这是列出质数的好算法吗？ [英] Is this a good algorithm for listing prime numbers?

问题描述

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;

这有点像是Eratosthenes筛网筛分法。

It's kind of a one pass Sieve of Eratosthenes approach.

我已经看过循环，存储过程之类的东西，以及伪代码和其他语言的实现，但是在我看来，这种基于素数的简单，基于集合的方法就足够了

I've seen loops, stored procedures and the like, as well as pseudo-code and other language implementations, but it seems to me that this simple, set-based approach stemming from the definition of prime numbers should suffice.

请记住，我目前不关心性能，内存消耗或优化，也没有对它进行较大的测试。我只想发布算法，让人们确认（或挑战）从列表中排除复合数字就足够了。

Please bear in mind I'm not concerned with performance or memory consumption or optimizations at this time, and I have not tested it with larger numbers. I just want to publish the algorithm and have people confirm (or challenge) that excluding composite numbers from the list is enough.

推荐答案

递归CTE（rCTE）很少是性能最好的解决方案。以下是使用计数表的方法，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 /

Recursive CTEs (rCTE) are very rarely the best performing solution. Below is an approach that uses a tally table, it's a slightly tweaked version of the approach that Hugo Kornelis posted here: https://sqlserverfast.com/blog/hugo/2006/09/the-prime-number-challenge-great-waste-of-time/

让我们将理货表解决方案与rCTE解决方案进行比较：

Let's compare the tally table solution to the rCTE solution:

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;

，结果为：

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.

如您所见，针对10,000的计数表方法快了5倍，并且也不会产生任何读取（rCTE会产生一吨！）

As you can see, the tally table approach against 10,000 was 5 times faster and also doesn't produce any reads (the rCTE produces a ton!)

如果您真的在使用质数，则绝对最快的方法是将它们存储在表中，这样就不会每次需要质数时都需要计算它们。

If you are really working with prime numbers the absolute fastest approach would be to store them in a table so you don't need to calculate them each time you need prime numbers.

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