The most elegant way to generate permutations in SQL server - sql

Question

Given a the following table:
Index | Element
---------------
1 | A
2 | B
3 | C
4 | D
We want to generate all the possible permutations (without repetitions) using the elements.
the final result (skipping some rows) will look like this:
Results
----------
ABCD
ABDC
ACBD
ACDB
ADAC
ADCA
...
DABC
DACB
DBCA
DBAC
DCAB
DCBA
(24 Rows)
How would you do it?

Answer 1

After making some perhaps snarky comments, this problem stuck in my brain all evening, and I eventually came up with the following set-based approach. I believe it definitely qualifies as "elegant", but then I also think it qualifies as "kinda dumb". You make the call.
First, set up some tables:
-- For testing purposes
DROP TABLE Source
DROP TABLE Numbers
DROP TABLE Results
-- Add as many rows as need be processed--though note that you get N! (number of rows, factorial) results,
-- and that gets big fast. The Identity column must start at 1, or the algorithm will have to be adjusted.
-- Element could be more than char(1), though the algorithm would have to be adjusted again, and each element
-- must be the same length.
CREATE TABLE Source
(
SourceId int not null identity(1,1)
,Element char(1) not null
)
INSERT Source (Element) values ('A')
INSERT Source (Element) values ('B')
INSERT Source (Element) values ('C')
INSERT Source (Element) values ('D')
--INSERT Source (Element) values ('E')
--INSERT Source (Element) values ('F')
-- This is a standard Tally table (or "table of numbers")
-- It only needs to be as long as there are elements in table Source
CREATE TABLE Numbers (Number int not null)
INSERT Numbers (Number) values (1)
INSERT Numbers (Number) values (2)
INSERT Numbers (Number) values (3)
INSERT Numbers (Number) values (4)
INSERT Numbers (Number) values (5)
INSERT Numbers (Number) values (6)
INSERT Numbers (Number) values (7)
INSERT Numbers (Number) values (8)
INSERT Numbers (Number) values (9)
INSERT Numbers (Number) values (10)
-- Results are iteratively built here. This could be a temp table. An index on "Length" might make runs
-- faster for large sets. Combo must be at least as long as there are characters to be permuted.
CREATE TABLE Results
(
Combo varchar(10) not null
,Length int not null
)
Here's the routine:
SET NOCOUNT on
DECLARE
#Loop int
,#MaxLoop int
-- How many elements there are to process
SELECT #MaxLoop = max(SourceId)
from Source
-- Initialize first value
TRUNCATE TABLE Results
INSERT Results (Combo, Length)
select Element, 1
from Source
where SourceId = 1
SET #Loop = 2
-- Iterate to add each element after the first
WHILE #Loop <= #MaxLoop
BEGIN
-- See comments below. Note that the "distinct" remove duplicates, if a given value
-- is to be included more than once
INSERT Results (Combo, Length)
select distinct
left(re.Combo, #Loop - nm.Number)
+ so.Element
+ right(re.Combo, nm.Number - 1)
,#Loop
from Results re
inner join Numbers nm
on nm.Number <= #Loop
inner join Source so
on so.SourceId = #Loop
where re.Length = #Loop - 1
-- For performance, add this in if sets will be large
--DELETE Results
-- where Length <> #Loop
SET #Loop = #Loop + 1
END
-- Show results
SELECT *
from Results
where Length = #MaxLoop
order by Combo
The general idea is: when adding a new element (say "B") to any string (say, "A"), to catch all permutations you would add B
to all possible positions (Ba, aB), resulting in a new set of strings. Then iterate: Add a new element (C) to each position in a string
(AB becomes Cab, aCb, abC), for all strings (Cba, bCa, baC), and you have the set of permutations. Iterate over each result set with
the next character until you run out of characters... or resources. 10 elements is 3.6 million permutations, roughly 48MB with the above algorithm, and 14 (unique) elements would hit 87 billion permutations and 1.163 terabytes.
I'm sure it could eventually be wedged into a CTE, but in the end all that would be is a glorified loop. The logic
is clearer this way, and I can't help but think the CTE execution plan would be a nightmare.

Answer 2

DECLARE #s VARCHAR(5);
SET #s = 'ABCDE';
WITH Subsets AS (
SELECT CAST(SUBSTRING(#s, Number, 1) AS VARCHAR(5)) AS Token,
CAST('.'+CAST(Number AS CHAR(1))+'.' AS VARCHAR(11)) AS Permutation,
CAST(1 AS INT) AS Iteration
FROM dbo.Numbers WHERE Number BETWEEN 1 AND 5
UNION ALL
SELECT CAST(Token+SUBSTRING(#s, Number, 1) AS VARCHAR(5)) AS Token,
CAST(Permutation+CAST(Number AS CHAR(1))+'.' AS VARCHAR(11)) AS
Permutation,
s.Iteration + 1 AS Iteration
FROM Subsets s JOIN dbo.Numbers n ON s.Permutation NOT LIKE
'%.'+CAST(Number AS CHAR(1))+'.%' AND s.Iteration < 5 AND Number
BETWEEN 1 AND 5
--AND s.Iteration = (SELECT MAX(Iteration) FROM Subsets)
)
SELECT * FROM Subsets
WHERE Iteration = 5
ORDER BY Permutation
Token Permutation Iteration
----- ----------- -----------
ABCDE .1.2.3.4.5. 5
ABCED .1.2.3.5.4. 5
ABDCE .1.2.4.3.5. 5
(snip)
EDBCA .5.4.2.3.1. 5
EDCAB .5.4.3.1.2. 5
EDCBA .5.4.3.2.1. 5
first posted a while ago here
However, it would be better to do it in a better language such as C# or C++.

Answer 3

Just using SQL, without any code, you could do it if you can crowbar yourself another column into the table. Clearly you need to have one joined table for each of the values to be permuted.
with llb as (
select 'A' as col,1 as cnt union
select 'B' as col,3 as cnt union
select 'C' as col,9 as cnt union
select 'D' as col,27 as cnt
)
select a1.col,a2.col,a3.col,a4.col
from llb a1
cross join llb a2
cross join llb a3
cross join llb a4
where a1.cnt + a2.cnt + a3.cnt + a4.cnt = 40

Answer 4

Am I correctly understanding that you built Cartesian product n x n x n x n, and then filter out unwanted stuff? The alternative would be generating all the numbers up to n! and then using factorial number system to map them via element encoding.

Answer 5

Simpler than a recursive CTE:
declare #Number Table( Element varchar(MAX), Id varchar(MAX) )
Insert Into #Number Values ( 'A', '01')
Insert Into #Number Values ( 'B', '02')
Insert Into #Number Values ( 'C', '03')
Insert Into #Number Values ( 'D', '04')
select a.Element, b.Element, c.Element, d.Element
from #Number a
join #Number b on b.Element not in (a.Element)
join #Number c on c.Element not in (a.Element, b.Element)
join #Number d on d.Element not in (a.Element, b.Element, c.Element)
order by 1, 2, 3, 4
For an arbitrary number of elements, script it out:
if object_id('tempdb..#number') is not null drop table #number
create table #number (Element char(1), Id int, Alias as '_'+convert(varchar,Id))
insert #number values ('A', 1)
insert #number values ('B', 2)
insert #number values ('C', 3)
insert #number values ('D', 4)
insert #number values ('E', 5)
declare #sql nvarchar(max)
set #sql = '
select '+stuff((
select char(13)+char(10)+'+'+Alias+'.Element'
from #number order by Id for xml path (''), type
).value('.','NVARCHAR(MAX)'),3,1,' ')
set #sql += '
from #number '+(select top 1 Alias from #number order by Id)
set #sql += (
select char(13)+char(10)+'join #number '+Alias+' on '+Alias+'.Id not in ('
+stuff((
select ', '+Alias+'.Id'
from #number b where a.Id > b.Id
order by Id for xml path ('')
),1,2,'')
+ ')'
from #number a where Id > (select min(Id) from #number)
order by Element for xml path (''), type
).value('.','NVARCHAR(MAX)')
set #sql += '
order by 1'
print #sql
exec (#sql)
To generate this:
select
_1.Element
+_2.Element
+_3.Element
+_4.Element
+_5.Element
from #number _1
join #number _2 on _2.Id not in (_1.Id)
join #number _3 on _3.Id not in (_1.Id, _2.Id)
join #number _4 on _4.Id not in (_1.Id, _2.Id, _3.Id)
join #number _5 on _5.Id not in (_1.Id, _2.Id, _3.Id, _4.Id)
order by 1

Answer 6

This method uses a binary mask to select the correct rows:
;with src(t,n,p) as (
select element, index, power(2,index-1)
from table
)
select s1.t+s2.t+s3.t+s4.t
from src s1, src s2, src s3, src s4
where s1.p+s2.p+s3.p+s4.p=power(2,4)-1
My original post:
declare #t varchar(4) = 'ABCD'
;with src(t,n,p) as (
select substring(#t,1,1),1,power(2,0)
union all
select substring(#t,n+1,1),n+1,power(2,n)
from src
where n < len(#t)
)
select s1.t+s2.t+s3.t+s4.t
from src s1, src s2, src s3, src s4
where s1.p+s2.p+s3.p+s4.p=power(2,len(#t))-1
This is one of those problems that haunts you. I liked the simplicity of my original answer but there was this issue where I was still building all the possible solutions and then selecting the correct ones. One more try to make this process more efficient by only building the solutions that were correct yielded this answer. Add a character to the string only if that character didn't exist in the string. Patindex seemed like the perfect companion for a CTE solution. Here it is.
declare #t varchar(10) = 'ABCDEFGHIJ'
;with s(t,n) as (
select substring(#t,1,1),1
union all
select substring(#t,n+1,1),n+1
from s where n<len(#t)
)
,j(t) as (
select cast(t as varchar(10)) from s
union all
select cast(j.t+s.t as varchar(10))
from j,s where patindex('%'+s.t+'%',j.t)=0
)
select t from j where len(t)=len(#t)
I was able to build all 3.6 million solutions in 3 minutes and 2 seconds. Hopefully this solution will not get missed just because it's not the first.

Answer 7

Current solution using a recursive CTE.
-- The base elements
Declare #Number Table( Element varchar(MAX), Id varchar(MAX) )
Insert Into #Number Values ( 'A', '01')
Insert Into #Number Values ( 'B', '02')
Insert Into #Number Values ( 'C', '03')
Insert Into #Number Values ( 'D', '04')
-- Number of elements
Declare #ElementsNumber int
Select #ElementsNumber = COUNT(*)
From #Number;
-- Permute!
With Permutations( Permutation, -- The permutation generated
Ids, -- Which elements where used in the permutation
Depth ) -- The permutation length
As
(
Select Element,
Id + ';',
Depth = 1
From #Number
Union All
Select Permutation + ' ' + Element,
Ids + Id + ';',
Depth = Depth + 1
From Permutations,
#Number
Where Depth < #ElementsNumber And -- Generate only the required permutation number
Ids Not like '%' + Id + ';%' -- Do not repeat elements in the permutation (this is the reason why we need the 'Ids' column)
)
Select Permutation
From Permutations
Where Depth = #ElementsNumber

Answer 8

Assuming your table is named Elements and has 4 rows, this is as simple as:
select e1.Element + e2.Element + e3.Element + e4.Element
from Elements e1
join Elements e2 on e2.Element != e1.Element
join Elements e3 on e3.Element != e2.Element AND e3.Element != e1.Element
join Elements e4 on e4.Element != e3.Element AND e4.Element != e2.Element AND e4.Element != e1.Element

Answer 9

Way too much rust on my SQL skills, but i took a different tack for a similar problem and thought it worth sharing.
Table1 - X strings in a single field Uno
Table2 - Y strings in a single field Dos
(SELECT Uno, Dos
FROM Table1
CROSS JOIN Table2 ON 1=1)
UNION
(SELECT Dos, Uno
FROM Table1
CROSS JOIN Table2 ON 1=1)
Same principle for 3 tables with an added CROSS JOIN
(SELECT Tres, Uno, Dos
FROM Table1
CROSS JOIN Table2 ON 1=1
CROSS JOIN Table3 ON 1=1)
although it takes 6 cross-join sets in the union.

Answer 10

--Hopefully this is a quick solution, just change the values going into #X
IF OBJECT_ID('tempdb.dbo.#X', 'U') IS NOT NULL DROP TABLE #X; CREATE table #X([Opt] [nvarchar](10) NOT NULL)
Insert into #X values('a'),('b'),('c'),('d')
declare #pSQL NVarChar(max)='select * from #X X1 ', #pN int =(select count(*) from #X), #pC int = 0;
while #pC<#pN begin
if #pC>0 set #pSQL = concat(#pSQL,' cross join #X X', #pC+1);
set #pC = #pC +1;
end
execute(#pSQL)
--or as single column result
IF OBJECT_ID('tempdb.dbo.#X', 'U') IS NOT NULL DROP TABLE #X; CREATE table #X([Opt] [nvarchar](10) NOT NULL)
Insert into #X values('a'),('b'),('c'),('d')
declare #pSQL NVarChar(max)=' as R from #X X1 ',#pSelect NVarChar(Max)=' ',#pJoin NVarChar(Max)='', #pN int =(select count(*) from #X), #pC int = 0;
while #pC<#pN begin
if #pC>0 set #pJoin = concat(#pJoin ,' cross join #X X', #pC+1) set #pSelect = concat(#pSelect ,'+ X', #pC+1,'.Opt ')
set #pC = #pC +1;
end
set #pSQL = concat ('select X1.Opt', #pSelect,#pSQL ,#pJoin)
exec(#pSQL)

Answer 11

create function GeneratePermutations (#string nvarchar(4000))
RETURNS #Permutations
TABLE(
name nVARCHAR(500)
)
AS
begin
declare #SplitedString table(name nvarchar(500))
insert into #SplitedString
select *
from string_split(#string,' ')
declare #CountOfWords as int
set #CountOfWords = (select count(*) from #SplitedString)
;with cte_Permutations (name, level) as (
select convert(nvarchar(500), name), 1 as level from #SplitedString
union all
select convert(nvarchar(500),splited.name+','+cte_Permutations.name),level+1
from #SplitedString splited ,cte_Permutations
where level < #CountOfWords
)
insert into #Permutations
select name
from cte_Permutations
where level = #CountOfWords
order by name
return
end
select *
From (
select 1 id,'a b c' msg
union all
select 2 id,'d e' msg
) p
cross apply dbo.GeneratePermutations(p.msg)