It's been longer than I hoped since my last installment on

bitmask / big number handling.
Life caught up with me and I've had many thankless tasks to catch up
on. But that's over now and I'm back to the general slacking that
typifies my days, so welcome to Part 3, handling logical operators.

I'll be discussing four operators in this post. AND, OR, XOR,
and NOT. The first three are extremely easy given the framework already
built out in the previous two posts. The last one has some problems --
so I'll discuss those first.

I haven't been able to find any mathematical or computer
science texts that discuss how to deal with variable-length bitmasks,
which is what I'm attempting to define here. Most texts keep the
discussion to, at most, two or four bytes, and even then those two or
four bytes are stable. But the problem with using a variable number is
that it creates some inconsistencies with regards to signing. Take this
example:

SELECT 1 AS Oops

WHERE

-1 <>

CONVERT(INT, CONVERT(VARBINARY, CONVERT(SMALLINT, -1)))

Oops

----

1

So what's happening here? The -1 on the right side of the comparison
is being cast into a 2-byte SMALLINT, then converted to VARBINARY
(which produces 0xFFFF). Then it's being converted back to INT. But
guess what..?

SELECT CONVERT(INT, 0xFFFF) AS Arrgh

Arrgh

-----

65535

So what is SQL Server telling us? That a small -1 is not the same as a bigger -1. **-1 <> -1**!!!

You're probably asking yourself, "what is this guy talking
about, and what does any of this have to do with the topic at hand,
implementing a binary NOT operation?" And if that's what you were
asking yourself, then good job, because you've asked the correct
question. So what do they have to do with each other?

The truth-table for NOT is quite simple:

But let's delve a little deeper. The SMALLINT representation of
the decimal number 1 in hex is 0x0001. In binary, that's
0000000000000001. If you run that through a logical NOT, the result is
1111111111111110, or 0xFFFE. That's -2. But remember, that's only -2 *if you're a SMALLINT*.
If you're either an INT or a BIGINT, it's 65534. And that's just not
consistent. I want to know that any equivalent number into my function
yeilds an equivalent number on the way out. So NOT(0x000001) should
yeild the same result as NOT(0x000000000001).

... and that result, in the function I provide, will be: 0xFE.
One byte in, one byte out. Similar to some prison gang mottos, but
that's a topic for a later post.

You'll notice that this is similar to the way I handled the bitmask re-constitution in the last post. So I feel pretty good about this. Sparsity is a good thing.

But this has a very important *side-effect*. These
numbers are now officially and permanently un-signed. We can't deal
with sign because we can't know which byte corresponds to the highest
byte -- that's variable. And since we can't determine the highest byte,
we also can't determine the highest bit in that byte, and so can't know
whether or not our number is negative.

But I can live with that. And I hope you can, too. And if you can't, write a solution and send it to me and I'll post it.

So on that note, and without further ado, here's how you figure out which bit positions should be output by a **NOT** operation:

DECLARE @Bitmask VARBINARY(4096)

SET @Bitmask = 0x01

SELECT x.Number

FROM BitmaskNumbers x

LEFT JOIN dbo.SplitBitmask(@Bitmask) y ON y.Number = x.Number

WHERE y.Number IS NULL

AND x.Byte <=

(SELECT MAX(Byte)

FROM BitmaskNumbers z

WHERE z.Number =

(SELECT MAX(Number)

FROM dbo.SplitBitmask(@Bitmask)))

Number

-------

2

3

4

5

6

7

8

Pretty simple, really: Split the bitmask and take any numbers within
the same byte range that aren't in the bitmask. To reconstitute it,
simply modify the reconstitution pattern a bit, stuff it all into a
function, and you get:

CREATE FUNCTION bitwiseNot

(

@Bitmask VARBINARY(4096)

)

RETURNS VARBINARY(4096)

AS

BEGIN

DECLARE @BitsInBitmask TABLE(Number SMALLINT)

INSERT @BitsInBitmask

SELECT Number

FROM dbo.splitBitmask(@Bitmask)

SET @Bitmask = 0x

SELECT @Bitmask = @Bitmask +

CONVERT(VARBINARY(1),

SUM(CASE

WHEN x.Number IS NOT NULL THEN 0

ELSE BitmaskNumbers.BitValue

END)

)

FROM @BitsInBitmask x

RIGHT JOIN BitmaskNumbers ON BitmaskNumbers.Number = x.Number

WHERE BitmaskNumbers.Byte <=

(SELECT

CASE MAX(Number) % 8

WHEN 0 THEN (MAX(Number) - 1) / 8

ELSE MAX(Number) / 8

END + 1

FROM @BitsInBitmask)

GROUP BY BitmaskNumbers.Byte

ORDER BY BitmaskNumbers.Byte DESC

RETURN(@Bitmask)

END

GO

SELECT dbo.BitwiseNot(0x01) AS Not01

Not01

-----

0xFE

And of course, given the properties I described above:

SELECT dbo.BitwiseNot(0x0000000001) AS Not0000000001

Not0000000001

-------------

0xFE

And now on to the other three logical operations, which are much simpler...

The easiest is OR, which has the following truth table:

And what is that similar to, in relational parlance..? A UNION, perhaps?

SELECT Number

FROM

(

SELECT Number

FROM dbo.splitBitmask(0x01)

UNION

SELECT Number

FROM dbo.splitBitmask(0x03)

) x

... and how about exclusive OR (XOR)?

Similar to the UNION, but we only want intersections with exactly one bitmask position... Luckily, SQL is equipped for that:

SELECT Number

FROM

(

SELECT Number FROM dbo.SplitBitmask(0x01)

UNION ALL

SELECT Number FROM dbo.SplitBitmask(0x02)

) x

GROUP BY Number

HAVING COUNT(*) = 1

Finally, the AND operation:

Just like XOR, but you need exactly *two* bit positions in each intersection:

SELECT Number

FROM

(

SELECT Number FROM dbo.SplitBitmask(0x01)

UNION ALL

SELECT Number FROM dbo.SplitBitmask(0x02)

) x

GROUP BY Number

HAVING COUNT(*) = 2

Putting it all together, I present the following OR, XOR, and AND UDFs:

**OR**

CREATE FUNCTION bitwiseOr

(

@Bitmask1 VARBINARY(4096),

@Bitmask2 VARBINARY(4096)

)

RETURNS VARBINARY(4096)

AS

BEGIN

DECLARE @BitsInBitmask TABLE(Number SMALLINT)

INSERT @BitsInBitmask

SELECT Number

FROM

(

SELECT Number

FROM dbo.splitBitmask(@Bitmask1)

UNION

SELECT Number

FROM dbo.splitBitmask(@Bitmask2)

) x

SET @Bitmask1 = 0x

SELECT @Bitmask1 = @Bitmask1 +

CONVERT(VARBINARY(1),

SUM(CASE

WHEN x.Number IS NULL THEN 0

ELSE BitmaskNumbers.BitValue

END)

)

FROM @BitsInBitmask x

RIGHT JOIN BitmaskNumbers ON BitmaskNumbers.Number = x.Number

WHERE BitmaskNumbers.Byte <=

(SELECT

CASE MAX(Number) % 8

WHEN 0 THEN (MAX(Number) - 1) / 8

ELSE MAX(Number) / 8

END + 1

FROM @BitsInBitmask)

GROUP BY BitmaskNumbers.Byte

ORDER BY BitmaskNumbers.Byte DESC

RETURN(@Bitmask1)

END

SELECT dbo.bitwiseOr(0x01, 0x03) AS Or_01_03

Or_01_03

--------

0x03

**XOR**

CREATE FUNCTION bitwiseXOr

(

@Bitmask1 VARBINARY(4096),

@Bitmask2 VARBINARY(4096)

)

RETURNS VARBINARY(4096)

AS

BEGIN

DECLARE @BitsInBitmask TABLE(Number SMALLINT)

INSERT @BitsInBitmask

SELECT Number

FROM

(

SELECT Number

FROM dbo.splitBitmask(@Bitmask1)

UNION ALL

SELECT Number

FROM dbo.splitBitmask(@Bitmask2)

) x

GROUP BY Number

HAVING COUNT(*) = 1

SET @Bitmask1 = 0x

SELECT @Bitmask1 = @Bitmask1 +

CONVERT(VARBINARY(1),

SUM(CASE

WHEN x.Number IS NULL THEN 0

ELSE BitmaskNumbers.BitValue

END)

)

FROM @BitsInBitmask x

RIGHT JOIN BitmaskNumbers ON BitmaskNumbers.Number = x.Number

WHERE BitmaskNumbers.Byte <=

(SELECT

CASE MAX(Number) % 8

WHEN 0 THEN (MAX(Number) - 1) / 8

ELSE MAX(Number) / 8

END + 1

FROM @BitsInBitmask)

GROUP BY BitmaskNumbers.Byte

ORDER BY BitmaskNumbers.Byte DESC

RETURN(@Bitmask1)

END

SELECT dbo.bitwiseXOr(0x01, 0x03) AS XOr_01_03

XOr_01_03

--------

0x02

... And finally, **AND**

CREATE FUNCTION bitwiseAnd

(

@Bitmask1 VARBINARY(4096),

@Bitmask2 VARBINARY(4096)

)

RETURNS VARBINARY(4096)

AS

BEGIN

DECLARE @BitsInBitmask TABLE(Number SMALLINT)

INSERT @BitsInBitmask

SELECT Number

FROM

(

SELECT Number

FROM dbo.splitBitmask(@Bitmask1)

UNION ALL

SELECT Number

FROM dbo.splitBitmask(@Bitmask2)

) x

GROUP BY Number

HAVING COUNT(*) = 2

SET @Bitmask1 = 0x

SELECT @Bitmask1 = @Bitmask1 +

CONVERT(VARBINARY(1),

SUM(CASE

WHEN x.Number IS NULL THEN 0

ELSE BitmaskNumbers.BitValue

END)

)

FROM @BitsInBitmask x

RIGHT JOIN BitmaskNumbers ON BitmaskNumbers.Number = x.Number

WHERE BitmaskNumbers.Byte <=

(SELECT

CASE MAX(Number) % 8

WHEN 0 THEN (MAX(Number) - 1) / 8

ELSE MAX(Number) / 8

END + 1

FROM @BitsInBitmask)

GROUP BY BitmaskNumbers.Byte

ORDER BY BitmaskNumbers.Byte DESC

RETURN(@Bitmask1)

END

SELECT dbo.bitwiseAnd(0x01, 0x03) AS And_01_03

And_01_03

--------

0x01

... And that's enough for today's installment. Enjoy..!

## Comments

## Lars Söderholm said:

Hello,it appears to be a small bug in your logical or funtion, bitwiseor.

The last order by "ORDER BY BitmaskNumbers.Byte DESC"

cannot be order by desc, because if you use the function in a nestled manner it will reverse the order of the bytes causing problematic behaviour.

Otherwise the parts i've tried works fine.

Cheers!

## Adam Machanic said:

Quick installment this time. Left-shift and right-shift operators. Left-shift and right-shift are integral

## Etienne Divina said:

Great article, making code for bitwise operation with large binary data easy to read !

But when i tested it, i noticed performances issues.

It is lot more efficient to work on slices of 2bytes and iterate.

The algo is like that :

- calculate the max length of data and number of iteration

- loop the number of slices (number of slice = CEILING(maxLength/CAST(2 AS FLOAT)))

- Work on slice of data : SET @bin = CAST(SUBSTRING(@Source, DATALENGTH(@Source) + 1 - (@sliceIndex + 1) * @sliceBytes, 2) AS SMALLINT)

I can post more if needed

## Adam Machanic said:

Etienne:

If you feel like it, you're more than welcome to share your more optimized code. Perhaps someone will be able to benefit. Personally I've never had a use case for any of this stuff. I just did it for fun :-)

## Oleg said:

Thank you for the interesting articles.

Could I ask a question.

create table t(id int, b varbinary(128))

Every bit in binary means something and I need to have possibility to check if one of bits from list (i1, i2, .., iN) checked.

I use some of your functions, all is ok. But naturally when table is large then checking subset is slow.

For example simply saying:

select * from t inner join idlist on t.id=idlist.id and substring(b, n, 1)&mask<>0

Thank you

How I can optimize this kind of tables and queries?