PostgreSQL 生成任意基数数独 - 2

标签

PostgreSQL , 数独

背景

《PostgreSQL 生成任意基数数独 - 1》 提供了一种方法，计算一个未完成的数独矩阵每个像素在XYB方向上还有多少个未填充的像素。

通过XYB的值，进行各种排序，选出下一个要填充的像素，进行随机填充。

可以通过调整规则，实现不同的填充位置选择，从而达到生成可解数独的目的。

创建一个生成以N为基数的数独的函数

函数输入条件为N（基数），例如81个像素的数独，基数为3。(3*3)平方。

返回一个数独（如果无解的话，raise出来）.

create or replace function gen_sudoku(    
  dim int  -- 基数    
) returns int[] as $$    
declare    
  res int[];           -- 结果  
  dims int := dim^2;   -- X,Y,BOX集合元素个数  
  
  vxyb xyb[];          -- 存储每个像素在XYB方向上未填充的元素个数  
  x int;               -- 从xyb[]集合中，按指定方法选中一个像素。  X坐标  
  y int;               -- 从xyb[]集合中，按指定方法选中一个像素。  Y坐标  
  
  vloops int := 2*dims;     -- 计算N次(实际上就是随机多少次能覆盖到所有的值，值的取值空间为dims，通常来说执行DIMS次，能覆盖到所有的随机数)  
  vloop int :=0;            -- 计算N次计数器  
  
  cnt int := 0;             -- 统计当前数独总共填充了多少个元素  
  
  rand int;                 -- 随机值  
begin    
  
  -- 初始化矩阵    
  select array( select (select array_agg(0) from generate_series(1,dims)) from generate_series(1,dims)) into res;    
      
  loop  
    -- 生成每个像素X,Y,B方向的未知值个数  
    select comp_xyb(res, dim) into vxyb;  
  
    -- 选择下一个要填充的像素（根据未知值个数排行，从总未知值最多，按单轴最多的位置中随机取一个位置）  
    select ax,ay into x,y from   
      unnest(vxyb) t   
    where   
      t.x+t.y+t.b <> 0   
    order by   
      (t.x+t.y+t.b) desc ,   
      greatest(t.x,t.y,t.b) desc   
    limit 1;    
  
    -- 如果全部为0,0,0，说明已解完，返回res。  
    if not found then  
      raise notice '计算有解，计算%次，结束。', cnt;  
      return res;  
    end if;  
  
    -- 初始化以下计算循环次数  
    vloop := 0;  
    loop    
      -- 生成随机值    
      rand := 1+(random()*(dims-1))::int;    
  
      -- 这轮循环无法生成并返回空   
      if vloop >= vloops then    
        raise notice '本像素已循环%次，计算无解。已填充%个元素。无解数独如下: %', vloop, cnt, res;  
	-- return res;  
	return null;  
      end if;    
  
      -- 循环次数+1  
      vloop := vloop+1;    
  
      -- 横向验证    
      perform 1 where array(select res[x][generate_series(1,dims)]) && array[rand];    
      if found then    
        continue;    
      end if;    
          
      -- 纵向验证    
      perform 1 where array(select res[generate_series(1,dims)][y]) && array[rand];    
      if found then    
        continue;    
      end if;    
          
      -- BOX验证    
      perform 1 where   
        array(  
          select res[xx][yy] from   
            (select generate_series(((((x-1)/dim)::int)*dim)+1, ((((x-1)/dim)::int)*dim)+dim) xx) t1,   
            (select generate_series(((((y-1)/dim)::int)*dim)+1, ((((y-1)/dim)::int)*dim)+dim) yy) t2  
        ) && array[rand];    
      if found then    
        continue;    
      end if;    
          
      -- 这个像素值，通过验证    
      res[x][y] := rand;    
      -- raise notice 'res[%][%] %', x, y, rand;    
        
      -- 通过验证并跳出循环，找下一个需要填充的像素  
      cnt := cnt+1;  
      exit;    
    end loop;    
  
  end loop;    
end;  
$$ language plpgsql strict volatile;   

生成数独测试

1、生成基数为2的数独，16个像素。

postgres=# select sudo from (select gen_sudoku(2) as sudo from generate_series(1,50)) t where sudo is not null limit 1;  
NOTICE:  计算有解，计算16次，结束。  
                   sudo                      
-------------------------------------------  
 {{3,4,2,1},{2,1,4,3},{1,2,3,4},{4,3,1,2}}  
(1 row)  
  
Time: 30.798 ms  

2、生成基数为3的数独，81个像素。

但是非常的遗憾，填充个数50个左右，后面就没法符合速度条件进行填充了。

postgres=# select sudo from (select gen_sudoku(3) as sudo from generate_series(1,10)) t where sudo is not null limit 1;  
NOTICE:  本像素已循环18次，计算无解。已填充45个元素。无解数独如下: {{5,3,6,2,0,0,0,8,0},{0,9,0,5,8,4,6,0,0},{7,0,0,6,0,0,5,2,4},{6,4,5,3,0,0,0,9,0},{0,8,0,9,7,5,0,4,0},{1,0,0,0,2,0,3,7,8},{0,7,4,0,5,6,0,0,3},{0,0,3,0,1,2,8,0,5},{9,0,8,0,0,3,4,0,6}}  
NOTICE:  本像素已循环18次，计算无解。已填充46个元素。无解数独如下: {{8,3,9,2,4,0,0,5,0},{0,2,0,6,3,9,8,0,0},{1,0,0,5,0,0,4,7,6},{7,8,2,3,0,0,0,1,0},{0,9,0,4,2,6,0,8,0},{3,0,0,0,8,0,5,4,7},{0,4,7,0,1,5,0,0,8},{0,0,6,0,7,4,2,0,3},{5,0,8,0,0,3,7,0,1}}  
NOTICE:  本像素已循环18次，计算无解。已填充49个元素。无解数独如下: {{8,4,6,1,9,0,0,2,0},{9,2,0,7,5,6,8,0,0},{3,0,0,4,0,0,1,5,6},{5,7,3,2,0,1,0,4,0},{0,9,4,3,8,7,0,1,0},{6,0,0,0,4,0,3,9,8},{0,5,8,0,6,4,0,0,7},{0,0,1,0,3,2,5,0,4},{4,0,2,0,0,5,6,0,1}}  
NOTICE:  本像素已循环18次，计算无解。已填充45个元素。无解数独如下: {{6,8,2,3,0,0,0,4,0},{0,5,0,7,2,1,9,0,0},{7,0,0,9,0,0,1,5,3},{1,3,4,6,0,0,0,9,0},{0,9,0,8,1,3,0,6,0},{8,0,0,0,4,0,3,2,7},{0,2,7,0,5,6,0,0,4},{0,0,9,0,8,2,7,0,5},{4,0,3,0,0,7,8,0,2}}  
NOTICE:  本像素已循环18次，计算无解。已填充45个元素。无解数独如下: {{1,6,7,9,0,0,0,5,0},{0,2,0,4,3,8,7,0,0},{4,0,0,5,0,0,8,6,3},{3,5,8,7,0,0,0,2,0},{0,7,0,1,2,6,0,3,0},{6,0,0,0,9,0,5,4,1},{0,8,1,0,5,2,0,0,4},{0,0,5,0,7,3,6,0,2},{2,0,4,0,0,1,9,0,8}}  
NOTICE:  本像素已循环18次，计算无解。已填充50个元素。无解数独如下: {{2,3,5,6,9,0,0,7,0},{6,7,0,2,8,4,1,0,0},{4,0,0,7,0,0,9,8,5},{1,4,7,3,0,8,0,5,0},{0,9,3,5,7,6,0,4,0},{5,0,0,0,4,0,2,6,3},{0,2,8,4,3,7,0,0,1},{0,0,1,0,6,5,3,0,2},{3,0,6,0,0,2,8,0,7}}  
NOTICE:  本像素已循环18次，计算无解。已填充46个元素。无解数独如下: {{2,6,7,9,5,0,0,8,0},{0,5,0,2,4,1,3,0,0},{1,0,0,6,0,0,4,5,7},{8,7,9,4,0,0,0,3,0},{0,1,0,3,2,6,0,4,0},{3,0,0,0,8,0,6,2,9},{0,3,5,0,1,7,0,0,2},{0,0,8,0,6,3,9,0,5},{4,0,6,0,0,9,8,0,1}}  
NOTICE:  本像素已循环18次，计算无解。已填充29个元素。无解数独如下: {{4,3,2,5,0,0,0,0,0},{0,0,0,6,8,7,0,0,0},{0,0,0,0,0,0,1,9,4},{6,5,0,4,0,0,0,7,0},{0,1,0,8,3,0,0,0,0},{2,0,0,0,0,0,3,5,0},{0,0,3,0,0,8,0,0,5},{0,0,4,0,5,9,0,0,0},{9,0,0,0,0,0,2,0,3}}  
NOTICE:  本像素已循环18次，计算无解。已填充46个元素。无解数独如下: {{4,8,3,5,9,0,0,1,0},{0,5,0,7,4,1,2,0,0},{1,0,0,2,0,0,7,3,8},{9,6,8,3,0,0,0,4,0},{0,2,0,4,1,5,0,7,0},{7,0,0,0,8,0,3,5,6},{0,4,9,0,5,3,0,0,7},{0,0,2,0,7,6,5,0,3},{6,0,7,0,0,4,8,0,2}}  
NOTICE:  本像素已循环18次，计算无解。已填充56个元素。无解数独如下: {{5,6,3,8,4,0,7,2,0},{4,2,0,9,7,1,5,0,0},{8,9,0,2,0,0,1,4,3},{3,7,5,4,0,9,0,6,2},{0,4,9,1,2,8,0,5,0},{2,0,6,0,5,0,9,3,8},{0,3,2,5,1,7,0,0,6},{0,5,8,0,6,4,3,0,9},{6,0,1,0,0,3,4,8,7}}  
 sudo   
------  
(0 rows)  
  
Time: 1037.195 ms (00:01.037)  

调整选取填充像素的方法

1、从各维度冲突最大的开始填充

    select ax,ay into x,y from   
      unnest(vxyb) t   
    where   
      t.x+t.y+t.b <> 0   
    order by   
      (t.x+t.y+t.b) ,   
      greatest(t.x,t.y,t.b)    
    limit 1;    

使用这种选择像素的方法，从填充像素个数来看，很快就会发现无解，因为冲突最大化了。

NOTICE:  本像素已循环18次，计算无解。已填充35个元素。无解数独如下: {{5,7,2,4,6,3,1,8,9},{8,1,4,5,7,9,2,3,6},{6,9,3,2,8,1,5,4,7},{9,4,6,0,0,0,0,0,0},{3,5,7,0,0,0,0,0,0},{1,8,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0}}  
NOTICE:  本像素已循环18次，计算无解。已填充17个元素。无解数独如下: {{1,2,6,7,9,8,0,0,0},{3,8,4,1,5,2,0,0,0},{5,7,9,4,6,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0}}  
NOTICE:  本像素已循环18次，计算无解。已填充17个元素。无解数独如下: {{1,2,5,9,3,4,0,0,0},{7,9,6,2,1,8,0,0,0},{3,8,4,6,5,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0}}  
NOTICE:  本像素已循环18次，计算无解。已填充17个元素。无解数独如下: {{7,5,6,8,4,9,0,0,0},{4,3,8,5,2,6,0,0,0},{2,1,9,7,3,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0}}  
NOTICE:  本像素已循环18次，计算无解。已填充17个元素。无解数独如下: {{7,2,6,9,5,3,0,0,0},{3,4,1,6,8,2,0,0,0},{5,8,9,7,4,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0}}  
NOTICE:  本像素已循环18次，计算无解。已填充17个元素。无解数独如下: {{8,3,7,6,4,2,0,0,0},{1,6,5,8,9,3,0,0,0},{4,9,2,7,5,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0}}  
NOTICE:  本像素已循环18次，计算无解。已填充17个元素。无解数独如下: {{8,7,4,9,6,5,0,0,0},{3,5,2,7,4,8,0,0,0},{1,6,9,2,3,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0}}  
NOTICE:  本像素已循环18次，计算无解。已填充8个元素。无解数独如下: {{7,4,5,0,0,0,0,0,0},{9,3,2,0,0,0,0,0,0},{8,6,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0}}  
NOTICE:  本像素已循环18次，计算无解。已填充17个元素。无解数独如下: {{3,6,7,5,1,8,0,0,0},{5,1,2,9,6,3,0,0,0},{4,8,9,7,2,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0}}  
NOTICE:  本像素已循环18次，计算无解。已填充8个元素。无解数独如下: {{3,2,8,0,0,0,0,0,0},{7,4,9,0,0,0,0,0,0},{6,5,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0}}  
 sudo   
------  
(0 rows)  
  
Time: 486.963 ms  

2、你还可与根据其他的想法来选择每次需要填充的像素。

从BOX维度冲突最小，x,y维度冲突最小的像素开始填充

    select ax,ay into x,y from   
      unnest(vxyb) t   
    where   
      t.x+t.y+t.b <> 0   
    order by   
      t.b desc ,   
      greatest(t.x,t.y)  desc  
    limit 1;    

小结

暂时使用这几种方法，经过少量的计算，无法生成有解的数独。

1、选择下一个要填充的像素（根据未知值个数排行，从总未知值最多，按单轴最多的位置中随机取一个位置）

2、从BOX维度冲突最小，x,y维度冲突最小的像素开始填充

3、从各维度冲突最大的开始填充

随机的方法生成数独，效率比较低，维度越高，生成成功的概率越低。需要寻找更高效的生成数独的方法。

参考

NP完全问题近似求解。

《PostgreSQL 生成任意基数数独 - 1》

《PostgreSQL 生成任意基数数独 - 2》

《PostgreSQL 生成任意基数数独 - 3》