Finding Primes
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Comments posted here are about the content posted at http://www.sqlservercentral.com/columnists/kKellenberger/2782.asp
Aunt Kathi Data Platform MVP
Author of Expert T-SQL Window Functions
Simple-Talk Editor -
Finding primes is something I worked on about 6 months ago (for fun, of course
). Here's a link to the work...http://www.sqlteam.com/forums/topic.asp?TOPIC_ID=69646
It uses a sieve of Atkin approach, and in my tests back then, the function could find all primes below 100,000 in 0.5 seconds.
Ryan Randall
Solutions are easy. Understanding the problem, now, that's the hard part.
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Cool. Looks like there are a lot of ways to solve this puzzle.
Aunt Kathi Data Platform MVP
Author of Expert T-SQL Window Functions
Simple-Talk Editor -
By checking the square root outside of the select statement, you will save the processor a number of extra calculations. I modified it to be this:
SELECT @NextIntSqrt = SQRT(@NextInt)
IF NOT EXISTS (SELECT Prime FROM Primes WHERE @NextIntSqrt >= Prime AND @NextInt % Prime = 0)
Cuts down the calculation time significantly. I also adjusted the order of the filters in the WHERE clause.
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Good idea.
Aunt Kathi Data Platform MVP
Author of Expert T-SQL Window Functions
Simple-Talk Editor -
More on primes at:
http://blogs.technet.com/wardpond/archive/2006/09/23/458344.aspx
http://blogs.technet.com/wardpond/archive/2006/09/23/458580.aspx
http://sqlblog.com/blogs/hugo_kornelis/archive/2006/09/23/Prime_numbers.aspx
http://sqlblog.com/blogs/hugo_kornelis/archive/2006/09/23/Prime_numbers.aspx#comments
And of course, at my _old_ blog at:
http://robfarley.blogspot.com/2006/09/primes.html
http://robfarley.blogspot.com/2006/09/more-on-primes.html
All lots of fun...
Rob Farley
LobsterPot Solutions & Adelaide SQL Server User Group
Company: http://www.lobsterpot.com.au
Blog: http://blogs.lobsterpot.com.au -
Fun stuff.
I tried to do something similar in VB several years ago...the trick was that I was trying to solve for primes bigger than any of the data types. I ran out of time before I solved it...but it sure was a fun exercise!
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Kathi
In the book you mentioned, Chris has Aspbergers Syndrome (rather than being autistic). There are many similarities, but people with Aspbergers for the most part can lead relatively 'normal' lives. There are varying degrees of both of course, and it's sometimes hard to make a clear distinction where one ends and the other begins.
Regards,
MItch
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Hi Ryan,
What did you come up with for the largest prime number found and how many prime numbers? I wanna make sure I'm doing it right...
--Jeff Moden
RBAR is pronounced "ree-bar" and is a "Modenism" for Row-By-Agonizing-Row.
First step towards the paradigm shift of writing Set Based code:
________Stop thinking about what you want to do to a ROW... think, instead, of what you want to do to a COLUMN.Change is inevitable... Change for the better is not.
Helpful Links:
How to post code problems
How to Post Performance Problems
Create a Tally Function (fnTally) -
My suggestion, on our SQL Server 2000 installation it finds all primes less than 5000000 in 54 seconds. But of course it depends on the hardware...
set nocount on
go
select top 5000000 identity(int, 1, 1) as Number into #Numbers
from syscomments c1 cross join syscomments c2
go
alter table #Numbers add constraint PK_Numbers primary key clustered (Number)
go
create table #Primes(prime int primary key)
go
declare @time datetime
select @time = getdate()
declare @i int
set @i = 1
while @i*@i < 5000000
begin
insert into #Primes
select n.Number
from #Numbers n
where
n.Number < (@i+1)*(@i+1) and n.Number > @i*@i
and not exists
(
select * from #Primes p
where p.prime < @i + 1 and n.Number % p.prime = 0
 
set @i = @i + 1
end
--select * from #Primes
select datediff(ms, @time, getdate())
drop table #Primes
go
drop table #Numbers
go
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Hi Jeff,
Here's some code so you can run it yourself. I'll add some run results in the next post.
--Inputs
DECLARE @MaxNumber INT
SET @MaxNumber = 5000000
--Preparation
SET NOCOUNT ON
DECLARE @Time DATETIME
SELECT @Time = GETDATE()
--Preparation - Numbers table
DECLARE @SqrtMaxNumber INT
SET @SqrtMaxNumber = SQRT(@MaxNumber)
SET ROWCOUNT @SqrtMaxNumber
CREATE TABLE #Numbers (i BIGINT IDENTITY(1, 1) PRIMARY KEY CLUSTERED, j BIGINT, x bit)
INSERT INTO #Numbers (x) SELECT NULL FROM syscomments c1, syscomments c2, syscomments c3, syscomments c4
SET ROWCOUNT 0
UPDATE #Numbers SET j = i*i
PRINT 'Checkpoint A: ' + CAST(DATEDIFF(ms, @Time, GETDATE()) AS VARCHAR(10)) + ' ms'
--Preparation - Put candidate primes into a Primes table
-- (integers which have an odd number of representations by certain quadratic forms)
CREATE TABLE #Primes (i BIGINT PRIMARY KEY CLUSTERED)
INSERT #Primes
SELECT 2 UNION ALL SELECT 3 UNION ALL
SELECT k FROM (
SELECT k FROM (SELECT 4 * a.j + b.j AS k FROM #Numbers a, #Numbers b) c WHERE k <= @MaxNumber AND k % 12 IN (1, 5)
UNION ALL
SELECT k from (select 3 * a.j + b.j AS k FROM #Numbers a, #Numbers b) c WHERE k <= @MaxNumber AND k % 12 = 7
UNION ALL
SELECT k from (select 3 * a.j - b.j AS k FROM #Numbers a INNER JOIN #Numbers b ON a.i > b.i) c WHERE k <= @MaxNumber AND k % 12 = 11
) d GROUP BY k HAVING COUNT(*) IN (1, 3)
PRINT 'Checkpoint B: ' + CAST(DATEDIFF(ms, @Time, GETDATE()) AS VARCHAR(10)) + ' ms'
--Calculation - Eliminate composites by sieving
DECLARE @i BIGINT
SET @i = 5
WHILE @i * @i < @MaxNumber
BEGIN
DELETE #Primes WHERE i > @i and i % @i = 0
SELECT @i = min(i) FROM #Primes WHERE i > @i
PRINT @i
END
--Show results
PRINT 'Finish: ' + CAST(DATEDIFF(ms, @Time, GETDATE()) AS VARCHAR(10)) + ' ms'
SELECT COUNT(*) AS 'Number of primes', MAX(i) AS 'Max prime', AVG(i) AS 'Average of primes' FROM #Primes
SELECT * FROM #Primes ORDER BY i
--Tidy up
DROP TABLE #Primes
DROP TABLE #Numbers
Ryan Randall
Solutions are easy. Understanding the problem, now, that's the hard part.
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Here's a selection of results when I run the above code on a server here.
39 seconds to find all primes below 5 million, and 18 minutes to find all primes below 100 million.
DECLARE @Results TABLE (MaxNumber INT, CheckpointA INT, CheckpointB INT, Finish INT, NumberOfPrimes INT,
MaxPrime INT, AverageOfPrimes INT,
Seconds AS CAST(Finish AS DECIMAL) / 1000, Minutes AS CAST(Finish AS DECIMAL) / 60000,
Time AS CAST(Finish / 60000 AS VARCHAR(5)) + ':' + RIGHT('0' + CAST(Finish / 1000 % 60 AS VARCHAR(2)), 2))
INSERT @Results
SELECT 1000, 60, 93, 106, 168, 997, 453
UNION ALL SELECT 5000, 76, 123, 140, 669, 4999, 2314
UNION ALL SELECT 10000, 76, 156, 186, 1229, 9973, 4667
UNION ALL SELECT 50000, 76, 280, 360, 5133, 49999, 23575
UNION ALL SELECT 100000, 76, 466, 656, 9592, 99991, 47372
UNION ALL SELECT 500000, 76, 1250, 2576, 41538, 499979, 238678
UNION ALL SELECT 1000000, 93, 2203, 5516, 78498, 999983, 478361
UNION ALL SELECT 5000000, 110, 10156, 39280, 348513, 4999999, 2406213
UNION ALL SELECT 10000000, 93, 20030, 93403, 664579, 9999991, 4820081
UNION ALL SELECT 50000000, 123, 208930, 843080, 3001134, 49999991, 24197369
UNION ALL SELECT 100000000, 140, 466970, 1089933, 5761455, 99999989, 48461680
SELECT * FROM @Results
/* Results
MaxNumber CheckpointA CheckpointB Finish NumberOfPrimes MaxPrime AverageOfPrimes Seconds Minutes Time
--------- ----------- ----------- ------- -------------- -------- --------------- ----------- ---------- -----
1000 60 93 106 168 997 453 0.106000 0.001766 0:00
5000 76 123 140 669 4999 2314 0.140000 0.002333 0:00
10000 76 156 186 1229 9973 4667 0.186000 0.003100 0:00
50000 76 280 360 5133 49999 23575 0.360000 0.006000 0:00
100000 76 466 656 9592 99991 47372 0.656000 0.010933 0:00
500000 76 1250 2576 41538 499979 238678 2.576000 0.042933 0:02
1000000 93 2203 5516 78498 999983 478361 5.516000 0.091933 0:05
5000000 110 10156 39280 348513 4999999 2406213 39.280000 0.654666 0:39
10000000 93 20030 93403 664579 9999991 4820081 93.403000 1.556716 1:33
50000000 123 208930 843080 3001134 49999991 24197369 843.080000 14.051333 14:03
100000000 140 466970 1089933 5761455 99999989 48461680 1089.933000 18.165550 18:09
*/
Ryan Randall
Solutions are easy. Understanding the problem, now, that's the hard part.
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Good stuff Ryan!
Did you compare the runtime against any of the other code posted on this thread?
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I have tested my code against Ryan's code on SQL Server 2000 and it seems to be equally fast (with the same output).
If the Numbers table is preconstructed, I guess my code is faster (because I need a larger Numbers table). On the other hand, if I am forced to use bigints (as Ryan does), my code will
be a bit slower.
I have used the following in my test:
/*************** MY CODE ******************/
set nocount on
go
declare @time datetime
select @time = getdate()
DECLARE @MaxNumber INT
SET @MaxNumber = 5000000
SET ROWCOUNT @MaxNumber
select identity(int, 1, 1) as Number into #Numbers
from syscomments c1 cross join syscomments c2 cross join syscomments c3
SET ROWCOUNT 0
alter table #Numbers add constraint PK_Numbers primary key clustered (Number)
create table #Primes(prime int primary key)
declare @i int, @limit int
set @i = 1
while @i*@i < @MaxNumber
begin
set @limit = (@i+1)*(@i+1)
if (@i+1)*(@i+1) > @MaxNumber
set @limit = @MaxNumber + 1
insert into #Primes
select n.Number
from #Numbers n
where
n.Number < @limit and n.Number > @i*@i
and not exists
(select * from #Primes p where p.prime < @i + 1 and n.Number % p.prime = 0)
set @i = @i + 1
end
--select * from #Primes
PRINT 'Finish: ' + CAST(DATEDIFF(ms, @Time, GETDATE()) AS VARCHAR(10)) + ' ms'
SELECT COUNT(*) AS 'Number of primes', MAX(prime) AS 'Max prime' FROM #Primes
drop table #Primes
go
drop table #Numbers
go
/******** RYAN'S CODE************************/
--Inputs
DECLARE @MaxNumber INT
SET @MaxNumber = 5000000
--Preparation
SET NOCOUNT ON
DECLARE @Time DATETIME
SELECT @Time = GETDATE()
--Preparation - Numbers table
DECLARE @SqrtMaxNumber INT
SET @SqrtMaxNumber = SQRT(@MaxNumber)
SET ROWCOUNT @SqrtMaxNumber
CREATE TABLE #Numbers (i BIGINT IDENTITY(1, 1) PRIMARY KEY CLUSTERED, j BIGINT, x bit)
INSERT INTO #Numbers (x) SELECT NULL FROM syscomments c1 CROSS JOIN syscomments c2
SET ROWCOUNT 0
UPDATE #Numbers SET j = i*i
PRINT 'Checkpoint A: ' + CAST(DATEDIFF(ms, @Time, GETDATE()) AS VARCHAR(10)) + ' ms'
--Preparation - Put candidate primes into a Primes table
-- (integers which have an odd number of representations by certain quadratic forms)
CREATE TABLE #Primes (i BIGINT PRIMARY KEY CLUSTERED)
INSERT #Primes
SELECT 2 UNION ALL SELECT 3 UNION ALL
SELECT k FROM (
SELECT k FROM (SELECT 4 * a.j + b.j AS k FROM #Numbers a, #Numbers b) c WHERE k <= @MaxNumber AND k % 12 IN (1, 5)
UNION ALL
SELECT k from (select 3 * a.j + b.j AS k FROM #Numbers a, #Numbers b) c WHERE k <= @MaxNumber AND k % 12 = 7
UNION ALL
SELECT k from (select 3 * a.j - b.j AS k FROM #Numbers a INNER JOIN #Numbers b ON a.i > b.i) c WHERE k <= @MaxNumber AND k % 12 = 11
) d GROUP BY k HAVING COUNT(*) IN (1, 3)
PRINT 'Checkpoint B: ' + CAST(DATEDIFF(ms, @Time, GETDATE()) AS VARCHAR(10)) + ' ms'
--Calculation - Eliminate composites by sieving
DECLARE @i BIGINT
SET @i = 5
WHILE @i * @i < @MaxNumber
BEGIN
DELETE #Primes WHERE i > @i and i % @i = 0
SELECT @i = min(i) FROM #Primes WHERE i > @i
-- PRINT @i
END
--Show results
PRINT 'Finish: ' + CAST(DATEDIFF(ms, @Time, GETDATE()) AS VARCHAR(10)) + ' ms'
SELECT COUNT(*) AS 'Number of primes', MAX(i) AS 'Max prime' FROM #Primes
--SELECT * FROM #Primes ORDER BY i
--Tidy up
DROP TABLE #Primes
DROP TABLE #Numbers
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Ryan, I modified your script to use INT temp tables, instead of BIGINT, and got almost an 18% reduction in runtime. The following is the result of finding all primes below 100 million.
Using BIGINT for the data types doesn't seem to be needed unless you are going over 2 billion in your prime number search.
Test with INT temp tables Checkpoint A: 250 ms Checkpoint B: 230573 ms Finish: 652583 ms Number of primes Max prime Average of primes ---------------- ----------- -------------------- 5761455 99999989 48461680
Test with BIGINT temp tables Checkpoint A: 250 ms Checkpoint B: 280590 ms Finish: 791850 ms Number of primes Max prime Average of primes ---------------- -------------------- -------------------- 5761455 99999989 48461680
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