Question

美好的一天，我想为Set Cover Problem实现T-SQL查询，但无法找到有关如何在SQL中执行此操作的任何提示。

在我的情况下，我的表只有两列（IDnumber和Mut），我希望找到IDNumber的最小数量来获取Mut之一。我真的想要为每个IDnumbers获得三个Mut，但我认为我最好从一个开始，因为这可能更容易。

DECLARE @myTable TABLE (
    IDnumber int,
    Mut varchar(1))

INSERT @myTable VALUES 
 (1,'C'), (1,'N'), (1,'Z'), (1,'M'), (1,'E'), (2,'E'), (3,'B'), (3,'N'), (3,'D'), (3,'K'), 
(3,'W'), (4,'O'), (4,'G'), (4,'N'), (4,'B'), (4,'U'), (4,'C'), (5,'Q'), (5,'H'), (6,'K'), 
(6,'Y'), (6,'M'), (6,'A'), (6,'O'), (6,'U'), (6,'J'), (7,'H'), (7,'U'), (7,'M'), (7,'L'), 
(8,'B'), (8,'K'), (8,'P'), (9,'Y'), (9,'K'), (10,'Z'), (11,'R'), (12,'X'), (12,'R'), 
(12,'O'), (12,'Z'), (4,'C'), (1,'Z'), (4,'S'), (6,'E'), (5,'G'), (4,'C'), (4,'S'), (4,'H'), 
(6,'D'), (7,'W'), (3,'U'), (6,'N'), (7,'Y'), (6,'N'), (6,'F'), (4,'C'), (4,'I'), (7,'P'), 
(10,'H'), (10,'Z'), (10,'S'), (7,'Z'), (6,'B'), (7,'Z'), (8,'X'), (8,'J'), (8,'P'), (10,'K'), 
(8,'K'), (12,'P'), (8,'W'), (10,'M'), (12,'F'), (9,'T'), (9,'D'), (14,'Y'), (12,'P'), 
(14,'J'), (13,'D'), (15,'H'), (12,'J'), (6,'H'), (2,'Z'), (8,'G'), (10,'Q'), (6,'D'), 
(5,'X'), (9,'T'), (6,'W'), (6,'K'), (10,'W'), (7,'J'), (11,'W'), (12,'V'), (9,'F'), (7,'F'), 
(4,'M'), (5,'K'), (12,'G'), (12,'T'), (15,'T'), (13,'W'), (7,'J'), (9,'T'), (10,'U'), (9,'S'), 
(10,'L'), (10,'J'), (10,'H'), (11,'H'), (12,'S'), (12,'A'), (14,'L'), (13,'K'), (13,'D'), 
(4,'M'), (3,'N'), (4,'F'), (7,'M'), (7,'V'), (5,'R'), (4,'K'), (5,'F'), (7,'G'), (8,'M'), 
(4,'X'), (7,'F'), (9,'S'), (7,'N'), (6,'W'), (6,'W'), (5,'S'), (9,'Z'), (10,'I'), (11,'Y'), 
(11,'D'), (9,'X'), (7,'G'), (9,'S'), (9,'H'), (9,'T'), (8,'J'), (10,'U'), (9,'F'), (9,'S'), 
(7,'D'), (14,'R'), (10,'F'), (7,'E'), (15,'M'), (12,'F'), (5,'C'), (8,'E'), (16,'G'), (11,'V'),
(10,'I'), (12,'I'), (11,'Y'), (12,'I'), (14,'J'), (15,'D'), (19,'J'), (16,'B'), (12,'G'), 
(9,'J'), (18,'J'), (18,'C'), (16,'Q'), (18,'P'), (13,'F'), (19,'T'), (15,'J'), (15,'R'), 
(15,'Q'), (15,'O'), (11,'A'), (24,'B'), (19,'S'), (22,'I'), (15,'X'), (20,'T'), (15,'E'), 
(9,'V'), (8,'H'), (16,'N'), (17,'H')
--  Since the above list was generated by a bunch of random numbers/letters I need to
-- delete the duplicates

;WITH cte AS (
  SELECT IDnumber, mut, 
     row_number() OVER(PARTITION BY IDNumber, mut ORDER BY IDNumber) AS [rn]
  FROM @myTable
)
DELETE cte WHERE [rn] > 1


SELECT *
FROM ( SELECT IDnumber, Mut FROM @myTable) AS S
PIVOT
(COUNT(Mut) FOR mut IN ([A],[B],[C],[D],[E],[F],[G],[H],[I],[J],[K],[L],[M],[N],[O],[P],
[Q],[R],[S],[T],[U],[V],[W],[X],[Y],[Z])) AS pvt

所以你可以从数据透视表中看到最小IDnumbers将是3,5,7和12。

如何实施算法？在我看来，我可以找到所有组合（2 ^ 6），然后确定哪些组具有所有Muts。具有最小IDnumber数量的集合则是最小集合。

这种蛮力可能有效，但效率可能非常低。我的真实案例并不是很大，我有43个唯一Muts（不是示例中的九个）和~2000 IDnumbers，但我认为这需要一些时间才能运行，因为2 ^ 2000是相当大...

谢谢！

Answer 1

我想要一个更大的数据集进行测试，但这至少与给定数据集的结果相符...

DECLARE @myTable TABLE (
        IDnumber INT,
    Mut VARCHAR(1))

DECLARE @results TABLE (
    IDnumber INT)

INSERT @myTable VALUES 
    (1,'A'),(1,'B'),(1,'C'),(1,'D'),
    (2,'A'),(2,'C'),(2,'D'),
    (3,'A'),(3,'B'),(3,'F'),(3,'Z'),
    (4,'A'),(4,'B'),(4,'E'),(4,'F'),
    (5,'Y'),
    (6,'X'),(6,'Z')

DECLARE @IDnumber INT

WHILE EXISTS (SELECT 1 FROM @myTable)
BEGIN

    WITH MutRank AS
    (
        -- Find how many IDNumbers are associated with each mut
        SELECT Mut,
            COUNT(DISTINCT IDnumber) AS IDnumberCnt
        FROM @myTable
        GROUP BY Mut
    ), MutRarity AS
    (
        -- Translate the IDNumberCnt into a weighted rarity so dupes at  
        -- a IDNumberCnt level reduce their rarity compared to the next level 
        SELECT Mut,
            RANK() OVER (ORDER BY IDnumberCnt DESC) AS MutRarity
        FROM MutRank
    )
    -- Grab the IDNumber that is associated to the most "rare" muts
    SELECT @IDnumber = n.IDnumber
    FROM (SELECT TOP 1 m.IDnumber,
            SUM(MutRarity) AS IDNumberValue
        FROM @myTable m
        JOIN MutRarity mr
            ON m.Mut = mr.Mut
        GROUP BY m.IDnumber
        ORDER BY IDNumberValue DESC, IDnumber) n

    -- Save the number in our results
    INSERT @results (IDnumber)
    SELECT @IDnumber

    -- Purge all records for the "covered" muts and the selected IDNumber
    DELETE m2
    FROM @myTable m1
    JOIN @myTable m2
        ON m1.Mut = m2.Mut
        AND m1.IDnumber = @IDnumber
END

SELECT *
FROM @results
ORDER BY IDnumber

Answer 2

这是一种为每个Mut计算IDNumber值的位掩码的方法，然后使用递归CTE生成完成该集合的所有可能组合。代码输出给出最终结果的所有可能的IdNumber组合，不包括组合中一个或多个IdNumber冗余的组合（例如样本数据集中的1,2,3,4,5,6）。 / p>

要将其转换为适用于您的实际数据，主要区别在于您几乎肯定需要找到另一种机制来将位值分配给Mut值。（我可以在这里作弊，通过操纵每个Mut字母 - POWER(2,ASCII(Mut) - 64)）的十进制ASCII代码来计算位值。

DECLARE @myTable TABLE (
    IDnumber int,
    Mut varchar(1))

INSERT @myTable VALUES 
    (1,'A'),(1,'B'),(1,'C'),(1,'D'),
    (2,'A'),(2,'C'),(2,'D'),
    (3,'A'),(3,'B'),(3,'F'),(3,'Z'),
    (4,'A'),(4,'B'),(4,'E'),(4,'F'),
    (5,'Y'),
    (6,'X'),(6,'Z')

-- calculate the target bitmask
DECLARE @target bigint = (  SELECT SUM(POWER(2,ASCII(Mut) - 64))
                            FROM    (SELECT DISTINCT Mut FROM @myTable) AS x
                         )

;WITH baseCTE
AS
(
    --calculate a starting bitmask for each ID
    SELECT IDnumber, sum(mutbit) AS bitmask 
    FROM    (SELECT DISTINCT IDnumber, CAST(POWER(2,ASCII(Mut) - 64) AS bigint) AS mutbit FROM @myTable) AS x
    GROUP BY IDnumber
)
,recCTE
AS
(
    SELECT IDnumber, bitmask, CAST(IdNumber AS varchar(max)) AS trail, 1 AS lev
    FROM baseCTE
    UNION ALL
    SELECT b.IDnumber, b.bitmask | r.bitmask, CAST(CONCAT(r.trail,',',b.IDnumber) AS varchar(max)), r.lev + 1
    FROM recCTE AS r
    JOIN baseCTE AS b
    ON b.IDnumber > r.IDnumber
    WHERE b.bitmask | r.bitmask <> r.bitmask --ignore combinations which don't add any new Mut values
)
SELECT trail, lev 
FROM recCTE
WHERE bitmask = @target
ORDER BY lev

NB。 SQL Server按位运算符仅在一个或其他值为整数类型的情况下工作，因此此解决方案仅限于64个不同的Mut值（掩码为bigint） - 使其超出工作范围那个，你必须使用自定义的按位比较器（可能使用CLR）。但是，由于问题提到您有43 Mut个值，因此它现在可能对您有用。

查询集合封面

2 个答案: