This seems, at least upon first read, analogous to global value numbering (GVN). Or, depending on how you look at it, common subexpression elimination (CSE). I am mostly wondering why they are not mentioned in the article.
The canonical algorithm to do that is to compute the dominance relation. A node X dominates Y if every path to Y must go through X. Once you have computed the dominance relation, if a common subexpression is located at nodes N1, N2, N3, you can place the computation at some shared dominator of N1, N2, and N3. Because dominance is a statement about /all/ paths, there is a unique lowest dominator [1]. This is exactly the "lowest single common ancestor."
Note that dominance is also defined for cyclic graphs. There may be faster algorithms to compute dominance for acyclic graphs. Expressions in non-lazy programming languages are almost always acyclic (e.g. in Haskell, you can write cyclic expressions).
[1] Claim. Let A, B, and C be reachable nodes. Suppose A and B both dominate C. Then either A dominates B or B dominates A.
Proof. We prove the contrapositive. If neither A dominates B nor B dominates A, then there exist paths a, b from the root such that path a passes through A but not B and path b passes through B but not A. If there is no path from A to C, then A cannot dominate C as C is reachable. Similarly, if there is no path from B to C, then B cannot dominate C. So assume there are paths a' from A to C and b' B to C. Then the path b.b' witnesses that A does not dominate C, and the path a.a' witnesses that B does not dominate C.
(There might be a bug in the proof; I think I proved something too strong, but I'm going to bed.)
The algorithm in the article does O(1) queries with O(V+E) preprocessing (assuming linear-preprocessing LCA, which, yeah). What’s the best algorithm for dominator trees? People usually talk about Lengauer–Tarjan[2], which is linear in practice (linear except for UNION-FIND), and not the linear one by Georgiadis[3,4]. Unfortunately, I’m not a compiler person.
[1] https://doi.org/10.1016/j.ipl.2010.02.014
[2] https://maskray.me/blog/2020-12-11-dominator-tree
let $0 = f a b in
let $1 = g $0 c in
...
For an example, consider a three-node binary tree where R is the root, A is the left child, and B is the right child. A valid topological sort is R A B, but it is not the case that whenever B is computed, A has already been computed.
I don't think you necessarily have to compute the dominance relation because it can pop out implicitly.
If CSE is done on intermediate code, the dominance relation will pop out from thedirecton in which the instructions are followed around the basic blocks.
E.g. simple case: we see t3 <= t2 + t1. So we make a note in some CSE hash table that we had a t2 + t1, and that the result is cached in t3. Then if we see t2 + t1 again in the same basic block, we can replace that with t3. The dominance relation is that earlier instructions in the basic block dominate later ones, but we don't have to explicitly calculate it.
SSA transformations are essentially equivalent to what the author appears to be doing in terms of let-bindings [0].
I love this fact about the internet! Thanks guys! Keep it up! Including the snarkyness. It’s part of what makes it great!
(I am aware this is not a novel idea. Posting the wrong solution is better than asking for help.. It is just fun to see it in action)