LCA Merge

by Aaron Bentley

Essential characteristics

In the general case (no criss-cross), it is a three-way merge. When there is a criss-cross at the tree level, but not for the particular file, it is still a three-way merge. When there’s a file-level criss-cross, it’s superior to a three-way merge.


First, we compare the files we are trying to merge, and find the lines that differ. Next, we try to determine why they differ; this is essential to the merge operation, because it affects how we resolve the differences. In this merger, there are three possible outcomes:

  1. The line was added in this version: “new-this”

  2. The line was deleted in the other version: “killed-other”

  3. The line was preserved as part of merge resolution in this version, but deleted in the other version: “conflicted-this”

Option 3 is new, but I believe it is essential. When each side has made a conflicting merge resolution, we should let the user decide how to combine the two resolutions, i.e. we should emit a conflict. We cannot silently drop the line, or silently keep the line, which can happen if we choose options 1 or 2. If we choose options 1 or 2, there’s also a possibility that a conflict will be produced, but no guarantee. We need a guarantee, which is why we need a new possible outcome.

To decide whether a line is “new-this”, “killed-other” or “conflicted-this”, we compare this version against the versions from each “least common ancestor” (LCA), in graph terminology. For each LCA version, if the line is not present in the LCA version, we add it to the “new” set. If the line is present in the LCA version, we add it to the “killed” set.

When we are done going through each LCA version, each unique line will be in at least one of the sets. If it is only in the “new” set, it’s handled as “new-this”. If it is only in the “killed” set, it’s handled as “killed-other”. If it is in both sets, it’s handled as “conflicted-this”.

The logic here is a bit tricky: first, we know that the line is present in some, but not all, LCAs. We can assume that all LCAs were produced by merges of the same sets of revisions. That means that in those LCAs, there were different merge resolutions. Since THIS and OTHER disagree about whether the line is present, those differences have propagated into THIS and OTHER. Therefore, we should declare that the lines are in conflict, and let the user handle the issue.

LCA merge and Three-way merge

Now, in the common case, there’s a single LCA, and LCA merge behaves as a three-way merge. Since there’s only one LCA, we cannot get the “conflicted-this” outcome, only “new-this” or “killed-other. Let’s look at the typical description of three-way merges:

























Now, let’s assume that BASE is a common ancestor, as is typically the case. In fact, for best-case merges, BASE is the sole LCA.

We always pick the version that represents a change from BASE, if there is one. For the AAAA line, there is no change, so the output is rightfully BASE/THIS/OTHER. For ABAA, the THIS and OTHER are changes from BASE, and they are the same change so they both win. (This case is sometimes called convergence.) For ABBA, THIS is a change from BASE, so THIS wins. For AABB, OTHER is a change from BASE, so OTHER wins. For ABC*, THIS and OTHER are both changes to BASE, but they are different changes, so they can’t both win cleanly. Instead, we have a conflict.

Now in three-way merging, we typically talk about regions of text. In weave/knit/newness/lca merge, we also have regions. Each contiguous group of “unchanged” lines is a region, and the areas between them are also regions.

Let’s assign a to THIS and b to OTHER. “unchanged” regions represent the AAAA or ABAA cases; it doesn’t matter which, because the outcome is the same regardless. Regions which consist of only “new-a” or “killed-a” represent the ABBA case. Regions which consist of only “new-b” or “killed-b” represent the AABB case. Regions which have (new-a or killed-a) AND (new-b or killed-b) are the ABC* case– both sides have made changes, and they are different changes, so a conflict must be emitted.

This is what I mean when I say that it is a three-way merge in the common case; if there is only one LCA, then it is merely an alternative implementation of three-way. (One that happens to automatically do --reprocess, ftw).

Exception to three-way behavior

There is a special case of three-way merge which LCA merge handles differently from our default “merge3” algorithm: BASE has content X, THIS deletes the content, and OTHER changes X to Y. In this case, LCA merge emits Y in its output and does not indicate a conflict. merge3 would output Y, but would also indicate a conflict. (This is also the behavior in the inverse case where OTHER has nothing and THIS has Y.)

This behavior is due the way LCA determines basic conflicts; they can only be emitted when THIS and OTHER each have unique lines between common lines. If THIS does not have unique lines in this position, conflicts will not be emitted, even if its (lack of) content is unique.

This behavior difference is shared with “weave” merge. I hope that a future revision of LCA merge will handle this case as merge3 would.

Why a new name

  1. It was time. Although knit / annotate merge and newness merge have tried to emulate the behavior of the original weave merge algorithm, --merge-type=weave hasn’t been based on weaves for a long time.

  2. Behavior differences. This algorithm should behave like a three-way merge in the common case, while its predecessors did not. It also has explicit support for handling conflicting merge resolutions, so it should behave better in criss-cross merge scenarios.


Unlike the current “weave” merge implementation, lca merge does not perform any whole-history operations. LCA selection should scale with the number of uncommon revisions. Text comparison time should scale mO(n2), where m is the number of LCAs, and n is the number of lines in the file. The current weave merge compares each uncommon ancestor, potentially several times, so it is >= kO(n2), where k is the number of uncommon ancestors. So “lca” should beat “weave” both in history analysis time and in text comparison time.

Possible flaws

  1. Inaccurate LCA selection. Our current LCA algorithm uses Graph.heads(), which is known to be flawed. It may occasionally give bad results. This risk is mitigated by the fact that the per-file graphs tend to be simpler than the revision graph. And since we’re already using this LCA algorithm, this is not an additional risk. I hope that John Meinel will soon have a fixed version of Graph.heads for us.

  2. False matches. Weaves have a concept of line identity, but knits and later formats do not. So a line may appear to be common to two files, when in fact it was introduced separately into each for entirely different reasons. This risk is the same for three-way merging. It is mitigated by using Patience sequence matching, which a longest-common-subsequence match.


I think this could be a great merge algorithm, and a candidate to make our default, but this work would not have been possible without the work of others, especially:

  • Martin Pool’s weave merge and knit/annotate merge algorithms.

  • Bram Cohen’s discussions of merge algorithms

  • Andrew Tridgell’s dissection of BitKeeper merge

  • Nathaniel Smith’s analysis of why criss-cross histories necessarily produce poor three-way merges.