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.

Algorithm
---------

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 propogated
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:

+-----+------+-------+------------+
|THIS | BASE | OTHER | OUT        |
+-----+------+-------+------------+
|A    | A    | A     | A          |
+-----+------+-------+------------+
|A    | B    | A     | A          |
+-----+------+-------+------------+
|A    | B    | B     | A          |
+-----+------+-------+------------+
|A    | A    | B     | B          |
+-----+------+-------+------------+
|A    | B    | C     |\*conflict\*|
+-----+------+-------+------------+

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.

Performance
-----------

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(n\ :sup:`2`\ ), 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(n\ :sup:`2`\ ), 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.

Acknowledgements
================

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.
