BTree Index Prefetch

This document outlines how we decide to pre-read extra nodes in the btree index.

Rationale

Because of the latency involved in making a request, it is often better to make fewer large requests, rather than more small requests, even if some of the extra data will be wasted.

Example

Using my connection as an example, I have a max bandwidth of 160kB/s, and a latency of between 100-400ms to London, I’ll use 200ms for this example. With this connection, in 200ms you can download 32kB. So if you make 10 requests for 4kB of data, you spend 10*.2s = 2s sending the requests, and 4*10/160 = .25s actually downloading the data. If, instead, you made 3 requests for 32kB of data each, you would take 3*.2s = .6s for requests, and 32*3/160 = .6s for downloading the data. So you save 2.25 - 1.2 = 1.05s even though you downloaded 32*3-4*10 = 56kB of data that you probably don’t need. On the other hand, if you made 1 request for 480kB, you would take .2s for the request, and 480/160=3s for the data. So you end up taking 3.2s, because of the wasted 440kB.

BTree Structure

This is meant to give a basic feeling for how the btree index is laid out on disk, not give a rigorous discussion. For that look elsewhere[ref?].

The basic structure is that we have pages of 4kB. Each page is either a leaf, which holds the final information we are interested in, or is an internal node, which contains a list of references to the next layer of nodes. The layers are structured such that all nodes for the top layer come first, then the nodes for the next layer, linearly in the file.

Example 1 layer

In the simplest example, all the data fits into a single page, the root node. This means the root node is a leaf node.

Example 2 layer

As soon as the data cannot fit in a single node, we create a new internal node, make that the root, and start to create multiple leaf nodes. The root node then contains the keys which divide the leaf pages. (So if leaf node 1 ends with ‘foo’ and leaf node 2 starts with ‘foz’, the root node would hold the key ‘foz’ at position 0).

Example 3 layer

It is possible for enough leaf nodes to be created, that we cannot fit all there references in a single node. In this case, we again split, creating another layer, and setting that as the root. This layer then references the intermediate layer, which references the final leaf nodes.

In all cases, the root node is a single page wide. The next layer can have 2-N nodes.

Current Info

Empirically, we’ve found that the number of references that can be stored on a page varies from about 60 to about 180, depending on how much we compress, and how similar the keys are. Internal nodes also achieve approximately the same compression, though they seem to be closer to 80-100 and not as variable. For most of this discussion, we will assume each page holds 100 entries, as that makes the math nice and clean.

So the idea is that if you have <100 keys, they will probably all fit on the root page. If you have 100 - 10,000 keys, we will have a 2-layer structure, if you have 10,000 - 1,000,000 keys, you will have a 3-layer structure. 10^6-10^8 will be 4-layer, etc.

Data and Request

It is important to be aware of what sort of data requests will be made on these indexes, so that we know how to optimize them. This is still a work in progress, but generally we are searching through ancestry. The final information (in the leaf nodes) is stored in sorted order. Revision ids are generally of the form “prefix:committer@email-timestamp-randomtail”. This means that revisions made by the same person around the same time will be clustered, but revisions made by different people at the same time will not be clustered. For files, the keys are (file-id, revision-id) tuples. And file-ids are generally basename-timestamp-random-count (depending on the converter). This means that all revisions for a given file-id will be grouped together, and that files with similar names will be grouped together. However, files committed in the same revisions will not be grouped together in the index.[1]_

[1]

One interesting possibility would be to change file-ids from being ‘basename-…’, to being ‘containing-dirname-filename-…’, which would group files in the similarly named directories together.

In general, we always start with a request for the root node of the index, as it tells us the final structure of the rest of the index. How many total pages, what pages are internal nodes and what layer, which ones are leaves. Before this point, we do know the size of the index, because that is stored in the pack-names file.

Thoughts on expansion

This is just a bullet list of things to consider when expanding a request.

• We generally assume locality of reference. So if we are currently reading page 10, we are more likely to read page 9 or 11 than we are page 20.

• However, locality of reference only really holds within a layer. If we are reading the last node in a layer, we are unlikely to read the first node of the next layer. In fact, we are most likely to read the last node of the next layer.

  More directly, we are probably equally likely to read any of the nodes in the next layer, which could be referred to by this layer. So if we have a structure of 1 root node, 100 intermediate nodes, and 10,000 leaf nodes. They will have offsets: 0, 1-101, 102-10,102.

  If we read the root node, we are likely to want any of the 1-101 nodes (because we don’t know where the key points). If we are reading node 90, then we are likely to want a node somewhere around 9,100-9,200.

• When expanding a request, we are considering that we probably want to read on the order of 10 pages extra. (64kB / 4kB = 16 pages.) It is unlikely that we want to expand the requests by 100.

• At the moment, we assume that we don’t have an idea of where in the next layer the keys might fall. We could use a predictive algorithm assuming homogenous distribution. When reading the root node, we could assume an even distribution from ‘a-z’, so that a key starting with ‘a’ would tend to fall in the first few pages of the next layer, while a key starting with ‘z’ would fall at the end of the next layer. However, this is quite likely to fail in many ways. Specific examples:

  • Converters tend to use an identical prefix. So all revisions will start with ‘xxx:’, leading us to think that the keys fall in the last half, when in reality they fall evenly distributed.

  • When looking in text indexes. In the short term, changes tend to be clustered around a small set of files. Short term changes are unlikely to cross many pages, but it is unclear what happens in the mid-term. Obviously in the long term, changes have happened to all files.

  A possibility, would be to use this after reading the root node. And then using an algorithm that compares the keys before and after this record, to find what a distribution would be, and estimate the next pages.

  This is a lot of work for a potentially small benefit, though.

• When checking for N keys, we do sequential lookups in each layer. So we look at layer 1 for all N keys, then in layer 2 for all N keys, etc. So our requests will be clustered by layer.

• For projects with large history, we are probably more likely to end up with a bi-modal distribution of pack files. Where we have 1 pack file with a large index, and then several pack files with small indexes, several with tiny indexes, but no pack files with medium sized indexes. This is because a command like bzr pack will combine everything into a single large file. Commands like bzr commit will create an index with a single new record, though these will be packaged together by autopack. Commands like bzr push and bzr pull will create indexes with more records, but these are unlikely to be a significant portion of the history. Consider bzr has 20,000 revisions, a single push/pull is likely to only be 100-200 revisions, or 1% of the history.

  Note that there will always be cases where things are evenly distributed, but we probably shouldn’t optimize for that case.

• 64kB is 16 pages. 16 pages is approximately 1,600 keys.

• We are considering an index with 1 million keys to be very large. 10M is probably possible, and maybe 100M, but something like 1 billion keys is unlikely. So a 3-layer index is fairly common (it exists already in bzr), but a 4-layer is going to be quite rare, and we will probably never see a 5-layer.

• There are times when the second layer is going to be incompletely filled out. Consider an index with 101 keys. We found that we couldn’t fit everything into a single page, so we expanded the btree into a root page and a leaf page, and started a new leaf page. However, the root node only has a single entry. There are 3 pages, but only one of them is “full”. This happens again when we get near the 10,000 node barrier. We found we couldn’t fit the index in a single page, so we split it into a higher layer, and 1 more sub-layer. So we have 1 root node, 2 layer-2 nodes, and N leaf nodes (layer 3). If we read the first 3 nodes, we will have read all internal nodes.

  It is certainly possible to detect this for the first-split case (when things no-longer fit into just the root node), as there will only be a few nodes total. Is it possible to detect this from only the ‘size’ information for the second-split case (when the index no longer fits in a single page, but still fits in only a small handful of pages)?

  This only really works for the root + layer 2. For layers 3+ they will always be too big to read all at once. However, until we’ve read the root, we don’t know the layout, so all we have to go on is the size of the index, though that also gives us the explicit total number of pages. So it doesn’t help to read the root page and then decide. However, on the flip side, if we read before the split, then we don’t gain much, as we are reading pages we aren’t likely to be interested in.

  For example:

  We have 100 keys, which fits onto 100 pages, with a single root node. At 1,100 keys, it would be 101 leaf pages, which would then cause us to need 2 index pages, triggering an extra layer. However, this is very sensitive to the number of keys we fit per-page, which depends on the compression. Although, we could consider 2,000 keys. Which would be 200 leaf nodes, and 2 intermediate nodes, and a single root node. It is unlikely that we would ever be able to fit 200 references into a single root node.

  So if we pretend that we split at 1 page, 100 pages, and 10,000 pages. We might be able to say, at 1-5 pages, read all pages, for 5-100 pages, read only the root. At 100 - 500 pages, read 1-5 pages, for 500-10,000 read only the root. At 10,000-50,000 read 1-5 pages again, but above 50,000 read only the root. We could bias this a bit smaller, say at powers of 80, instead of powers of 100, etc. The basic idea is that if we are close to a layer split, go ahead and read a small number of extra pages.

• The previous discussion applies whenever we have an upper layer that is not completely full. So the pages referenced by the last node from the upper layer will often not have a full 100-way fan out. Probably not worthwhile very often, though.

• Sometimes we will be making a very small request for a very small number of keys, we don’t really want to bloat tiny requests. Hopefully we can find a decent heuristic to determine when we will be wanting extra nodes later, versus when we expect to find all we want right now.

Algorithm

This is the basic outline of the algorithm.

1. If we don’t know the size of the index, don’t expand as we don’t know what is available. (This only really applies to the pack-names file, which is unlikely to ever become larger than 1 page anyway.)

2. If a request is already wide enough to be greater than the number of recommended pages, don’t bother trying to expand. This only really happens with LocalTransport which recommends a single page.

3. Determine what pages have already been read (if any). If the pages left to read can fit in a single request, just request them. This tends to happen on medium sized indexes (ones with low hundreds of revisions), and near the end when we’ve read most of the whole index already.

4. If we haven’t read the root node yet, and we can’t fit the whole index into a single request, only read the root node. We don’t know where the layer boundaries are anyway.

5. If we haven’t read “tree depth” pages yet, and are only requesting a single new page don’t expand. This is meant to handle the ‘lookup 1 item in the index’ case. In a large pack file, you’ll read only a single page at each layer and then be done. When spidering out in a search, this will cause us to take a little bit longer to start expanding, but once we’ve started we’ll be expanding at full velocity. This could be improved by having indexes inform each other that they have already entered the ‘search’ phase, or by having a hint from above to indicate the same.

   However, remember the ‘bi-modal’ distribution. Most indexes will either be very small, or very large. So either we’ll read the whole thing quickly, or we’ll end up spending a lot of time in the index. Which makes a small number of extra round trips to large indexes a small overhead. For 2-layer nodes, this only ‘wastes’ one round trip.

6. Now we are ready to expand the requests. Expand by looking for more pages next to the ones requested that fit within the current layer. If you run into a cached page, or a layer boundary, search further only in the opposite direction. This gives us proper locality of reference, and also helps because when a search goes in a single direction, we will continue to prefetch pages in that direction.