How does 'git merge' work in details?

You might be best off looking for a description of a 3-way merge algorithm. A high-level description would go something like this:

1. Find a suitable merge base B - a version of the file that is an ancestor of both of the new versions (X and Y), and usually the most recent such base (although there are cases where it will have to go back further, which is one of the features of gits default recursive merge)
2. Perform diffs of X with B and Y with B.
3. Walk through the change blocks identified in the two diffs. If both sides introduce the same change in the same spot, accept either one; if one introduces a change and the other leaves that region alone, introduce the change in the final; if both introduce changes in a spot, but they don't match, mark a conflict to be resolved manually.

The full algorithm deals with this in a lot more detail, and even has some documentation (https://github.com/git/git/blob/master/Documentation/technical/trivial-merge.txt for one, along with the git help XXX pages, where XXX is one of merge-base, merge-file, merge, merge-one-file and possibly a few others). If that's not deep enough, there's always source code...

How does git perform when there are multiple common bases for merging branches?

This article was very helpful: http://codicesoftware.blogspot.com/2011/09/merge-recursive-strategy.html (here is part 2).

Recursive uses diff3 recursively to generate a virtual branch which will be used as the ancestor.

E.g.:

(A)----(B)----(C)-----(F)        |      |       |        |      |   +---+        |      |   |        |      +-------+        |          |   |        |      +---+   |        |      |       |        +-----(D)-----(E)

Then:

git checkout Egit merge F

There are 2 best common ancestors (common ancestors that are not ancestors of any other), C and D. Git merges them into a new virtual branch V, and then uses V as the base.

(A)----(B)----(C)--------(F)        |      |          |        |      |      +---+        |      |      |        |      +----------+        |      |      |   |        |      +--(V) |   |        |          |  |   |        |      +---+  |   |        |      |      |   |        |      +------+   |        |      |          |        +-----(D)--------(E)

I suppose Git would just continue with the if there were more best common ancestors, merging V with the next one.

The article says that if there is a merge conflict while generating the virtual branch Git just leaves the conflict markers where they are and continues.

What happens when I merge multiple branches at once?

As @Nevik Rehnel explained, it depends on the strategy, it is well explained on man git-merge MERGE STRATEGIES section.

Only octopus and ours / theirs support merging multiple branches at once, recursive for example does not.

octopus refuses to merge if there would be conflicts, and ours is a trivial merge so there can be no conflicts.

Those commands generate a new commit will have more than 2 parents.

I did one merge -X octopus on Git 1.8.5 without conflicts to see how it goes.

Initial state:

   +--B   |A--+--C   |   +--D

Action:

git checkout Bgit merge -Xoctopus C D

New state:

   +--B--+   |     |A--+--C--+--E   |     |   +--D--+

As expected, E has 3 parents.

TODO: how exactly octopus operates on a single file modifications. Recursive two-by-two 3-way merges?

How does git perform when there is no common base for merging branches?

@Torek mentions that since 2.9, merge fails without --allow-unrelated-histories.

I tried it out empirically on on Git 1.8.5:

git initprintf 'a\nc\n' > agit add .git commit -m agit checkout --orphan bprintf 'a\nb\nc\n' > agit add .git commit -m bgit merge master

a contains:

a<<<<<<< oursb=======>>>>>>> theirsc

Then:

git checkout --conflict=diff3 -- .

a contains:

<<<<<<< oursabc||||||| base=======ac>>>>>>> theirs

Interpretation:

• the base is empty
• when the base is empty, it is not possible to resolve any modification on a single file; only things like new file addition can be resolved. The above conflict would be solved on a 3-way merge with base a\nc\n as a single line addition
• I think that a 3-way merge without a base file is called a 2-way merge, which is just a diff

I'm interested too. I don't know the answer, but...

A complex system that works is invariably found to have evolved from a simple system that worked

I think git's merging is highly sophisticated and will be very difficult to understand - but one way to approach this is from its precursors, and to focus on the heart of your concern. That is, given two files that don't have a common ancestor, how does git merge work out how to merge them, and where conflicts are?

Let's try to find some precursors. From git help merge-file:

git merge-file is designed to be a minimal clone of RCS merge; that is,       it implements all of RCS merge's functionality which is needed by       git(1).

From wikipedia: http://en.wikipedia.org/wiki/Git_%28software%29 -> http://en.wikipedia.org/wiki/Three-way_merge#Three-way_merge -> http://en.wikipedia.org/wiki/Diff3 -> http://www.cis.upenn.edu/~bcpierce/papers/diff3-short.pdf

That last link is a pdf of a paper describing the diff3 algorithm in detail. Here's a google pdf-viewer version. It's only 12 pages long, and the algorithm is only a couple of pages - but a full-on mathematical treatment. That might seem a bit too formal, but if you want to understand git's merge, you'll need to understand the simpler version first. I haven't checked yet, but with a name like diff3, you'll probably also need to understand diff (which uses a longest common subsequence algorithm). However, there may be a more intuitive explanation of diff3 out there, if you have a google...

Now, I just did an experiment comparing diff3 and git merge-file. They take the same three input files version1 oldversion version2 and mark conflicts the way same, with <<<<<<< version1, =======, >>>>>>> version2 (diff3 also has ||||||| oldversion), showing their common heritage.

I used an empty file for oldversion, and near-identical files for version1 and version2 with just one extra line added to version2.

Result: git merge-file identified the single changed line as the conflict; but diff3 treated the whole two files as a conflict. Thus, sophisticated as diff3 is, git's merge is even more sophisticated, even for this simplest of cases.

Here's the actual results (I used @twalberg's answer for the text). Note the options needed (see respective manpages).

$ git merge-file -p fun1.txt fun0.txt fun2.txt

You might be best off looking for a description of a 3-way merge algorithm. Ahigh-level description would go something like this:    Find a suitable merge base B - a version of the file that is an ancestor ofboth of the new versions (X and Y), and usually the most recent such base(although there are cases where it will have to go back further, which is oneof the features of gits default recursive merge) Perform diffs of X with B andY with B.  Walk through the change blocks identified in the two diffs. If bothsides introduce the same change in the same spot, accept either one; if oneintroduces a change and the other leaves that region alone, introduce thechange in the final; if both introduce changes in a spot, but they don't match,mark a conflict to be resolved manually.<<<<<<< fun1.txt=======THIS IS A BIT DIFFERENT>>>>>>> fun2.txtThe full algorithm deals with this in a lot more detail, and even has somedocumentation (/usr/share/doc/git-doc/technical/trivial-merge.txt for one,along with the git help XXX pages, where XXX is one of merge-base, merge-file,merge, merge-one-file and possibly a few others). If that's not deep enough,there's always source code...

$ diff3 -m fun1.txt fun0.txt fun2.txt

<<<<<<< fun1.txtYou might be best off looking for a description of a 3-way merge algorithm. Ahigh-level description would go something like this:    Find a suitable merge base B - a version of the file that is an ancestor ofboth of the new versions (X and Y), and usually the most recent such base(although there are cases where it will have to go back further, which is oneof the features of gits default recursive merge) Perform diffs of X with B andY with B.  Walk through the change blocks identified in the two diffs. If bothsides introduce the same change in the same spot, accept either one; if oneintroduces a change and the other leaves that region alone, introduce thechange in the final; if both introduce changes in a spot, but they don't match,mark a conflict to be resolved manually.The full algorithm deals with this in a lot more detail, and even has somedocumentation (/usr/share/doc/git-doc/technical/trivial-merge.txt for one,along with the git help XXX pages, where XXX is one of merge-base, merge-file,merge, merge-one-file and possibly a few others). If that's not deep enough,there's always source code...||||||| fun0.txt=======You might be best off looking for a description of a 3-way merge algorithm. Ahigh-level description would go something like this:    Find a suitable merge base B - a version of the file that is an ancestor ofboth of the new versions (X and Y), and usually the most recent such base(although there are cases where it will have to go back further, which is oneof the features of gits default recursive merge) Perform diffs of X with B andY with B.  Walk through the change blocks identified in the two diffs. If bothsides introduce the same change in the same spot, accept either one; if oneintroduces a change and the other leaves that region alone, introduce thechange in the final; if both introduce changes in a spot, but they don't match,mark a conflict to be resolved manually.THIS IS A BIT DIFFERENTThe full algorithm deals with this in a lot more detail, and even has somedocumentation (/usr/share/doc/git-doc/technical/trivial-merge.txt for one,along with the git help XXX pages, where XXX is one of merge-base, merge-file,merge, merge-one-file and possibly a few others). If that's not deep enough,there's always source code...>>>>>>> fun2.txt

If you are truly interested in this, it's a bit of a rabbit hole. To me, it seems as deep as regular expressions, the longest common subsequence algorithm of diff, context free grammars, or relational algebra. If you want to get to the bottom of it, I think you can, but it will take some determined study.