[PATCH] D35851: [Dominators] Include infinite loops in PostDominatorTree

[Daniel Berlin via Phabricator via llvm-commits]

dberlin added a comment.

A few things, just so they end up here:
The NTSCD folks, unfortunately, seem to prove all of the algorithms for weak order dependence are equivalent to those that require with a unique end node.  See, http://people.cs.ksu.edu/~tamtoft/Papers/Amt+al:WODvsCD/short.pdf .  This means, i believe, outside of the NTSCD algorithms, we won't have a lot of luck with infinite loops.

The NTSCD paper mentions the augmentation we perform here in section 3.2 as being standard (see http://people.cs.ksu.edu/~tamtoft/Papers/Ran+Amt+Ban+Dwy+Hat:ESOP-2005/long.pdf).
They sadly point out, also, that "All definitions of control dependences that we are aware of require that CFGs satisfy the unique end node requirement"
(IE there is no magic we can use prior to this paper)

On the nice side, the NTSCD algorithms do not look that hard to implement.
On the not nice side, all the algorithms i can find are N^3 or N^4.

There was one paper author who believed significantly more efficient algorithms may exist (http://www.sciencedirect.com/science/article/pii/S0304397511007377).
He is, unfortunately, dead (not kidding).

I found one real world implementation of NTSCD (that doesn't use a published algorithm) (Joana, an information flow tool. Source on github)

However, now for some mixed news:
I'm positive you can prove any NTSCD algorithm based on all-pairs reachability of the CFG must be at least O(N*E) (IE N^2).
CFGs are a regular language (AFAIK!). The fastest known algorithms for all-pairs reachability on regular languages are (N*E)

Proof:
CFGs are trivially convertible to deterministic finite automata (uniquely label, generation transition table from successors, done)
All DFAs are trivially convertible to CFGs (again, AFAIK)
The language accepted by DFAs are exactly the set of regular languages.
Yannakakis gives a N*E algorithm for all-pairs reachability on regular languages (there is no faster known algorithm)

The good news is that it should be trivial to improve that part of the NTSCD algorithm (which is the N^3 part), *but* until they have algorithms that do not require all-pairs reachability, i do not believe you can make it faster than N^2.

I feel like you can also make this practically faster through various tricks that would not change complexity.  IE you should be able to avoid all pairs reachability for the parts of the graph that don't go through loops (IE generate condensation graph with marked condensed regions to get the maximal part of the cfg that does not go through a loop, and handle these separately)


https://reviews.llvm.org/D35851