用C++实现的数独解题程序 SudokuSolver 2.7 及实例分析

引言：一个 bug 的发现

在 MobaXterm 上看到有内置的 Sudoku 游戏，于是拿 SudokuSolver 求解，随机出题，一上来是个 medium 级别的题：

073 000 060
980 460 000
000 007 304
000 000 578
010 000 090
248 000 000
105 900 000
000 034 081
090 000 230

运行过程信息如下：

Order please:
load-quiz h:\s.txt
Quiz loaded.

Order please:
guess-mode 1
In interactive guessing mode:1 (0:no; other:yes)

Order please:
run
7) row 1 complete shrunken by group
9) row 3 complete shrunken by group
11) row 3 complete shrunken by group
13) row 3 complete shrunken by group
15) row 4 complete shrunken by group
1GW: 6 shrunken out of [4,3]
1GW: 3 shrunken out of [4,4]
1GW: 6 shrunken out of [4,4]
1GW: 6 shrunken out of [4,6]
17) row 4 incomplete shrunken by group
Take a guess please, by default it will be [3,1]=5:

18) Guess [3,1] level 1 at 1 out of 2
18) shrinking 2 from [8,2] went wrong
19) Forward guess [3,1] level 1 at 2 out of 2
473 215 869
982 463 157
651 897 324

369 142 578
517 386 492
248 759 613

135 928 746
726 534 981
894 671 235

Fine [steps:23, solution sum:1].
Run time: 1958 milliseconds; steps: 23, solution sum: 1.
 Biggest level on this run(til): 1

Order please:
step
24) No more solution (solution sum is 1).
...

Order please:

可以看到走到 17 步时做了一次交互式猜测。好奇这么一个 medium 级别的题是否真的需要猜测，重新进到 16 步时的上下文做仔细考察：

Order please:
runtil 16
7) row 1 complete shrunken by group
9) row 3 complete shrunken by group
11) row 3 complete shrunken by group
13) row 3 complete shrunken by group
15) row 4 complete shrunken by group
473 000 869
982 463 157
001 897 324

000 040 578
010 000 492
248 000 613

105 900 746
000 034 981
090 000 235

Steps:16
Candidates:
[1,4]: 1 2 5    [1,5]: 1 2 5    [1,6]: 1 2 5
[3,1]: 5 6    [3,2]: 5 6    [4,1]: 3 6
[4,2]: 3 6    [4,3]: 6 9    [4,4]: 1 2 3 6
[4,6]: 1 2 6 9    [5,1]: 3 5 6 7    [5,3]: 6 7
[5,4]: 3 5 6 7    [5,5]: 5 7 8    [5,6]: 5 6 8
[6,4]: 5 7    [6,5]: 5 7    [6,6]: 5 9
[7,2]: 2 3    [7,5]: 2 8    [7,6]: 2 8
[8,1]: 6 7    [8,2]: 2 6    [8,3]: 6 7
[8,4]: 2 5 6 7    [9,1]: 6 7 8    [9,3]: 4 6 7
[9,4]: 1 6 7    [9,5]: 1 7 8    [9,6]: 1 6 8
The foremost cell with 2 candidate(s) at [3,1]
Run time: 49 milliseconds; steps: 16, solution sum: 0.
 Biggest level on this run(til): 0

再走一步就要对 [3,1] 位置进行猜测，这时已填数的分布情况为：

473 000 869
982 463 157
001 897 324
000 040 578
010 000 492
248 000 613
105 900 746
000 034 981
090 000 235

考察第 7 行，三个空位的候选值分布情况为：

[7,2]: 2 3    [7,5]: 2 8    [7,6]: 2 8

候选值 3 只出现在 [7,2] 位置，因而必有 [7,2] = 3。而这是早期版本就已经实现了的算法，怎么到这里会做一次猜测？一定是程序有 bug。

调试程序很快发现问题出在 filterCandidates 接口实现如下标黄的语句：

u8 CQuizDealer::filterCandidates()
{
    incSteps();
    u8 ret = RET_PENDING;
    if (m_mode != 2) {
        for (u8 row = 0; row < 9; ++row)
            if (ret = filterRowGroup(row))
                return ret;
        for (u8 col = 0; col < 9; ++col)
            if (ret = filterColGroup(col))
                return ret;
        for (u8 blk = 0; blk < 9; ++blk)
            if (ret = filterBlkGroup(blk))
                return ret;
    }
    ...return ret;
}

从第一行起逐行去调用 filterRowGroup 接口对该行做收缩处理，只要返回值不为 0，即出错或有收缩，就结束当前这一步的收缩处理。出问题时，第 4 行发生了不完全收缩，对应如下输出信息：

1GW: 6 shrunken out of [4,3]
1GW: 3 shrunken out of [4,4]
1GW: 6 shrunken out of [4,4]
1GW: 6 shrunken out of [4,6]
17) row 4 incomplete shrunken by group

这时就结束整个 filterCandidates 的处理，就会导致上面所示的发生在第 7 行的完全收缩 [7,2] = 3 直接被跳过了，随后就出现了不必要的猜测。

SudokuSolver 2.7 版代码修改

filterCandidates 接口实现修改：

 1 u8 CQuizDealer::filterCandidates()
 2 {
 3     incSteps();
 4     u8 ret = RET_PENDING;
 5     bool partlyShrunken = false;
 6     if (m_mode != 2) {
 7         for (u8 row = 0; row < 9; ++row) {
 8             ret = filterRowGroup(row);
 9             if (IsDone(ret))
10                 return ret;
11             if (!partlyShrunken && ret == RET_SHRUNKEN)
12                 partlyShrunken = true;
13         }
14         for (u8 col = 0; col < 9; ++col) {
15             ret = filterColGroup(col);
16             if (IsDone(ret))
17                 return ret;
18             if (!partlyShrunken && ret == RET_SHRUNKEN)
19                 partlyShrunken = true;
20         }
21         for (u8 blk = 0; blk < 9; ++blk) {
22             ret = filterBlkGroup(blk);
23             if (IsDone(ret))
24                 return ret;
25             if (!partlyShrunken && ret == RET_SHRUNKEN)
26                 partlyShrunken = true;
27         }
28     }
29     if (m_mode != 1) {
30         for (u8 row = 0; row < 9; ++row) {
31             ret = filterRowCandidatesEx(row);
32             if (IsDone(ret))
33                 return ret;
34             if (!partlyShrunken && ret == RET_SHRUNKEN)
35                 partlyShrunken = true;
36         }
37         for (u8 col = 0; col < 9; ++col) {
38             ret = filterColCandidatesEx(col);
39             if (IsDone(ret))
40                 return ret;
41             if (!partlyShrunken && ret == RET_SHRUNKEN)
42                 partlyShrunken = true;
43         }
44     }
45     if (partlyShrunken) {
46         printf("%d) incomplete shrink met, filter again\n", m_steps);
47         return filterCandidates();
48     }
49     if (ret == RET_PENDING)
50         printf("%d) no shrink happened\n", m_steps);
51     return ret;
52 }

修改了上述的 bug，末尾部分也增加了未能收缩时输出 no shrink happened 提示信息的处理。

版本信息修改：

// 2.6 2021/10/30
#define STR_VER "Sudoku Solver 2.7 2021/11/1 by readalps\n\n"

用 2.7 版求解“最难”数独题

800 000 000
003 600 000
070 090 200
050 007 000
000 045 700
000 100 030
001 000 068
008 500 010
090 000 400

Order please:
load-quiz h:\s.txt
Quiz loaded.

Order please:
run
2) no shrink happened
3) Guess [8,7] level 1 at 1 out of 2
4) no shrink happened
5) Guess [7,7] level 2 at 1 out of 2
...
380) Guess [1,8] level 9 at 1 out of 2
382) no shrink happened
383) Guess [2,5] level 10 at 1 out of 2
386) shrinking 4 from [6,3] went wrong
387) Forward guess [2,5] level 10 at 2 out of 2
812 753 649
943 682 175
675 491 283

154 237 896
369 845 721
287 169 534

521 974 368
438 526 917
796 318 452

Fine [steps:392, solution sum:1].
Run time: 733 milliseconds; steps: 392, solution sum: 1.
 Biggest level on this run(til): 10

Order please:

第二次 run 的输出：

787) No more solution (solution sum is 1).
800 000 306
003 600 800
670 893 201

352 967 184
186 345 700
040 128 635

001 479 568
068 530 910
090 286 473

Invalid quiz [steps:787] - no more solution (solution sum is 1)
Run time: 573 milliseconds; steps: 787, solution sum: 1.
 Biggest level on this run(til): 8

2.4 版的两次 run 用了两千多步，这里只有 787 步；最大猜测级别也由 12 降到了 10。

用 2.7 版求解另一道难题

005 300 000
800 000 020
070 010 500
400 005 300
010 070 006
003 200 080
060 500 009
004 000 030
000 009 700

因为上述的 bug，此前 2.4 版求解这道数独题时，最大猜测级别达到了 11。来看 2.7 版求解的效果：

H:\Read\num\Release>sudoku.exe

Order please:
load-quiz h:\s.txt
Quiz loaded.

Order please:
run
2) row 2 complete shrunken by group
4) row 5 complete shrunken by group
2GW: 2 shrunken out of [4,3]
2GW: 8 shrunken out of [4,3]
2GW: 9 shrunken out of [4,3]
2GW: 5 shrunken out of [6,1]
2GW: 9 shrunken out of [6,1]
CelSet: [4,3] [6,1] ValSet: 6 7
6) blk 4 incomplete shrunken by group
6) incomplete shrink met, filter again
7) no shrink happened
8) Guess [6,2] level 1 at 1 out of 2
...
137) col 8 complete shrunken by group
145 327 698
839 654 127
672 918 543

496 185 372
218 473 956
753 296 481

367 542 819
984 761 235
521 839 764

Fine [steps:141, solution sum:1].
Run time: 168 milliseconds; steps: 141, solution sum: 1.
 Biggest level on this run(til): 10

Order please:
run
142) Forward guess [1,7] level 5 at 2 out of 2
144) shrinking 8 from [9,4] went wrong
...
287) col 7 complete shrunken by group
288) shrinking 6 from [2,4] went wrong
289) No more solution (solution sum is 1).
145 300 900
839 050 127
672 018 543

426 005 371
518 473 296
793 261 485

067 500 819
954 000 632
081 609 750

Invalid quiz [steps:289] - no more solution (solution sum is 1)
Run time: 188 milliseconds; steps: 289, solution sum: 1.
 Biggest level on this run(til): 6

Order please:

最大猜测级别也达到了 10。