SQL, Auxiliary table of numbers - sql

Question

For certain types of sql queries, an auxiliary table of numbers can be very useful. It may be created as a table with as many rows as you need for a particular task or as a user defined function that returns the number of rows required in each query.
What is the optimal way to create such a function?

Answer 1

Heh... sorry I'm so late responding to an old post. And, yeah, I had to respond because the most popular answer (at the time, the Recursive CTE answer with the link to 14 different methods) on this thread is, ummm... performance challenged at best.
First, the article with the 14 different solutions is fine for seeing the different methods of creating a Numbers/Tally table on the fly but as pointed out in the article and in the cited thread, there's a very important quote...
"suggestions regarding efficiency and
performance are often subjective.
Regardless of how a query is being
used, the physical implementation
determines the efficiency of a query.
Therefore, rather than relying on
biased guidelines, it is imperative
that you test the query and determine
which one performs better."
Ironically, the article itself contains many subjective statements and "biased guidelines" such as "a recursive CTE can generate a number listing pretty efficiently" and "This is an efficient method of using WHILE loop from a newsgroup posting by Itzik Ben-Gen" (which I'm sure he posted just for comparison purposes). C'mon folks... Just mentioning Itzik's good name may lead some poor slob into actually using that horrible method. The author should practice what (s)he preaches and should do a little performance testing before making such ridiculously incorrect statements especially in the face of any scalablility.
With the thought of actually doing some testing before making any subjective claims about what any code does or what someone "likes", here's some code you can do your own testing with. Setup profiler for the SPID you're running the test from and check it out for yourself... just do a "Search'n'Replace" of the number 1000000 for your "favorite" number and see...
--===== Test for 1000000 rows ==================================
GO
--===== Traditional RECURSIVE CTE method
WITH Tally (N) AS
(
SELECT 1 UNION ALL
SELECT 1 + N FROM Tally WHERE N < 1000000
)
SELECT N
INTO #Tally1
FROM Tally
OPTION (MAXRECURSION 0);
GO
--===== Traditional WHILE LOOP method
CREATE TABLE #Tally2 (N INT);
SET NOCOUNT ON;
DECLARE #Index INT;
SET #Index = 1;
WHILE #Index <= 1000000
BEGIN
INSERT #Tally2 (N)
VALUES (#Index);
SET #Index = #Index + 1;
END;
GO
--===== Traditional CROSS JOIN table method
SELECT TOP (1000000)
ROW_NUMBER() OVER (ORDER BY (SELECT 1)) AS N
INTO #Tally3
FROM Master.sys.All_Columns ac1
CROSS JOIN Master.sys.ALL_Columns ac2;
GO
--===== Itzik's CROSS JOINED CTE method
WITH E00(N) AS (SELECT 1 UNION ALL SELECT 1),
E02(N) AS (SELECT 1 FROM E00 a, E00 b),
E04(N) AS (SELECT 1 FROM E02 a, E02 b),
E08(N) AS (SELECT 1 FROM E04 a, E04 b),
E16(N) AS (SELECT 1 FROM E08 a, E08 b),
E32(N) AS (SELECT 1 FROM E16 a, E16 b),
cteTally(N) AS (SELECT ROW_NUMBER() OVER (ORDER BY N) FROM E32)
SELECT N
INTO #Tally4
FROM cteTally
WHERE N <= 1000000;
GO
--===== Housekeeping
DROP TABLE #Tally1, #Tally2, #Tally3, #Tally4;
GO
While we're at it, here's the numbers I get from SQL Profiler for the values of 100, 1000, 10000, 100000, and 1000000...
SPID TextData Dur(ms) CPU Reads Writes
---- ---------------------------------------- ------- ----- ------- ------
51 --===== Test for 100 rows ============== 8 0 0 0
51 --===== Traditional RECURSIVE CTE method 16 0 868 0
51 --===== Traditional WHILE LOOP method CR 73 16 175 2
51 --===== Traditional CROSS JOIN table met 11 0 80 0
51 --===== Itzik's CROSS JOINED CTE method 6 0 63 0
51 --===== Housekeeping DROP TABLE #Tally 35 31 401 0
51 --===== Test for 1000 rows ============= 0 0 0 0
51 --===== Traditional RECURSIVE CTE method 47 47 8074 0
51 --===== Traditional WHILE LOOP method CR 80 78 1085 0
51 --===== Traditional CROSS JOIN table met 5 0 98 0
51 --===== Itzik's CROSS JOINED CTE method 2 0 83 0
51 --===== Housekeeping DROP TABLE #Tally 6 15 426 0
51 --===== Test for 10000 rows ============ 0 0 0 0
51 --===== Traditional RECURSIVE CTE method 434 344 80230 10
51 --===== Traditional WHILE LOOP method CR 671 563 10240 9
51 --===== Traditional CROSS JOIN table met 25 31 302 15
51 --===== Itzik's CROSS JOINED CTE method 24 0 192 15
51 --===== Housekeeping DROP TABLE #Tally 7 15 531 0
51 --===== Test for 100000 rows =========== 0 0 0 0
51 --===== Traditional RECURSIVE CTE method 4143 3813 800260 154
51 --===== Traditional WHILE LOOP method CR 5820 5547 101380 161
51 --===== Traditional CROSS JOIN table met 160 140 479 211
51 --===== Itzik's CROSS JOINED CTE method 153 141 276 204
51 --===== Housekeeping DROP TABLE #Tally 10 15 761 0
51 --===== Test for 1000000 rows ========== 0 0 0 0
51 --===== Traditional RECURSIVE CTE method 41349 37437 8001048 1601
51 --===== Traditional WHILE LOOP method CR 59138 56141 1012785 1682
51 --===== Traditional CROSS JOIN table met 1224 1219 2429 2101
51 --===== Itzik's CROSS JOINED CTE method 1448 1328 1217 2095
51 --===== Housekeeping DROP TABLE #Tally 8 0 415 0
As you can see, the Recursive CTE method is the second worst only to the While Loop for Duration and CPU and has 8 times the memory pressure in the form of logical reads than the While Loop. It's RBAR on steroids and should be avoided, at all cost, for any single row calculations just as a While Loop should be avoided. There are places where recursion is quite valuable but this ISN'T one of them.
As a side bar, Mr. Denny is absolutely spot on... a correctly sized permanent Numbers or Tally table is the way to go for most things. What does correctly sized mean? Well, most people use a Tally table to generate dates or to do splits on VARCHAR(8000). If you create an 11,000 row Tally table with the correct clustered index on "N", you'll have enough rows to create more than 30 years worth of dates (I work with mortgages a fair bit so 30 years is a key number for me) and certainly enough to handle a VARCHAR(8000) split. Why is "right sizing" so important? If the Tally table is used a lot, it easily fits in cache which makes it blazingly fast without much pressure on memory at all.
Last but not least, every one knows that if you create a permanent Tally table, it doesn't much matter which method you use to build it because 1) it's only going to be made once and 2) if it's something like an 11,000 row table, all of the methods are going to run "good enough". So why all the indigination on my part about which method to use???
The answer is that some poor guy/gal who doesn't know any better and just needs to get his or her job done might see something like the Recursive CTE method and decide to use it for something much larger and much more frequently used than building a permanent Tally table and I'm trying to protect those people, the servers their code runs on, and the company that owns the data on those servers. Yeah... it's that big a deal. It should be for everyone else, as well. Teach the right way to do things instead of "good enough". Do some testing before posting or using something from a post or book... the life you save may, in fact, be your own especially if you think a recursive CTE is the way to go for something like this. ;-)
Thanks for listening...

Answer 2

The most optimal function would be to use a table instead of a function. Using a function causes extra CPU load to create the values for the data being returned, especially if the values being returned cover a very large range.

Answer 3

This article gives 14 different possible solutions with discussion of each. The important point is that:
suggestions regarding efficiency and
performance are often subjective.
Regardless of how a query is being
used, the physical implementation
determines the efficiency of a query.
Therefore, rather than relying on
biased guidelines, it is imperative
that you test the query and determine
which one performs better.
I personally liked:
WITH Nbrs ( n ) AS (
SELECT 1 UNION ALL
SELECT 1 + n FROM Nbrs WHERE n < 500 )
SELECT n FROM Nbrs
OPTION ( MAXRECURSION 500 )

Answer 4

This view is super fast and contains all positive int values.
CREATE VIEW dbo.Numbers
WITH SCHEMABINDING
AS
WITH Int1(z) AS (SELECT 0 UNION ALL SELECT 0)
, Int2(z) AS (SELECT 0 FROM Int1 a CROSS JOIN Int1 b)
, Int4(z) AS (SELECT 0 FROM Int2 a CROSS JOIN Int2 b)
, Int8(z) AS (SELECT 0 FROM Int4 a CROSS JOIN Int4 b)
, Int16(z) AS (SELECT 0 FROM Int8 a CROSS JOIN Int8 b)
, Int32(z) AS (SELECT TOP 2147483647 0 FROM Int16 a CROSS JOIN Int16 b)
SELECT ROW_NUMBER() OVER (ORDER BY z) AS n
FROM Int32
GO

Answer 5

From SQL Server 2022 you will be able to do
SELECT Value
FROM GENERATE_SERIES(START = 1, STOP = 100, STEP=1)
In the public preview of SQL Server 2022 (CTP2.0) there are some very promising elements and other less so. Hopefully the negative aspects can be addressed before the actual release.
✅ Execution time for number generation
The below generates 10,000,000 numbers in 700 ms in my test VM (the assigning to a variable removes any overhead from sending results to the client)
DECLARE #Value INT
SELECT #Value =[value]
FROM GENERATE_SERIES(START=1, STOP=10000000)
✅ Cardinality estimates
It is simple to calculate how many numbers will be returned from the operator and SQL Server takes advantage of this as shown below.
❌ Unnecessary Halloween Protection
The plan for the below insert has a completely unnecessary spool - presumably as SQL Server does not currently have logic to determine the source of the rows is not potentially the destination.
CREATE TABLE dbo.NumberHeap(Number INT);
INSERT INTO dbo.Numbers
SELECT [value]
FROM GENERATE_SERIES(START=1, STOP=10);
When inserting into a table with a clustered index on Number the spool may be replaced by a sort instead (that also provides the phase separation)
❌ Unnecessary sorts
The below will return the rows in order anyway but SQL Server apparently does not yet have the properties set to guarantee this and take advantage of it in the execution plan.
SELECT [value]
FROM GENERATE_SERIES(START=1, STOP=10)
ORDER BY [value]
RE: This last point Aaron Bertrand indicates that this is not a box currently ticked but that this may be forthcoming.

Answer 6

Using SQL Server 2016+ to generate numbers table you could use OPENJSON :
-- range from 0 to #max - 1
DECLARE #max INT = 40000;
SELECT rn = CAST([key] AS INT)
FROM OPENJSON(CONCAT('[1', REPLICATE(CAST(',1' AS VARCHAR(MAX)),#max-1),']'));
LiveDemo
Idea taken from How can we use OPENJSON to generate series of numbers?

Answer 7

edit: see Conrad's comment below.
Jeff Moden's answer is great ... but I find on Postgres that the Itzik method fails unless you remove the E32 row.
Slightly faster on postgres (40ms vs 100ms) is another method I found on here adapted for postgres:
WITH
E00 (N) AS (
SELECT 1 UNION ALL SELECT 1 UNION ALL SELECT 1 UNION ALL SELECT 1 UNION ALL SELECT 1 UNION ALL
SELECT 1 UNION ALL SELECT 1 UNION ALL SELECT 1 UNION ALL SELECT 1 UNION ALL SELECT 1 ),
E01 (N) AS (SELECT a.N FROM E00 a CROSS JOIN E00 b),
E02 (N) AS (SELECT a.N FROM E01 a CROSS JOIN E01 b ),
E03 (N) AS (SELECT a.N FROM E02 a CROSS JOIN E02 b
LIMIT 11000 -- end record 11,000 good for 30 yrs dates
), -- max is 100,000,000, starts slowing e.g. 1 million 1.5 secs, 2 mil 2.5 secs, 3 mill 4 secs
Tally (N) as (SELECT row_number() OVER (ORDER BY a.N) FROM E03 a)
SELECT N
FROM Tally
As I am moving from SQL Server to Postgres world, may have missed a better way to do tally tables on that platform ... INTEGER()? SEQUENCE()?

Answer 8

Still much later, I'd like to contribute a slightly different 'traditional' CTE (does not touch base tables to get the volume of rows):
--===== Hans CROSS JOINED CTE method
WITH Numbers_CTE (Digit)
AS
(SELECT 0 UNION ALL SELECT 1 UNION ALL SELECT 2 UNION ALL SELECT 3 UNION ALL SELECT 4 UNION ALL SELECT 5 UNION ALL SELECT 6 UNION ALL SELECT 7 UNION ALL SELECT 8 UNION ALL SELECT 9)
SELECT HundredThousand.Digit * 100000 + TenThousand.Digit * 10000 + Thousand.Digit * 1000 + Hundred.Digit * 100 + Ten.Digit * 10 + One.Digit AS Number
INTO #Tally5
FROM Numbers_CTE AS One CROSS JOIN Numbers_CTE AS Ten CROSS JOIN Numbers_CTE AS Hundred CROSS JOIN Numbers_CTE AS Thousand CROSS JOIN Numbers_CTE AS TenThousand CROSS JOIN Numbers_CTE AS HundredThousand
This CTE performs more READs then Itzik's CTE but less then the Traditional CTE.
However, it consistently performs less WRITES then the other queries.
As you know, Writes are consistently quite much more expensive then Reads.
The duration depends heavily on the number of cores (MAXDOP) but, on my 8core, performs consistently quicker (less duration in ms) then the other queries.
I am using:
Microsoft SQL Server 2012 - 11.0.5058.0 (X64)
May 14 2014 18:34:29
Copyright (c) Microsoft Corporation
Enterprise Edition (64-bit) on Windows NT 6.3 <X64> (Build 9600: )
on Windows Server 2012 R2, 32 GB, Xeon X3450 #2.67Ghz, 4 cores HT enabled.