# Part 5: Pseudo-edges

This is the fifth (and final planned) post in a series on some new ideas in version control. To start at the beginning, go here.

The goal of this post is to describe pseudo-edges: what they are, how to compute them efficiently, and how to update them efficiently upon small changes. To recall the important points from the last post:

• We (pretend, for now, that we) represent the state of the repository as a graph in memory: one node for every line, with a directed edges that enforce ordering constraints between two lines. Each line has a flag that says whether it is deleted or not.
• The current output of the repository consists of just those nodes that are not deleted, and there is an ordering constraint between two nodes if there is a path in the graph between them, but note that the path is allowed to go through deleted nodes.
• Applying a patch to the repository is very efficient: the complexity of applying a patch is proportional to the number of changes it makes.
• Rendering a the current output to a file is potentially very expensive: its complexity requires traversing the entire graph, including nodes that are marked as deleted. To the extent we can, we’d like to reduce this complexity to the number of live nodes in the graph.

The main idea for solving this is to add “pseudo-edges” to the graph: for every path that connects two live nodes through a sequence of deleted nodes, add a corresponding edge to the graph. Once this is done, we can render the current output without traversing the deleted parts of the graph, because every ordering contraint that used to depend on some deleted parts is now represented by some pseudo-edge. Here’s an example: the deleted nodes are in gray, and the pseudo-edge that they induce is the dashed arrow. We haven’t really solved anything yet, though: once we have the pseudo-edges, we can efficiently render the output, but how do we compute the pseudo-edges? The naive algorithm (look at every pair of live nodes, and check if they’re connected by a path of deleted nodes) still depends on the number of deleted nodes. Clearly, what we need is some sort of incremental way to update the pseudo-edges.

# Deferring pseudo-edges

The easiest way that we can reduce the amount of time required for computing pseudo-edges is simply to do it rarely. Specifically, remember that applying a patch can be very fast, and that pseudo-edges only need to be computed when outputting a file. So, obviously, we should only update the pseudo-edges when it’s time to actually output the file. This sounds trivial, but it can actually be significant. Imagine, for example, that you’re cloning a repository that has a long history; let’s say it has `n` patches, each of which has a constant size, and let’s assume that computing pseudo-edges takes time `O(m)`, where `m` is the size of the history. Cloning a repository involves downloading all of those patches, and then applying them one-by-one. If we recompute the pseudo-edges after every patch application, the total amount of time required to clone the repository is `O(n^2)`; if we apply all the patches first and only compute the pseudo-edges at the end, the total time is `O(n)`.

You can see how `ojo` implements this deferred pseudo-edge computation here: first, it applies all of the patches; then it recomputes the pseudo-edges.

# Connected deleted components

Deferring the pseudo-edge computation certainly helps, but we’d also like to speed up the computation itself. The main idea is to avoid unnecessary recomputation by only examining parts of the graph that might have actually changed. At this point, I need to admit that I don’t know whether what I’m about to propose is the best way of updating the pseudo-edges. In particular, its efficiency rests on a bunch of assumptions about what sort of graphs we’re likely to encounter. I haven’t made any attempt to test these assumptions on actual large repositories (although that’s something I’d like to try in the future).

The main assumption is that while there may be many deleted nodes, they tend to be collected into a large number of connected components, each of which tends to be small. What’s more, each patch (I’ll assume) tends to only affect a small number of these connected components. In other words, the plan will be:

• keep track (incrementally) of connected components made up of deleted nodes,
• when applying or reverting a patch, figure out which connected components were touched, and only recompute paths among the live nodes that are on the boundary of one of the dirty connected components.

Before talking about algorithms, here are some pictures that should help unpack what it is that I actually mean. Here is a graph containing three connected components of deleted nodes (represented by the rounded rectangles): When I delete node `h`, it gets added to one of the connected components, and I can update relevant pseudo-edges without looking at the other two connected components: If I delete node `d` then it will cause all of the connected components to merge: This isn’t hard to handle, it just means that we should run our pseudo-edge-checking algorithm on the merged component.

# Maintaining the components

To maintain the partition of deleted nodes into connected components, we use a disjoint-set data structure. This is very fast (pretty close to constant time) when applying patches, because applying patches can only enlarge deleted components. It’s slower when reverting patches, because the disjoint-set algorithm doesn’t allow splitting: when reverting patches, connected components could split into smaller ones. Our approach is to defer the splitting: we just mark the original connected component as dirty. When it comes time to compute the pseudo-edges, we explore the original component, and figure out what the new connected pieces are.

The disjoint-set data structure is implemented in the `ojo_partition` subcrate. It appears in the `Graggle` struct; note also the `dirty_reps` member: that’s for keeping track of which parts in the partition have been modified by a patch and require recomputing pseudo-edges.

We recompute the components here. Specifically, we consider the subgraph consisting only of nodes that belong to one of the dirty connected components. We run Tarjan’s algorithm on that subgraph to find out what the new connected components are. On each of those components, we recompute the pseudo-edges.

# Recomputing the pseudo-edges

The algorithm for this is: after deleting the node, look at the deleted connected component that it belongs to, including the “boundary” consisting of live nodes: Using depth-first search, check which of the live boundary nodes (in this case, just `a` and `i`) are connected by a path within that component (in this case, they are). If so, add a pseudo-edge. The complexity of this algorithm is `O(nm)`, where `n` is the number of boundary nodes, and `m` is the total number of nodes in the component, including the boundary (because we need to run `n` DFSes, and each one takes `O(m)` time). The hope here is that `m` and `n` are small, even for large histories. For example, I hope that `n` is almost always 2; at least, this is the case if the final live graph is totally ordered.

This algorithm is implemented `here`.

# Unapplying, and pseudo-edge reasons

There’s one more wrinkle in the pseudo-edge computation, and it has to do with reverting patches: if applying a patch created a pseudo-edge, removing a patch might cause that pseudo-edge to get deleted. But we have to be very careful when doing so, because a pseudo-edge might have multiple reasons for existing. You can see why in this example from before: The pseudo-edge from `a` to `d` is caused independently by the both the `b -> c` component and the `cy -> cl -> e` component. If by unapplying some patch we destroy the `b -> c` component but leave the `cy -> cl -> e` component untouched, we have to be sure not to delete the pseudo-edge from `a` to `d`.

The solution to this is to track to “reasons” for pseudo-edges, where each “reason” is a deleted connected component. This is a many-to-many mapping between connected deleted components and pseudo-edges, and it’s stored in the `pseudo_edge_reasons` and `reason_pseudo_edges` members of the `GraggleData` struct. Once we store pseudo-edge reasons, it’s easy to figure out when a pseudo-edge needs deleting: whenever its last reason becomes obsolete.

# Pseudo-edge spamming: an optimization

We’ve finished describing `ojo`’s algorithm for keeping pseudo-edges up to date, but there’s stil room for improvement. Here, I’ll describe a potential optimization that I haven’t implemented yet. It’s based on a simple, but non-quite-correct, algorithm for adding pseudo-edges incrementally: every time you mark a node as deleted, add a pseudo-edge from each of its in-neighbors to each of its out-neighbors. I call this “pseudo-edge spamming” because it just eagerly throws in as many pseudo-edges as needed. In pictures, if we have this graph and we delete the “deleted” line, then we’ll add a pseudo-edge from the in-neighbor of “deleted” (namely, “first”) to the out-neighbor of “deleted” (namely, “last”).

This algorithm has two problems. The first is that it isn’t complete: you might also need to add pseudo-edges when adding an edge where at least one end is deleted. Consider this example, where our graph consists of two disconnected parts. If we add an edge from “deleted 1” to “deleted 2”, clearly we also need to add a pseudo-edge between each of the “first” nodes and each of the “last” nodes. In order to handle this case, we really do need to explore the deleted connected component (which could be slow).

The second problem with our pseudo-edge spamming algorithm is that it doesn’t handle reverting patches: it only describes how to add pseudo-edges, not delete them.

The nice thing about pseudo-edge spamming is that even if it isn’t completely correct, it can be used as a fast-path in the correct algorithm: when applying a patch, if it modifies the boundary of a deleted connected component that isn’t already dirty, use pseudo-edge spamming to update the pseudo-edges (and don’t mark the component as dirty). In every other case, fall back to the previous algorithm.