Buffered writes and compressed spines in qp tries

Based on a conversation with @tef on Twitter, inspired by https://www.percona.com/files/presentations/percona-live/PLMCE2012/PLMCE2012-The_Right_Read_Optimization_is_Actually_Write_Optimization.pdf

@tef wrote,

so there's this idea of a cache-oblivious b-trie

and the idea is that you sorta batch writes to get better performance

that and "linear search of arrays is very fast if you've already loaded it into memory"

so this would be "take a b-trie, add a fixed length insert buffer" "write into the buffer first and cascade on overflow" "reads go normally but check insert buffer at each depth"

so you could imagine having a qp-trie where there's a fixed size buffer attached to each node

writes cascade down eventually, which reduces allocation/resizing traffic.

so i was wondering, what if you used the trailing space in the pop count array for inserts into the trie

insert becomes: append to root node, then on full, do real insert into child node & so on

inserting the first 16 elements, it would just fill up the root bucket, adding would flip it into a qp-trie node, and allocate childs

you could even do something like having "fixed width nodes" i.e an array with a popcount at the front, the child nodes as normal, and then a write buffer growing from the other end of the array

the idea being you normally insert into the write buffer until it's full, then merge it into the popcount'd array at the front, clearing the merge buffer

well for a start: small tries are kept short, there's much less allocation dancing for things in the insert buffer

the next idea is that you can probably allocate nodes in "power of 2" sizes, free list for each

so when you hit a full buffer, you can pull out something from the free list, update it, and return the old one to the free list

so there might be less pain on the allocator, or smoother progress

but the real idea is that this would make a trie much more cache friendly

but i have no idea if that would be true in practice

There are lots of good ideas in here! Before fleshing it out, a couple of background comments.

Allocation

One of the weaknesses of qp-tries, at least in my proof-of-concept implementation, is the allocator is called for every insert or delete. I have not bothered to implement a custom allocator, since internally malloc() already maintains per-node-size free lists and rounds up allocation sizes, and I haven't had any good ideas for improving on that.

There's also a trade-off between small memory size, and realloc() calls. I have erred on the side of optimizing for small size at the cost of greater allocator stress.

@tef's suggestion of adding write buffers could be a great way to change this trade-off.

Search trees vs. radix tries

To search a qp-trie, you only need to look at the trie nodes and the key you are searching for. You do not need to look at any other keys until you get to a leaf, where you compare the final key from the trie with your search key. Stored keys are off to the side of the trie, and cold keys remain untouched.

However in search trees you need to compare the search key against the tree keys at each point as you traverse the tree. In B-trees, keys are kept inline in the tree, and keys near the root of the tree are hot even if they are never queried.

It would be nice to avoid adding lots of extra full-key comparisons to a qp-trie search.

Terminology

Quick reminder of qp-trie structure to make sure we're using consistent terminology.

A "node" is a two-word structure.

A "leaf" node pairs a key pointer and a value pointer.

A "branch" node contains an index (an offset of a nybble in a key), a bitmap denoting which nybble values are present in the trie, and a pointer to an array of nodes, called the branch's "twigs".

    :           :       :
    |           |       |
    |           |       |         +----------+----------+
    | index bmp | twigs |  ---->  |          |          |
    |           |       |         |          |          |
    |           |       |         :          :          :
    :           :       :

Tricky points

The branch node and its twig array are logically a single structure, except for the oddity that the metadata about the twig array is put next to the pointer rather than next to the array. It would perhaps have been more helpful (specially for comparison with other kinds of tree) if I had uses the term "node" for this logical structure as a whole. I'll call it a "logical node" below.

There's an important invariant that the prefixes of all of the keys are equal up to the index of their common parent branch node.

Write buffers

Let's add extra space to each twig array to use as a write buffer.

There are two different kinds of insert into a qp-trie:

When inserting a new logical node, we need four extra words:

With a write buffer, we need to use three words:

The latter will be used to speed up searching and later to expand the write buffer into normal qp nodes.

It is OK that three words doesn't match the uniform two word qp leaf/branch node size, because the write buffer does not need to be polymorphic.

The index of the new logical node is between the indexes of its would-be parent and child. It's best to add it to the child's write buffer; it would be more awkward to add it to the parent's write buffer, since then we would have to record and maintain which twig each write buffer entry relates to.

In a normal branch node, the bitmap relates nybble values to twig array entries. The metadata needs to be different in a write buffer entry: we need to know which nybble value belongs to the write buffer leaf key, and which nybble value matches the child twigs. (A bitmap would tell us the values but not which is which.) The twig nybble isn't needed when searching, but it is needed when building the branch bitmap when the write buffer is expanded to proper branch nodes.

Search algorithm

When we are following a branch, and before we look at its twig array, scan the entries in its write buffer. Write buffer entries have smaller index values than the index of the twig array, so need checking first. Write buffer entries need to be scanned in index order.

For each write buffer entry, get a nybble from the search key corresponding to the write buffer's index.

After running out of write buffer entries, index the twig array as in a normal qp trie search.

NOTE: for efficient memory pipelining, the actual order of operations should be:

An insertion gotcha

When adding an entry to the write buffer, there is a problem if the new entry has the same index and leaf nybble as an existing entry.

This would require the search code to do a full comparison against more than one key, making the innermost search loop more complicated.

After the write buffer is expanded, these two leaves will be off on a side branch: they aren't on the spine between the parent and child nodes. A three-word write buffer entry contains enough information to construct a branch node on the spine when that becomes necessary. But there isn't enough space to record the information about side branches. It would be possible to reconstruct side-branch indexes and bitmaps by re-scanning the keys; this is unnecessary in other situations.

It's probably preferable to avoid this gotcha by expanding the write buffer when a new entry collides with an existing one, as if there were no space for the new entry.

How to expand a write buffer

Some observations:

This suggests that it's usually better to keep a spine in write buffer layout as much as possible. When we run out of space in an allocation (too many twigs or too many write buffer entries), just make it bigger, without changing the layout.

The layout needs to change when the spine grows a side-branch, as noted in the previous section.

Another case where it could make sense to change the layout is when there are many twigs at one point on the spine. e.g.

    +--------------+-------+
    | i=4 bmp wb=3 | twigs |
    +--------------+-------+
                       |
                       V
              +-----------+-------+
              |   child   |  ptr  |
              |   child   |  ptr  |
              +-----------+-------+-------+
    wr buf    | i=2 nib=1 |  key  | value |
              | i=2 nib=3 |  key  | value |
              | i=2 nib=5 |  key  | value |
              +-----------+-------+-------+

At some point when the fanout gets wide enough a classic qp indexed indirection will be faster than a linear scan. (benchmarks needed!) After expansion the layout becomes:

    +--------------+-------+
    | i=2 bmp wb=0 | twigs |
    +--------------+-------+
                       |
                       V
         +--------------+-------+
         |     key      | value |
         | 1=4 bmp wb=0 | twigs | --> +-----------+-------+
         |     key      | value |     |   child   |  ptr  |
         |     key      | value |     |   child   |  ptr  |
         +--------------+-------+     +-----------+-------+

Alternative point of view

After the previous section I think we have strayed from tef's original suggestion to a different area of the design space.

Instead of a write buffer, we have a compressed spine.

This change of perspective is useful, because it allows us to separate the idea of spine compression from the idea of using larger allocations to avoid realloc() calls.

A compressed spine reduces the number of indirections needed to scan past a sequence of leaves.

It can also save space if the spine has at most one leaf at each index (not counting unused space in the write buffer). It uses the same space as a classic qp trie layout if there are two leaves at a particular index, and costs a word more each time the fanout increases.

So compressed spines can be better than classic qp tries, even when it is important to minimize space overhead. In this case, use the compressed spine layout, but do not leave any write buffer space, and realloc() eagerly.

Performance handwaving

Good points:

Bad points:

Overall my guess is this will favour update-heavy workloads at the expense of lookup-heavy workloads, though the change in depth is likely to lead to surprises.


Written by Tony Finch dot@dotat.at http://dotat.at/; You may do anything with this. It has no warranty. http://creativecommons.org/publicdomain/zero/1.0/