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TerminusDB Internals 3 - Sorting Every Sort of Thing

Some of the the original experiments with TerminusDB were in Postgres, where we built a table of IRIs and ids, and then created a multi-indexed table of triples. We then compared the speed of this to a library called HDT which created a compact representation of graphs, and found HDT to be extremely fast for large RDF databases (Think TTL files over 100GB).

This got us thinking seriously about succinct data structures, so we used the ideas in HDT as a starting point for TerminusDB.

One of the choices made by HDT is the front coded dictionary. This tends to work very well for IRIs, since we tend to share addresses as prefixes, leading to substantial compression and reasonable retrieval speed.

The dictionary relies on lexical sorting of data, as we want to branch at points that share prefixes. So sorting lexically helps us to maximally share prefixes.

Lexical ordering also allows us to start with a prefix, and iterate over everything that shares it, or to perform fast range queries, simply by finding the beginning point, and iterating until we are at the terminus of the range.

So we get:

  • Compression
  • Log like access
  • Range queries

But not everything is naturally designed to be stored lexically. Take the classic example of the directory in Linux:

					gavin@titan:~/tmp/sort$ touch 2
gavin@titan:~/tmp/sort$ touch 11
gavin@titan:~/tmp/sort$ touch 10
gavin@titan:~/tmp/sort$ touch 100
gavin@titan:~/tmp/sort$ touch 101
gavin@titan:~/tmp/sort$ touch 110
gavin@titan:~/tmp/sort$ ls -la
total 8
drwxrwxr-x  2 gavin gavin 4096 Okt 31 11:59 .
drwxrwxr-x 14 gavin gavin 4096 Okt 31 11:59 ..
-rw-rw-r--  1 gavin gavin    0 Okt 31 11:59 1
-rw-rw-r--  1 gavin gavin    0 Okt 31 11:59 10
-rw-rw-r--  1 gavin gavin    0 Okt 31 11:59 100
-rw-rw-r--  1 gavin gavin    0 Okt 31 11:59 101
-rw-rw-r--  1 gavin gavin    0 Okt 31 11:59 11
-rw-rw-r--  1 gavin gavin    0 Okt 31 11:59 110
-rw-rw-r--  1 gavin gavin    0 Okt 31 11:59 2

In this world 101 is less than 11, and 2 is greater than 110. Not what we typically want when we are sorting numbers.

Lexical Everything

But as it turns out, numbers can also be sorted lexically, provided we store them in a clever way. These lexical ordering tricks can give us id<->data conversion using dictionaries, which allows for compression, prefix queries and range queries.


So how do we get 100 to be larger than 2? If we have fixed size integers, such as Int32, the answer is relatively simple. We break Int32 into 4 bytes written out in big endian. We’re almost done save one complication. In most representations, we keep around a sign bit on integers, generally stored in the most significant position, which is 1 if the number is negative, and 0 if it is positive.

This is terrible, since all negative numbers are now larger than positive numbers. Further, integers are generally stored in a two’s complement, meaning that we flip every bit of a negative number. This is actually a good thing. Because it means that smaller numbers are bigger, and bigger numbers are smaller. Which is exactly how we expect negative numbers to sort! That is, -10 should be smaller than -1.

					-1 = 0bffff_fffe

To fix the sign problem is simple. We just flip the sign bit and we are done! We now have lexically sortable integers.


IEEE floating point numbers are also surprisingly simple to sort lexically. We have the same trick, requiring a sign flip, but in the case of negative numbers, we actually have to put them in the two’s complement representation, as floating point can’t avail of the same twos complement tricks used in integer arithmetic, so this is computed externally.

This is all there is to it in rust (using the bytes_order library to ensure we get a big endian representation).

					const F32_SIGN_MASK: u32 = 0x8000_0000;
const F32_COMPLEMENT: u32 = 0xffff_ffff;
fn float32_to_vec(f: &f32) -> Vec<u8> {
    let g: f32 = if f.to_bits() & F32_SIGN_MASK > 0 {
        f32::from_bits(f.to_bits() ^ F32_COMPLEMENT)
    } else {
        f32::from_bits(f.to_bits() ^ F32_SIGN_MASK)
    let mut wtr = Vec::with_capacity(4);

Notice, flipping all of the bits is just an xor with a complement mask which has every bit set.

Perhaps surprisingly, this trick also works for NaN and Negative and Positive Infinity!


But what about big integers? If the size is not fixed, what are we to do? Well, we could find the largest number and store everything in the number of bits required by this largest number, and use the representation above. However, this threatens to use up a lot of space.

Instead, we can simply prefix integers with their size. Conceptually we can rewrite our directory files from above as:

					gavin@titan:~/$ python
>>> x = ["0_1", "2_10","3_100","3_101","2_11","0_2"]
>>> x.sort()
>>> x
['0_1', '0_2', '2_10', '2_11', '3_100', '3_101']

Presto! It works!

But what about negative numbers? Well, we can perform the same sorts of trick with complements. Let’s do a complement in base 10 to see how it works.

First, let’s add a few more numbers to our list. How about -1, -2, and -10. Now, we can represent a negative with an - character which is less than every number in ascii. Then we can take our size, and complement it with 9, so that 9 is 0, and 0 is 9, and every other digit is in between.

Ok, so -1 becomes -9_9. -2 is -9_8. And -10 is -8_89.

					gavin@titan:~/$ python
>>> x = ["0_1", "2_10","3_100","3_101","2_11","0_2", "-9_9", "-9_8", "-8_89"]
>>> x.sort()
>>> x
['-8_89', '-9_8', '-9_9', '0_1', '0_2', '2_10', '2_11', '3_100', '3_101']

Great! We have an encoding that puts -10 on the bottom, and -1 on the top of the negatives.

Of course, if you’re paying close attention, you’ll see that we need a way to represent sizes that can go bigger than 9. We need a lexically sortable kind of size signifier. There are lots of ways to do this, with the simplest being unary. You simply take the size and represent it with a number of 1s and a self-delimiting zero. So the size 3 becomes 1110. This is pretty big, and a bit awkward, plus it doesn’t respect byte alignment, but is manageable in certain settings.

In TerminusDB we instead use an encoding which has a continuation bit. That is, we represent numbers up to 128 with the bits of the number in binary. So 3 would be 0b000_0011. We then stick a bit at the top which is zero if the number is less than 128 and one if we’re only representing the top 7 bits of our number, and our number is greater than 128 in which case we need another byte. Now, our encoded 3 becomes the entire byte: [0b0000_0011]. One can easily see that anything greater than 128 will already compare as larger than anything less than 128 because it will have its most significant bit set. 129 becomes the two bytes: [0b1000_0001, 0b0000_0001].

This kind of trick is called variable-byte encoding, and sorts the orders first, and the numbers inside of that order.


TerminusDB implements the XSD data types. And one of these data types is the rather unusual xsd:decimal which codes arbitrary precision decimal numbers. It’s unusual because most database do not store arbitrary precision floating points, and if they do, they generally do so in binary. While often times people want information recorded in the base-10 they are familiar with, a lot of our computing infrastructure makes this somewhat awkward.

As it turns out, you can also store these lexically, provided we do a bit of monkey-business.

In TerminusDB we store these with a pair of elements concatenated. The first is a bignum, which stores everything before the full stop. The second part is a v-byte style binary-coded decimal, which packs two decimal characters per byte. This represents no decimal as the lowest element, and a single digit decimal interleaved between two-digit decimals.

Decimal Encoding
none 0
0 1
00 2
01 3
02 4
1 12
10 13
9 100
90 101
99 111

We leave the last bit as a continuation bit for our v-byte encoding, as we need to compare as larger, only if the rest of the byte is the same.


    |     continuation bit
    |     |
| xxxxxxx c | xxxxxxx c | ...

This representation also allows us to keep significant digits, for instance 0.0 will be encoded differently from 0 and from 0.00, while retaining appropriate lexical sorting. This is important in scientific applications where significance should be recorded.


xsd:dateTime is a format that is based on ISO 8660, and is actually already mostly lexical. However it is broken in that the first element can be negative, can be larger than 4 digits, and can have a time-zone offset, and an arbitrary precision float for sub-second portions of the date.

Further, the format is a string, which is far too large to represent the data contained, efficiently.

In TerminusDB we deal with this simply by converting the number into a decimal encoding of the number of seconds since 1970 with an arbitrary precision. This can encode the femto-seconds of a laser pulse to the point at which the universe started in a format that is range queryable, without breaking the bank in terms of size!

Does any other database do this?

Other Types

We’ve also played around with some other types, including tuples and even dictionaries, all of which support a lexical ordering and range queries. We haven’t implemented them as we’re not sure about the use-cases, but it’s certainly interesting.

Even Smaller

Of course it’s possible to get smaller and faster on all of these datatypes with appropriate data structures, and we’re keen on plumbing the depths of tiny as TerminusDB matures. If you’ve a favourite encoding approach which is also fast and supports range queries, or can think of other interesting data types which can be supported, give us a shout!

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