Interference Graphs for SSA are Chordal
Background & Context
I ran into a very interesting idea a couple of weeks back: if you’re working with pure SSA (phi nodes and all) then you get to do optimal register allocation in polynomial time. You don’t get a free lunch though, because you will eventually have to translate out of SSA by mapping the phi nodes into copies; and doing that optimally is NP complete^{1}.
Please note that none of this is original work and optimal register allocation using SSA is a wellresearched topic. A set of references can be found in the bibliography section.
The key observation that allows this is that the interference graph^{2} you get from an SSA program^{3} is a chordal graph^{4}, and chordal graphs can be optimally colored in \(O\left(V + E\right)\) time. There are many ways to prove this, and one such way is presented here.
Proof
What are Chordal Graphs?
A graph is chordal if every cycle of length > 3 has a chord. Any (induced) subgraph of a chordal graph is again a chordal graph: if the vertices of the subgraph induce a cycle, they would also induce the chords present in the original chordal graph. Chordal graphs are useful because they are perfectly orderable^{5} – perfectly orderable graphs can be colored in polynomial time.
The Intersection Graph of Connected Subgraphs of a Tree
In “The intersection graphs of subtrees in trees are exactly the chordal graphs” Gavril^{6} completely characterizes chordal graphs as intereference graphs of subtrees of topological graphs^{7}. We do not need the full treatment of topological graphs since for our purposes it is sufficient to show that certain kinds of graphs are chordal (and not the other way around). So we’ll go for a restricted and simpler “proof” (essentially a simplified form of Gavril’s proof) that works with “normal” edgevertex graphs.
Consider a tree \(T\). We construct a graph \(G\) such that the vertices of that graph \(V(G)\) are connected subgraphs of \(T\) and two vertices \(n_1\) and \(n_2\) have an edge between them if the subtrees they correspond to are not disjoint (\(n_1 \cap n_2 \neq \emptyset\)). Then \(G\) is a chordal graph.
This construction of having sets as vertices, and edges between two vertices iff their intersection is nonempty is called an “intersection graph”. Another way of stating our assertion is “intersection graphs of connected subtrees of a tree are chordal”
I won’t present a full proof here, but a small example (that can be extended into a proof) to motivate why the statement above should be true.
Assume \(G\) (the intersection graph of connected subtrees of a tree, \(T\)) has a cycle of length 4 without a chord. Each of the edges, \(p_i\) correspond to a nonempty set of nodes in \(T\). We know that every vertex in \(p1 \cup p2\) belongs to the connected subtree corresponding to \(v2\). This means \(\forall n_{1} \in p1, \forall n_{2} \in p2\) there is a acyclic path between \(n_{1}\) and \(n_{2}\) in \(v2\). Same is true for \(\forall n_{2} \in p2, \forall n_{3} \in p3\) (the paths connecting these are in \(v3\)) and so on. Therefore, we can pick \(n_0 \in p0\), \(n_1 \in p1\), \(n_2 \in p2\) and \(n_3 \in p3\) such that there is a path \(n_0 \leftrightarrow n_1 \leftrightarrow n_2 \leftrightarrow n_3 \leftrightarrow n_0\). We will prove that the shortest of these paths is a cycle (i.e. has no repeated vertices), hence any tree containing \(v0\), \(v1\), \(v2\) and \(v3\) has a cycle. Since trees don’t contain cycles, we’ll have proved by contradiction that the above construct could not have been the intersection graph of connected subtrees of a tree. Hence all cycles of length 4 in an intersection graph of connected subtrees of a tree will have a chord. A full proof of the assertion can be constructed by extending this argument to cycles of length greater than 4; and a yet more general proof can be found in Gavril’s paper^{6} as mentioned earlier.
To prove that the path \(n_0 \leftrightarrow n_1 \leftrightarrow n_2 \leftrightarrow n_3 \leftrightarrow n_0\) has no repeated vertices (and hence is a cycle), first note that the path \(n_0 \leftrightarrow n_1\) and \(n_2 \leftrightarrow n_3\) are disjoint, as otherwise \(v1\) and \(v3\) would have a vertex in common and hence this 4cycle would have had a chord between \(v1\) and \(v3\). \(n_1 \leftrightarrow n_2\) and \(n_3 \leftrightarrow n_0\) are disjoint for the same reason. Therefore the only possible repeated vertex will have to appear in a path connecting two vertices \(n_i\) and \(n_{(i+1) \% 4}\). Let that repeated vertex be \(n_r\), making the path between \(n_i\) and \(n_{(i+1) \% 4}\) of the form \(n_i \leftrightarrow P_0 \leftrightarrow n_r \leftrightarrow P_1 \leftrightarrow n_r \leftrightarrow P_2 \leftrightarrow n_{(i+1) \% 4}\) where \(P_i\) are paths themselves. But this means that there is a shorter path between \(n_i\) and \(n_{(i+1) \% 4}\), \(n_i \leftrightarrow P_0 \leftrightarrow n_r \leftrightarrow P_2 \leftrightarrow n_{(i+1) \% 4}\), also contained in \(v(i+1\%4)\). Therefore \(n_0 \leftrightarrow n_1 \leftrightarrow n_2 \leftrightarrow n_3 \leftrightarrow n_0\) is not the shortest path for the given \(n_0 \in p0\), \(n_1 \in p1\), \(n_2 \in p2\) and \(n_3 \in p3\). Hence in the shortest possible \(n_0 \leftrightarrow n_1 \leftrightarrow n_2 \leftrightarrow n_3 \leftrightarrow n_0\) there can be no such repetition; and such a shortest path is a cycle.
Dominator Trees and Chordal Graphs
Consider the dominator tree graph^{8} of the SSA program being register allocated (with the tree denoting the usual “use dominates def” relation). The live range of a def is the union of all paths from that def to any use of that def^{9}. We denote such a path as the set of SSA instructions that would be executed if the program follows that control flow, including the def but excluding the use. Thus each live range corresponds to a subset of the dominator tree (where each vertex is an SSA instruction). If we can show that a live range corresponds to a connected subtree of the dominator tree, we can use the result from the previous section to state that the interference graph of the program being register allocated is chordal (since it is the intersection graph with connected subtrees as vertices) and can be colored in polynomial time. Note that while a dominator tree is semantically a directed tree we don’t need to use that additional structure to invoke the above result.
To show that the live range of an SSA value is always a connected subtree, we exploit two properties of SSA values:

the live range of a def only contains instructions it dominates. If it contained an instruction it does not dominate, then by the definition of a live range, we’ve discovered a path from a def to a use that contains an instruction the def does not dominate. This means there is a path form the entry of the control flow graph to the use that does not contain the def. This means the def does not dominate one of its uses, and this cannot happen in a wellstructured SSA control flow graph.

if the live range of a def \(D\) contains \(L_{end}\) and there is an SSA instruction \(L_{middle}\) such that \(L_{middle}\) dominates \(L_{end}\) then the live range of \(D\) contains \(L_{middle}\). This is true since every path from \(D\) to \(L_{end}\) contains \(L_{middle}\).
(1) tells us that the live range of a def when represented in the dominator tree is a subset of the section of the dominator tree rooted at def. (2) tells us that this subset is really a connected subtree – every instruction in the path leading down from the def to an instruction contained in the def’s live range belongs to the def’s live range.
Conclusion
There is an existing compiler IR that uses chordality of SSA’s interference graphs to do optimal register allocation: libFirm http://pp.ipd.kit.edu/firm/. I’d like to spend some time looking at that next.
Bibliography

Brisk, Philip, et al. “Optimal register sharing for highlevel synthesis of SSA form programs.” ComputerAided Design of Integrated Circuits and Systems, IEEE Transactions on 25.5 (2006): 772779.

Sebastian Hack’s PhD thesis: http://digbib.ubka.unikarlsruhe.de/volltexte/documents/6532

“SSAbased Register Allocation” http://www.cdl.unisaarland.de/projects/ssara/

libFirm has an SSAbased register allocator http://pp.ipd.kit.edu/firm/

Rastello, Fabrice, F. de Ferrière, and Christophe Guillon. “Optimizing translation out of SSA using renaming constraints.” Proceedings of the international symposium on Code generation and optimization: feedbackdirected and runtime optimization. IEEE Computer Society, 2004. ↩

https://en.wikipedia.org/wiki/Static_single_assignment_form ↩

Gavril, F. “The intersection graphs of subtrees in trees are exactly the chordal graphs.” Journal of Combinatorial Theory, Series B 16.1 (1974): 4756. ↩ ↩^{2}

This is assuming that the def is reachable from the entry of the control flow graph. We do not consider unreachable (dead) defs to keep the discussion simple. ↩