Question

DECLARE @c int = 1000;
DECLARE @numbers  TABLE (n int NOT NULL PRIMARY KEY);
DECLARE @products TABLE (p int NOT NULL PRIMARY KEY);
DECLARE @primes   TABLE (p int NOT NULL PRIMARY KEY);

-- The 'composite exclusion' approach

-- 1. list all n = 2, 3, 4, ... c
WITH numbers AS
(
    SELECT  2 AS n
    UNION ALL
    SELECT n + 1 FROM numbers
    WHERE   n <= @c - 1
)
INSERT INTO @numbers SELECT n FROM numbers OPTION(MAXRECURSION 0);

-- 2. find all products n x n <= c
WITH products AS
(
    SELECT  DISTINCT m.n * n.n AS p
    FROM    @numbers m LEFT OUTER JOIN
            @numbers n ON 1 = 1
    WHERE   m.n * n.n <= @c
)
INSERT INTO @products SELECT p FROM products;

-- 3. numbers with no matching products are not composite, i.e, they're prime numbers.
INSERT INTO @primes
SELECT n.n FROM @numbers n LEFT JOIN @products p ON n.n = p.p WHERE p.p IS NULL;

这是Eratosthenes方法的一次筛选。

我已经看过循环，存储过程等，以及伪代码和其他语言实现，但在我看来，源于素数定义的这种简单的，基于集合的方法应该足够了。 / p>

请记住，我现在不关心性能或内存消耗或优化，我还没有用更大的数字进行测试。我只是想发布算法并让人们确认（或挑战）从列表中排除复合数字就足够了。

Answer 1

递归CTE（rCTE）很少是性能最佳的解决方案。下面是一个使用计数表的方法，它是Hugo Kornelis在此处发布的方法的略微调整版本：https://sqlserverfast.com/blog/hugo/2006/09/the-prime-number-challenge-great-waste-of-time/

让我们将计数表解决方案与rCTE解决方案进行比较：

SET STATISTICS TIME ON;

PRINT 'tally table approach'+char(13)+char(10)+replicate('-',50);
DECLARE @primes   TABLE (p int NOT NULL PRIMARY KEY);
DECLARE @limit bigint = 10000;

WITH E(x) AS (SELECT * FROM (VALUES (1),(1),(1),(1),(1),(1),(1),(1),(1),(1)) t(x)),
iTally(N) AS (SELECT TOP(@limit) ROW_NUMBER() OVER (ORDER BY (SELECT 1)) FROM E a, E b, E c, E d, E f)
INSERT @primes
SELECT      n1.N
FROM        itally AS n1
WHERE       n1.N > 1
AND         n1.N < @Limit
AND NOT EXISTS
 (SELECT    *
  FROM      itally AS n2
  WHERE     n2.N < @limit
  AND       n2.N BETWEEN 2 AND n1.N-1
  AND       n1.n % n2.N = 0)
--ORDER BY N
GO

PRINT 'rCTE approach'+char(13)+char(10)+replicate('-',50);
DECLARE @c int = 10000;
DECLARE @numbers  TABLE (n int NOT NULL PRIMARY KEY);
DECLARE @products TABLE (p int NOT NULL PRIMARY KEY);
DECLARE @primes   TABLE (p int NOT NULL PRIMARY KEY);

WITH numbers AS
(
    SELECT  2 AS n
    UNION ALL
    SELECT n + 1 FROM numbers
    WHERE   n <= @c - 1
)
INSERT INTO @numbers SELECT n FROM numbers OPTION(MAXRECURSION 0);

-- 2. find all products n x n <= c
WITH products AS
(
    SELECT  DISTINCT m.n * n.n AS p
    FROM    @numbers m LEFT OUTER JOIN
            @numbers n ON 1 = 1
    WHERE   m.n * n.n <= @c
)
INSERT INTO @products SELECT p FROM products;

-- 3. numbers with no matching products are not composite, i.e, they're prime numbers.
INSERT INTO @primes
SELECT n.n FROM @numbers n LEFT JOIN @products p ON n.n = p.p WHERE p.p IS NULL;

SET STATISTICS TIME OFF;

和结果：

tally table approach
--------------------------------------------------

 SQL Server Execution Times:
   CPU time = 3042 ms,  elapsed time = 3241 ms.
SQL Server parse and compile time: 
   CPU time = 0 ms, elapsed time = 10 ms.

rCTE approach
--------------------------------------------------

 SQL Server Execution Times:
   CPU time = 14976 ms,  elapsed time = 15757 ms.

正如你所看到的，对10,000的计数表方法快了5倍，也没有产生任何读数（rCTE产生了吨！）

如果你真的使用素数，绝对最快的方法是将它们存储在一个表中，这样你就不需要在每次需要素数时计算它们。

这是列出素数的好算法吗？

1 个答案: