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Finding Gaps in a Sequential Number Sequence Expand / Collapse
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Posted Monday, March 20, 2006 1:58 PM
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Comments posted to this topic are about the content posted at http://www.sqlservercentral.com/columnists/slasham/findinggapsinasequentialnumbersequence.asp


Post #267061
Posted Monday, April 3, 2006 8:32 AM
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If you just need to identify IF a list of numbers is in sequence, but not where the gaps are, I believe you can do it in a single pass of the table.  (Warning -- to understand this algorithm, some high school math is required!)

-- Mathematical way of identifying if a list of numbers is sequential

-- Based on formula:

-- M + (M+1) + (M+2) + ... + (M+n-1) = n*M + n*(n-1)/2

-- Calling the left side S (for Sum), and rearranging yields:

-- n2 + (2*M - 1)*n - 2*S = 0

-- Then the quadratic formula produces:

-- n = ( (1-2*M) +- sqrt( (2*M -1)2 + 8*S) )/2

-- I believe it can be shown that a list of numbers is a sequence if

-- and only if the value in the sqrt() above is a perfect square.

-- The code below is based on that assumption

--

-- Mike Arney, 4/3/2006

set nocount on

create table Numbers (id int not null)

insert Numbers values (6)

insert Numbers values (7)

insert Numbers values (8)

insert Numbers values (9)

insert Numbers values (10)

insert Numbers values (11)

insert Numbers values (12)

-- insert Numbers values (14) -- uncomment for testing

declare @Sum bigint, @Count bigint, @Min bigint

select @Sum = sum(convert(bigint, id)), @Count = count(*), @Min = min(id) from Numbers

declare @sqrt float

select @sqrt = sqrt( power((2*@Min - 1), 2) + 8*@Sum)

if @sqrt = floor(@sqrt) select 'Sequence' else select 'Not Sequence'

go

drop table Numbers

go

 

Post #270556
Posted Monday, April 3, 2006 9:07 AM
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A self join solution:

select a.SeqNumber, max(b.SeqNumber)
from #SequenceTable a join #SequenceTable b on a.SeqNumber > b.SeqNumber
group by a.SeqNumber
having a.SeqNumber - max(b.SeqNumber) > 1

Post #270570
Posted Monday, April 3, 2006 11:19 AM
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This way is a few hundred times faster (on large datasets)...NOTE:the details are missing but it still identifies the key holes in the sequence, except beiging and end which are obvious...?

SELECT s1.SeqNumber
FROM #SequenceTable s1
LEFT JOIN #SequenceTable s2
ON s1.SeqNumber = s2.SeqNumber -1
WHERE s2.SeqNumber IS NULL

 

 

 

Post #270622
Posted Monday, April 3, 2006 2:34 PM
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I agree, the join on "where Seq2.SeqNumber < Seq1.SeqNumber" will be a killer with large tables.

This version uses a simple a = b-1 join to find the gaps, then a subquery to find the last value before the gap.  On 1 million rows it ran in 1/4 the elapsed time and 1/13 the CPU time of the original in the article.

select LastSeqNumber, NextSeqNumber
    , FirstAvailable = LastSeqNumber + 1
    , LastAvailable = NextSeqNumber - 1
    , NumbersAvailable = NextSeqNumber - (LastSeqNumber + 1)
from (
    select (SELECT TOP 1 SeqNumber FROM #SequenceTable WHERE SeqNumber < a.SeqNumber ORDER BY SeqNumber DESC) as LastSeqNumber,
    a.SeqNumber as NextSeqNumber
    from #SequenceTable a
    left join #SequenceTable b on a.SeqNumber = b.SeqNumber + 1
    where b.SeqNumber IS NULL
)
a
order by LastSeqNumber

 




Post #270685
Posted Wednesday, April 5, 2006 8:52 AM
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Nice solution!
I would like to contribute too. If we only want to find IF there is a gap in the sequence we only need to verify that COUNT(*) + MIN(ID) = MAX(ID), right?
In that case, if we have an index on the ID columns we only need a quick scan on it with very little additional CPU+Memory usage with:

create table Numbers (id int not null)
insert Numbers values (6)
insert Numbers values (7)
insert Numbers values (8)

-- insert Numbers values (14) -- uncomment for testing
-- DELETE Numbers WHERE id = 14
SELECT CASE WHEN COUNT(*) + MIN(ID) - 1 = MAX(ID) THEN 'Sequence' ELSE 'Not Sequence' END FROM NUMBERS


go

drop table Numbers



Post #271289
Posted Thursday, April 27, 2006 3:51 PM
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Thanks all for the great examples of other ways to get the same results.  It is interesting to see these different methods and the understandings of how SQL works under the hood so to speak, with different timings on the various methods.  For me timing was not a big issue, and my 31 second run over a table of 3.5 million rows seemed fair.  Timing becomes more significant if a process is being repetatively and frequently run. 

 

Over my dataset, Scott Coleman’s method ran in 25 seconds, being significantly slower than his own reported difference, indicating other factors come into play.

 

Brendan Smith’s method ran in 10 seconds, and although I accept this is faster, it does not do the computes for the additional rows in my output set, so is not a reliable comparison, but looks promising.

 

Mike Gress’s example did not complete, I bombed it off at over 14 minutes with no results (sorry Mike).

 

I didn’t try Hanslindgren’s method as it is not my database, so I was unable to add indexes to it. 

 

To Mike Arney, my high school maths was never that good, but I am sure there will be some that can work this one out from your example.

 

Joe Celko’s second method also did not complete, and I bombed this off at over 14 minutes with no results, and Joe’s first method took less than a second and produced no results (sorry Joe).

 

Looking deeper at my source dataset, the timing differences I get to your own results may be due to the extreme number of gaps.  My data set produced 1.1 million rows (of identified gaps).  Also within my data set are sequence number duplications, which may have contributed to the failure of some of the scripts.

 

Thanks you all for your feedback and suggestions.

 

Stephen




Post #276056
Posted Friday, April 28, 2006 4:02 AM
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Hi Stephen!

Thank you, it is not always you get such an exhausting feedback of how things turn out when you post replies and hope you can help.

Hanslindgren




Post #276127
Posted Tuesday, April 3, 2007 2:59 AM
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Hi,

I'm a latecomer to the party and didn't realize there had already been healthy debate, but here's what I came up with.

Looking through the previous submissions above, it's the same concept as Brendan's, but extended to support the full functionality of the original post.

For the original example (working on an unoptimized temp table) the time taken seems to be approx 0.25 secs/1000 records, so with 10000 records I'm getting something like 3 seconds run time. With the code below there seems to be no increase in time taken at all (around 0.25 secs for anything from 0 to 10000 records), but this is probably because for small datasets the main delay below is not due to the work itself, but rather the creation of the temp table.

I'd be interested to see if anyone can make something faster! (ignoring the fact that I could probably have made it a table variable...)

Thanks,

Tao

 

CREATE TABLE #StartsAndEnds (EventID Int, EventType NVarChar(20), IdentityTracker INT IDENTITY (1,1))
INSERT INTO #StartsAndEnds (EventID, EventType)
SELECT SeqNumber, Type
FROM (
 SELECT Seq1.SeqNumber, 'Start' AS Type
 FROM #SequenceTable Seq1
 LEFT JOIN #SequenceTable Seq2 ON Seq2.SeqNumber = Seq1.SeqNumber + 1
 WHERE Seq2.SeqNumber Is Null
 
 UNION ALL
 
 SELECT Seq1.SeqNumber, 'End' AS Type
 FROM #SequenceTable Seq1
 LEFT JOIN #SequenceTable Seq2 ON Seq2.SeqNumber = Seq1.SeqNumber - 1
 WHERE Seq2.SeqNumber Is Null
) AS PingPong
ORDER BY SeqNumber

SELECT IsNull(FL1.EventID, 0) AS StartGap,
 FL2.EventID As EndGap,
 IsNull(FL1.EventID, 0) + 1 AS FirstAvailable,
 FL2.EventID - 1 AS LastAvailable,
 FL2.EventID - IsNull(FL1.EventID, 0) - 1 As AvailableCount
FROM #StartsAndEnds FL2
LEFT JOIN #StartsAndEnds FL1 ON FL2.IdentityTracker = FL1.IdentityTracker + 1
WHERE FL2.IdentityTracker % 2 = 1

DROP TABLE #StartsAndEnds



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Post #355549
Posted Tuesday, April 3, 2007 2:11 PM
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Hi Tao, did you post the correct version of your code?

I copied your code above and did a find/replace on #SequenceTable and SeqNumber swapping these for my table name and sequence field respectively, then ran it.  It was certainly quick, very impressive, 16 seconds to produce 142874 identified breaks in the sequence from a table containing 3,827,625 rows. 

First few rows of output are as below

StartGapEndGapFirstAvailableLastAvailableAvailableCount
0NULL1NULLNULL
0NULL1NULLNULL
0NULL1NULLNULL
0NULL1NULLNULL
0NULL1NULLNULL
0NULL1NULLNULL
582246210758225621063882
780258008678026800852060

My original query over the same table takes 44 seconds and produces the following first few rows.

StartGapEndGapFirstAvailableLastAvailableAvailableCount
02111
206242062620625206251
520915209352092520921
555185552055519555191
582225822458223582231
6210778025621087802415917
800868008880087800871
801608016280161801611

I've checked the results and my query is showing the correct gaps.  Your result line showing 58224 to 62107 represents my end gap result line 5 above to my start gap result line 6 above.

Love to see your revision as certainly looks like it will be very fast.

Stephen




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