PostgreSQL生成任意基數數獨-2
標籤
PostgreSQL , 數獨
背景
《PostgreSQL 生成任意基數數獨 – 1》 提供了一種方法，計算一個未完成的數獨矩陣每個畫素在XYB方向上還有多少個未填充的畫素。
通過XYB的值，進行各種排序，選出下一個要填充的畫素，進行隨機填充。
可以通過調整規則，實現不同的填充位置選擇，從而達到生成可解數獨的目的。
建立一個生成以N為基數的數獨的函式
函式輸入條件為N（基數），例如81個畫素的數獨，基數為3。(3*3)平方。
返回一個數獨（如果無解的話，raise出來）.
create or replace function gen_sudoku(
dim int -- 基數
) returns int[] as $$
declare
res int[]; -- 結果
dims int := dim^2; -- X,Y,BOX集合元素個數
vxyb xyb[]; -- 儲存每個畫素在XYB方向上未填充的元素個數
x int; -- 從xyb[]集合中，按指定方法選中一個畫素。 X座標
y int; -- 從xyb[]集合中，按指定方法選中一個畫素。 Y座標
vloops int := 2*dims; -- 計算N次(實際上就是隨機多少次能覆蓋到所有的值，值的取值空間為dims，通常來說執行DIMS次，能覆蓋到所有的隨機數)
vloop int :=0; -- 計算N次計數器
cnt int := 0; -- 統計當前數獨總共填充了多少個元素
rand int; -- 隨機值
begin
-- 初始化矩陣
select array( select (select array_agg(0) from generate_series(1,dims)) from generate_series(1,dims)) into res;
loop
-- 生成每個畫素X,Y,B方向的未知值個數
select comp_xyb(res, dim) into vxyb;
-- 選擇下一個要填充的畫素（根據未知值個數排行，從總未知值最多，按單軸最多的位置中隨機取一個位置）
select ax,ay into x,y from
unnest(vxyb) t
where
t.x+t.y+t.b <> 0
order by
(t.x+t.y+t.b) desc ,
greatest(t.x,t.y,t.b) desc
limit 1;
-- 如果全部為0,0,0，說明已解完，返回res。
if not found then
raise notice `計算有解，計算%次，結束。`, cnt;
return res;
end if;
-- 初始化以下計算迴圈次數
vloop := 0;
loop
-- 生成隨機值
rand := 1+(random()*(dims-1))::int;
-- 這輪迴圈無法生成並返回空
if vloop >= vloops then
raise notice `本畫素已迴圈%次，計算無解。已填充%個元素。無解數獨如下: %`, vloop, cnt, res;
-- return res;
return null;
end if;
-- 迴圈次數+1
vloop := vloop+1;
-- 橫向驗證
perform 1 where array(select res[x][generate_series(1,dims)]) && array[rand];
if found then
continue;
end if;
-- 縱向驗證
perform 1 where array(select res[generate_series(1,dims)][y]) && array[rand];
if found then
continue;
end if;
-- BOX驗證
perform 1 where
array(
select res[xx][yy] from
(select generate_series(((((x-1)/dim)::int)*dim)+1, ((((x-1)/dim)::int)*dim)+dim) xx) t1,
(select generate_series(((((y-1)/dim)::int)*dim)+1, ((((y-1)/dim)::int)*dim)+dim) yy) t2
) && array[rand];
if found then
continue;
end if;
-- 這個畫素值，通過驗證
res[x][y] := rand;
-- raise notice `res[%][%] %`, x, y, rand;
-- 通過驗證並跳出迴圈，找下一個需要填充的畫素
cnt := cnt+1;
exit;
end loop;
end loop;
end;
$$ language plpgsql strict volatile;
生成數獨測試
1、生成基數為2的數獨，16個畫素。
postgres=# select sudo from (select gen_sudoku(2) as sudo from generate_series(1,50)) t where sudo is not null limit 1;
NOTICE: 計算有解，計算16次，結束。
sudo
-------------------------------------------
{{3,4,2,1},{2,1,4,3},{1,2,3,4},{4,3,1,2}}
(1 row)
Time: 30.798 ms
2、生成基數為3的數獨，81個畫素。
但是非常的遺憾，填充個數50個左右，後面就沒法符合速度條件進行填充了。
postgres=# select sudo from (select gen_sudoku(3) as sudo from generate_series(1,10)) t where sudo is not null limit 1;
NOTICE: 本畫素已迴圈18次，計算無解。已填充45個元素。無解數獨如下: {{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: 本畫素已迴圈18次，計算無解。已填充46個元素。無解數獨如下: {{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: 本畫素已迴圈18次，計算無解。已填充49個元素。無解數獨如下: {{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: 本畫素已迴圈18次，計算無解。已填充45個元素。無解數獨如下: {{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: 本畫素已迴圈18次，計算無解。已填充45個元素。無解數獨如下: {{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: 本畫素已迴圈18次，計算無解。已填充50個元素。無解數獨如下: {{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: 本畫素已迴圈18次，計算無解。已填充46個元素。無解數獨如下: {{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: 本畫素已迴圈18次，計算無解。已填充29個元素。無解數獨如下: {{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: 本畫素已迴圈18次，計算無解。已填充46個元素。無解數獨如下: {{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: 本畫素已迴圈18次，計算無解。已填充56個元素。無解數獨如下: {{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)
調整選取填充畫素的方法
1、從各維度衝突最大的開始填充
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;
使用這種選擇畫素的方法，從填充畫素個數來看，很快就會發現無解，因為衝突最大化了。
NOTICE: 本畫素已迴圈18次，計算無解。已填充35個元素。無解數獨如下: {{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: 本畫素已迴圈18次，計算無解。已填充17個元素。無解數獨如下: {{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: 本畫素已迴圈18次，計算無解。已填充17個元素。無解數獨如下: {{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: 本畫素已迴圈18次，計算無解。已填充17個元素。無解數獨如下: {{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: 本畫素已迴圈18次，計算無解。已填充17個元素。無解數獨如下: {{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: 本畫素已迴圈18次，計算無解。已填充17個元素。無解數獨如下: {{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: 本畫素已迴圈18次，計算無解。已填充17個元素。無解數獨如下: {{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: 本畫素已迴圈18次，計算無解。已填充8個元素。無解數獨如下: {{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: 本畫素已迴圈18次，計算無解。已填充17個元素。無解數獨如下: {{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: 本畫素已迴圈18次，計算無解。已填充8個元素。無解數獨如下: {{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、你還可與根據其他的想法來選擇每次需要填充的畫素。
從BOX維度衝突最小，x,y維度衝突最小的畫素開始填充
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;
小結
暫時使用這幾種方法，經過少量的計算，無法生成有解的數獨。
1、選擇下一個要填充的畫素（根據未知值個數排行，從總未知值最多，按單軸最多的位置中隨機取一個位置）
2、從BOX維度衝突最小，x,y維度衝突最小的畫素開始填充
3、從各維度衝突最大的開始填充
隨機的方法生成數獨，效率比較低，維度越高，生成成功的概率越低。需要尋找更高效的生成數獨的方法。
參考
NP完全問題近似求解。
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