Some parallel environments execute threads in groups that allow
communication within the group using special primitives called
convergent operations. The outcome of a convergent operation is
sensitive to the set of threads that executes it “together”, i.e.,
convergently.
A value is said to be uniform across a set of threads if it is the
same across those threads, and divergent otherwise. Correspondingly,
a branch is said to be a uniform branch if its condition is uniform,
and it is a divergent branch otherwise.
Whether threads are converged or not depends on the paths they take
through the control flow graph. Threads take different outgoing edges
at a divergent branch. Divergent branches constrain
program transforms such as changing the CFG or moving a convergent
operation to a different point of the CFG. Performing these
transformations across a divergent branch can change the sets of
threads that execute convergent operations convergently. While these
constraints are out of scope for this document, the described
uniformity analysis allows these transformations to identify
uniform branches where these constraints do not hold.
Convergence and
uniformity are inter-dependent: When threads diverge at a divergent
branch, they may later reconverge at a common program point.
Subsequent operations are performed convergently, but the inputs may
be non-uniform, thus producing divergent outputs.
Uniformity is also useful by itself on targets that execute threads in
groups with shared execution resources (e.g. waves, warps, or
subgroups):
Uniform outputs can potentially be computed or stored on shared
resources.
These targets must “linearize” a divergent branch to ensure that
each side of the branch is followed by the corresponding threads in
the same group. But linearization is unnecessary at uniform
branches, since the whole group of threads follows either one side
of the branch or the other.
This document presents a definition of convergence that is reasonable
for real targets and is compatible with the currently implicit
semantics of convergent operations in LLVM IR. This is accompanied by
a uniformity analysis that extends the existing divergence analysis
[DivergenceSPMD] to cover irreducible control-flow.
Julian Rosemann, Simon Moll, and Sebastian
Hack. 2021. An Abstract Interpretation for SPMD Divergence on
Reducible Control Flow Graphs. Proc. ACM Program. Lang. 5, POPL,
Article 31 (January 2021), 35 pages.
https://doi.org/10.1145/3434312
Two paths in a CFG are said to be disjoint if the only nodes common
to both are the start node or the end node, or both.
Join node
A join node of a branch is a node reachable along disjoint paths
starting from that branch.
Diverged path
A diverged path is a path that starts from a divergent branch and
either reaches a join node of the branch or reaches the end of the
function without passing through any join node of the branch.
Each occurrence of an instruction in the program source is called a
static instance. When a thread executes a program, each execution of
a static instance produces a distinct dynamic instance of that
instruction.
Each thread produces a unique sequence of dynamic instances:
The sequence is generated along branch decisions and loop
traversals.
Starts with a dynamic instance of a “first” instruction.
Continues with dynamic instances of successive “next”
instructions.
Threads are independent; some targets may choose to execute them in
groups in order to share resources when possible.
1
2
3
4
5
6
7
8
9
Thread 1
Entry1
H1
B1
L1
H3
L3
Exit
Thread 2
Entry1
H2
L2
H4
B2
L4
H5
B3
L5
Exit
In the above table, each row is a different thread, listing the
dynamic instances produced by that thread from left to right. Each
thread executes the same program that starts with an Entry node
and ends with an Exit node, but different threads may take
different paths through the control flow of the program. The columns
are numbered merely for convenience, and empty cells have no special
meaning. Dynamic instances listed in the same column are converged.
Converged-with is a transitive symmetric relation over dynamic
instances produced by different threads for the same static
instance. Informally, two threads that produce converged dynamic
instances are said to be converged, and they are said to execute
that static instance convergently, at that point in the execution.
Convergence order is a strict partial order over dynamic instances
that is defined as the transitive closure of:
If dynamic instance P is executed strictly before Q in the
same thread, then P is convergence-beforeQ.
If dynamic instance P is executed strictly before Q1 in the
same thread, and Q1 is converged-withQ2, then P is
convergence-beforeQ2.
If dynamic instance P1 is converged-withP2, and P2
is executed strictly before Q in the same thread, then P1
is convergence-beforeQ.
1
2
3
4
5
6
7
8
9
Thread 1
Entry
…
S2
T
…
Exit
Thread 2
Entry
…
Q2
R
S1
…
Exit
Thread 3
Entry
…
P
Q1
…
The above table shows partial sequences of dynamic instances from
different threads. Dynamic instances in the same column are assumed
to be converged (i.e., related to each other in the converged-with
relation). The resulting convergence order includes the edges P->Q2, Q1->R, P->R, P->T, etc.
The fact that convergence-before is a strict partial order is a
constraint on the converged-with relation. It is trivially satisfied
if different dynamic instances are never converged. It is also
trivially satisfied for all known implementations for which
convergence plays some role. Aside from the strict partial convergence
order, there are currently no additional constraints on the
converged-with relation imposed in LLVM IR.
Note
The convergent attribute on convergent operations does
constrain changes to converged-with, but it is expressed in
terms of control flow and does not explicitly deal with thread
convergence.
The convergence-before relation is not
directly observable. Program transforms are in general free to
change the order of instructions, even though that obviously
changes the convergence-before relation.
Converged dynamic instances need not be executed at the same
time or even on the same resource. Converged dynamic instances
of a convergent operation may appear to do so but that is an
implementation detail. The fact that P is convergence-before
Q does not automatically imply that P happens-before
Q in a memory model sense.
Future work: Providing convergence-related guarantees to
compiler frontends enables some powerful optimization techniques
that can be used by programmers or by high-level program
transforms. Constraints on the converged-with relation may
be added eventually as part of the definition of LLVM
IR, so that guarantees can be made that frontends can rely on.
For a proposal on how this might work, see D85603.
This section defines a constraint that may be used to
produce a maximal converged-with relation without violating the
strict convergence-before order. This maximal converged-with
relation is reasonable for real targets and is compatible with
convergent operations.
The maximal converged-with relation is defined in terms of cycle
headers, which are not unique to a given CFG. Each cycle hierarchy for
the same CFG results in a different maximal converged-with relation.
Maximal converged-with:
Dynamic instances X1 and X2 produced by different threads
for the same static instance X are converged in the maximal
converged-with relation if and only if for every cycle C with
header H that contains X:
every dynamic instance H1 of H that precedes X1 in
the respective thread is convergence-before X2, and,
every dynamic instance H2 of H that precedes X2 in
the respective thread is convergence-before X1,
without assuming that X1 is converged with X2.
Note
For brevity, the rest of the document restricts the term
converged to mean “related under the maximal converged-with
relation for the given cycle hierarchy”.
Maximal convergence can now be demonstrated in the earlier example as follows:
1
2
3
4
5
6
7
8
9
Thread 1
Entry1
H1
B1
L1
H3
L3
Exit
Thread 2
Entry2
H2
L2
H4
B2
L4
H5
B3
L5
Exit
Entry1 and Entry2 are converged.
H1 and H2 are converged.
B1 and B2 are not converged due to H4 which is not
convergence-before B1.
H3 and H4 are converged.
H3 is not converged with H5 due to H4 which is not
convergence-before H3.
L1 and L2 are converged.
L3 and L4 are converged.
L3 is not converged with L5 due to H5 which is not
convergence-before L3.
Contradictions in convergence order are possible only between two
nodes that are inside some cycle. The dynamic instances of such nodes
may be interleaved in the same thread, and this interleaving may be
different for different threads.
When a thread executes a node X once and then executes it again,
it must have followed a closed path in the CFG that includes X.
Such a path must pass through the header of at least one cycle — the
smallest cycle that includes the entire closed path. In a given
thread, two dynamic instances of X are either separated by the
execution of at least one cycle header, or X itself is a cycle
header.
In reducible cycles (natural loops), each execution of the header is
equivalent to the start of a new iteration of the cycle. But this
analogy breaks down in the presence of explicit constraints on the
converged-with relation, such as those described in future
work. Instead, cycle headers should be
treated as implicit points of convergence in a maximal
converged-with relation.
Consider a sequence of nested cycles C1, C2, …, Ck such
that C1 is the outermost cycle and Ck is the innermost cycle,
with headers H1, H2, …, Hk respectively. When a thread
enters the cycle Ck, any of the following is possible:
The thread directly entered cycle Ck without having executed
any of the headers H1 to Hk.
The thread executed some or all of the nested headers one or more
times.
The maximal converged-with relation captures the following intuition
about cycles:
When two threads enter a top-level cycle C1, they execute
converged dynamic instances of every node that is a child of C1.
When two threads enter a nested cycle Ck, they execute
converged dynamic instances of every node that is a child of
Ck, until either thread exits Ck, if and only if they
executed converged dynamic instances of the last nested header that
either thread encountered.
Note that when a thread exits a nested cycle Ck, it must follow
a closed path outside Ck to reenter it. This requires executing
the header of some outer cycle, as described earlier.
Consider two dynamic instances X1 and X2 produced by threads T1
and T2 for a node X that is a child of nested cycle Ck.
Maximal convergence relates X1 and X2 as follows:
If neither thread executed any header from H1 to Hk, then
X1 and X2 are converged.
Otherwise, if there are no converged dynamic instances Q1 and
Q2 of any header Q from H1 to Hk (where Q is
possibly the same as X), such that Q1 precedes X1 and
Q2 precedes X2 in the respective threads, then X1 and
X2 are not converged.
Otherwise, consider the pair Q1 and Q2 of converged dynamic
instances of a header Q from H1 to Hk that occur most
recently before X1 and X2 in the respective threads. Then
X1 and X2 are converged if and only if there is no dynamic
instance of any header from H1 to Hk that occurs between
Q1 and X1 in thread T1, or between Q2 and X2 in
thread T2. In other words, Q1 and Q2 represent the last
point of convergence, with no other header being executed before
executing X.
Example:
The above figure shows two nested irreducible cycles with headers
R and S. The nodes Entry and Q have divergent
branches. The table below shows the convergence between three threads
taking different paths through the CFG. Dynamic instances listed in
the same column are converged.
1
2
3
4
5
6
7
8
10
Thread1
Entry
P1
Q1
S1
P3
Q3
R1
S2
Exit
Thread2
Entry
P2
Q2
R2
S3
Exit
Thread3
Entry
R3
S4
Exit
P2 and P3 are not converged due to S1
Q2 and Q3 are not converged due to S1
S1 and S3 are not converged due to R2
S1 and S4 are not converged due to R3
Informally, T1 and T2 execute the inner cycle a different
number of times, without executing the header of the outer cycle. All
threads converge in the outer cycle when they first execute the header
of the outer cycle.
The output of two converged dynamic instances is uniform if and
only if it compares equal for those two dynamic instances.
The output of a static instance X is uniform for a given set
of threads if and only if it is uniform for every pair of
converged dynamic instances of X produced by those threads.
A non-uniform value is said to be divergent.
For a set S of threads, the uniformity of each output of a static
instance is determined as follows:
The semantics of the instruction may specify the output to be
uniform.
Otherwise, if it is a PHI node, its output is uniform if and only
if for every pair of converged dynamic instances produced by all
threads in S:
Both instances choose the same output from converged
dynamic instances, and,
That output is uniform for all threads in S.
Otherwise, the output is uniform if and only if the input
operands are uniform for all threads in S.
When a divergent branch occurs inside a cycle, it is possible that a
diverged path continues to an exit of the cycle. This is called a
divergent cycle exit. If the cycle is irreducible, the diverged path
may re-enter and eventually reach a join within the cycle. Such a join
should be examined for the diverged entry criterion.
Nodes along the diverged path that lie outside the cycle experience
temporal divergence, when two threads executing convergently inside
the cycle produce uniform values, but exit the cycle along the same
divergent path after executing the header a different number of times
(informally, on different iterations of the cycle). For a node N
inside the cycle the outputs may be uniform for the two threads, but
any use U outside the cycle receives a value from non-converged
dynamic instances of N. An output of U may be divergent,
depending on the semantics of the instruction.
Irreducible control flow results in different cycle hierarchies
depending on the choice of headers during depth-first traversal. As a
result, a static analysis cannot always determine the convergence of
nodes in irreducible cycles, and any uniformity analysis is limited to
those static instances whose convergence is independent of the cycle
hierarchy:
m-converged static instances:
A static instance X is m-converged for a given CFG if and only
if the maximal converged-with relation for its dynamic instances is
the same in every cycle hierarchy that can be constructed for that CFG.
Note
In other words, two dynamic instances X1 and X2 of an
m-converged static instance X are converged in some cycle
hierarchy if and only if they are also converged in every other
cycle hierarchy for the same CFG.
As noted earlier, for brevity, we restrict the term converged to
mean “related under the maximal converged-with relation for a given
cycle hierarchy”.
Each node X in a given CFG is reported to be m-converged if and
only if:
X is a top-level node, in which
case, there are no cycle headers to influence the convergence of
X.
Otherwise, if X is inside a cycle, then every cycle that
contains X satisfies the following necessary conditions:
A reducible cycle trivially satisfies the above conditions. In particular,
if the whole CFG is reducible, then all nodes in the CFG are
m-converged.
If a static instance is not m-converged, then every output is assumed
to be divergent. Otherwise, for an m-converged static instance, the
uniformity of each output is determined using the criteria
described earlier. The discovery of
divergent outputs may cause their uses (including branches) to also
become divergent. The analysis propagates this divergence until a
fixed point is reached.
The convergence inferred using these criteria is a safe subset of the
maximal converged-with relation for any cycle hierarchy. In
particular, it is sufficient to determine if a static instance is
m-converged for a given cycle hierarchy T, even if that fact is
not detected when examining some other cycle hierarchy T'.
This property allows compiler transforms to use the uniformity
analysis without being affected by DFS choices made in the underlying
cycle analysis. When two transforms use different instances of the
uniformity analysis for the same CFG, a “divergent value” result in
one analysis instance cannot contradict a “uniform value” result in
the other.
Generic transforms such as SimplifyCFG, CSE, and loop transforms
commonly change the program in ways that change the maximal
converged-with relations. This also means that a value that was
previously uniform can become divergent after such a transform.
Uniformity has to be recomputed after such transforms.
The above figure shows a divergent branch Q inside an irreducible
cyclic region. When two threads diverge at Q, the convergence of
dynamic instances within the cyclic region depends on the cycle
hierarchy chosen:
In an implementation that detects a single cycle C with header
P, convergence inside the cycle is determined by P.
In an implementation that detects two nested cycles with headers
R and S, convergence inside those cycles is determined by
their respective headers.
A conservative approach would be to simply report all nodes inside
irreducible cycles as having divergent outputs. But it is desirable to
recognize m-converged nodes in the CFG in order to maximize
uniformity. This section describes one such pattern of nodes derived
from closed paths, which are a property of the CFG and do not depend
on the cycle hierarchy.
Diverged Entry Criterion:
The dynamic instances of all the nodes in a closed path P are
m-converged only if for every divergent branch B and its
join node J that lie on P, there is no entry to P which
lies on a diverged path from B to J.
Consider the closed path P->Q->R->S in the above figure.
P and R are entries to the closed
path. Q is a divergent branch and S is a
join for that branch, with diverged paths Q->R->S and Q->S.
If a diverged entry R exists, then in some cycle hierarchy,
R is the header of the smallest cycle C containing the
closed path and a child cycleC'
exists in the set C-R, containing both branch Q and join
S. When threads diverge at Q, one subset M continues
inside cycle C', while the complement N exits C' and
reaches R. Dynamic instances of S executed by threads in set
M are not converged with those executed in set N due to the
presence of R. Informally, threads that diverge at Q
reconverge in the same iteration of the outer cycle C, but they
may have executed the inner cycle C' differently.
1
2
3
4
5
6
7
8
9
10
11
Thread1
Entry
P1
Q1
R1
S1
P3
…
Exit
Thread2
Entry
P2
Q2
S2
P4
Q4
R2
S4
Exit
In the table above, S2 is not converged with S1 due to R1.
If R does not exist, or if any node other than R is the
header of C, then no such child cycle C' is detected.
Threads that diverge at Q execute converged dynamic instances of
S since they do not encounter the cycle header on any path from
Q to S. Informally, threads that diverge at Q
reconverge at S in the same iteration of C.
1
2
3
4
5
6
7
8
9
10
Thread1
Entry
P1
Q1
R1
S1
P3
Q3
R3
S3
Exit
Thread2
Entry
P2
Q2
S2
P4
Q4
R2
S4
Exit
Note
In general, the cycle C in the above statements is not
expected to be the same cycle for different headers. Cycles and
their headers are tightly coupled; for different headers in the
same outermost cycle, the child cycles detected may be different.
The property relevant to the above examples is that for every
closed path, there is a cycle C that contains the path and
whose header is on that path.
The diverged entry criterion must be checked for every closed path
passing through a divergent branch B and its join J. Since
every closed path passes through the header of some
cycle, this amounts to checking every cycle
C that contains B and J. When the header of C
dominates the join J, there can be no entry to any path from the
header to J, which includes any diverged path from B to J.
This is also true for any closed paths passing through the header of
an outer cycle that contains C.
Thus, the diverged entry criterion can be conservatively simplified
as follows:
For a divergent branch B and its join node J, the nodes in a
cycle C that contains both B and J are m-converged only
if:
B strictly dominates J, or,
The header H of C strictly dominates J, or,
Recursively, there is cycle C' inside C that satisfies the
same condition.
When J is the same as H or B, the trivial dominance is
insufficient to make any statement about entries to diverged paths.
The figure shows two cycle hierarchies with a divergent branch in
Entry instead of Q. For two threads that enter the closed path
P->Q->R->S at P and R respectively, the convergence
of dynamic instances generated along the path depends on whether P
or R is the header.
Convergence when P is the header.
1
2
3
4
5
6
7
8
9
10
11
12
13
Thread1
Entry
P1
Q1
R1
S1
P3
Q3
S3
Exit
Thread2
Entry
R2
S2
P2
Q2
S2
P4
Q4
R3
S4
Exit
Convergence when R is the header.
1
2
3
4
5
6
7
8
9
10
11
12
Thread1
Entry
P1
Q1
R1
S1
P3
Q3
S3
Exit
Thread2
Entry
R2
S2
P2
Q2
S2
P4
…
Exit
Thus, when diverged paths reach different entries of an irreducible
cycle from outside the cycle, the static analysis conservatively
reports every node in the cycle as not m-converged.
If C is a reducible cycle with header H, then in any DFS,
Hmust be the header of some cycleC' that contains C. Independent of the DFS, there is no entry
to the subgraph C other than H itself. Thus, we have the
following:
The diverged entry criterion is trivially satisfied for a divergent
branch and its join, where both are inside subgraph C.
When diverged paths reach the subgraph C from outside, their
convergence is always determined by the same header H.
Clearly, this can be determined only in a cycle hierarchy T where
C is detected as a reducible cycle. No such conclusion can be made
in a different cycle hierarchy T' where C is part of a larger
cycle C' with the same header, but this does not contradict the
conclusion in T.