<set>
 13 #include<map>
 14 #include<vector>
 15 using namespace std;
 16 typedef bool boolean;
 17 #define smin(a, b)    (a) = min((a), (b))
 18 #define smax(a, b)    (a) = max((a), (b))
 19 template<typename T>
 20 inline void readInteger(T& u){
 21     char x;
 22     int aFlag = 1;
 23     while(!isdigit((x = getchar())) && x != '-');
 24     if(x == '-'){
 25         x = getchar();
 26         aFlag = -1;
 27     }
 28     for(u = x - '0'; isdigit((x = getchar())); u = (u << 3) + (u << 1) + x - '0');
 29     ungetc(x, stdin);
 30     u *= aFlag;
 31 }
 32 
 33 typedef class Cell {
 34     public:
 35         int x, y;
 36         int mask;
 37         
 38         Cell(const int x = 0, const int y = 0, const int mask = 0):x(x), y(y), mask(mask) {           }
 39         
 40         boolean operator < (Cell b) const {
 41             if(x != b.x)    return x > b.x;
 42             return y > b.y;
 43         }
 44 }Cell;
 45 
 46 const int score[9][9] = {{6, 6, 6, 6, 6, 6, 6, 6, 6}, 
 47                          {6, 7, 7, 7, 7, 7, 7, 7, 6},
 48                          {6, 7, 8, 8, 8, 8, 8, 7, 6},
 49                          {6, 7, 8, 9, 9, 9, 8, 7, 6},
 50                          {6, 7, 8, 9, 10,9, 8, 7, 6},
 51                          {6, 7, 8, 9, 9, 9, 8, 7, 6},
 52                          {6, 7, 8, 8, 8, 8, 8, 7, 6},
 53                          {6, 7, 7, 7, 7, 7, 7, 7, 6},
 54                          {6, 6, 6, 6, 6, 6, 6, 6, 6}};
 55 int result = -1;
 56 short bmat[9][9];
 57 int cells[9];
 58 int lines[9];
 59 int rows[9];
 60 int sum;
 61 Cell table[81];
 62 
 63 void search(int s, int t){
 64     if(t == 81){
 65         smax(result, s);
 66         return;
 67     }
 68     if(sum * 9 + s <= result)    return;
 69     int x = table[t].x, y = table[t].y;
 70     if(bmat[x][y] == 0){
 71         int ce = (x / 3) * 3 + y / 3;
 72         int seced = (rows[x] | lines[y]) | cells[ce];
 73         for(int i = 8; i >= 0; i--){
 74             if(!(seced & (1 << i))){
 75                 int mask = 1 << i;
 76                 rows[x] |= mask, lines[y] |= mask, cells[ce] |= mask, sum -= i + 1;
 77                 search(s + (i + 1) * score[x][y], t + 1);
 78                 rows[x] ^= mask, lines[y] ^= mask, cells[ce] ^= mask, sum += i + 1;
 79             }
 80         }
 81     }else{
 82         sum -= bmat[x][y];
 83         search(s + bmat[x][y] * score[x][y], t + 1);
 84         sum += bmat[x][y];
 85     }
 86 }
 87 
 88 inline void init(){
 89     for(int i = 0; i < 9; i++){
 90         for(int j = 0; j < 9; j++){
 91             readInteger(bmat[i][j]);
 92             if(bmat[i][j] != 0){
 93                 int mask = (1 << (bmat[i][j] - 1));
 94                 rows[i] |= mask, lines[j] |= mask, cells[(i / 3) * 3 + j / 3] |= mask;
 95             }
 96         }
 97     }
 98     sum = 405;
 99 }
100 
101 const int full = (1 << 9) - 1;
102 
103 inline void solve(){
104     for(int i = 0; i < 9; i++) {
105         for(int j = 0; j < 9; j++) {
106             if(bmat[i][j] != 0)
107                 table[i * 9 + j] = Cell(i, j, full);
108             else table[i * 9 + j] = Cell(i, j, (rows[i] | lines[j]) | cells[(i / 3) * 3 + j / 3]);
109         }
110     }
111     sort(table, table + 81);
112     search(0, 0);
113     printf("%d", result);
114 }
115 
116 int main(){
117     freopen("sudoku.in", "r", stdin);
118     freopen("sudoku.out", "w", stdout);
119     init();
120     solve();
121     return 0;
122 }
倒着搜索
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原文地址：https://www.cnblogs.com/yyf0309/p/6501427.html