.. _convergence-and-uniformity: | |

========================== | |

Convergence And Uniformity | |

========================== | |

.. contents:: | |

:local: | |

Introduction | |

============ | |

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 previous work on divergence analysis | |

[DivergenceSPMD]_ to cover irreducible control-flow. | |

.. [DivergenceSPMD] 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 | |

Terminology | |

=========== | |

Cycles | |

Described in :ref:`cycle-terminology`. | |

Closed path | |

Described in :ref:`cycle-closed-path`. | |

Disjoint paths | |

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. | |

.. _convergence-dynamic-instances: | |

Threads and Dynamic Instances | |

============================= | |

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. | |

.. figure:: convergence-natural-loop.png | |

:name: convergence-natural-loop | |

.. table:: | |

:name: convergence-thread-example | |

:align: left | |

+----------+--------+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+ | |

| | | 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. | |

.. _convergence-definition: | |

Convergence | |

=========== | |

*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-before* is a strict partial order over dynamic instances | |

that is defined as the transitive closure of: | |

1. If dynamic instance ``P`` is executed strictly before ``Q`` in the | |

same thread, then ``P`` is *convergence-before* ``Q``. | |

2. If dynamic instance ``P`` is executed strictly before ``Q1`` in the | |

same thread, and ``Q1`` is *converged-with* ``Q2``, then ``P`` is | |

*convergence-before* ``Q2``. | |

3. If dynamic instance ``P1`` is *converged-with* ``P2``, and ``P2`` | |

is executed strictly before ``Q`` in the same thread, then ``P1`` | |

is *convergence-before* ``Q``. | |

.. table:: | |

:name: convergence-order-example | |

:align: left | |

+----------+-------+-----+-----+-----+-----+-----+-----+-----+------+ | |

| | 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. | |

.. _convergence-note-convergence: | |

.. note:: | |

1. 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. | |

2. 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. | |

3. The fact that ``P`` is convergence-before | |

``Q`` does not automatically imply that ``P`` happens-before | |

``Q`` in a memory model sense. | |

.. _convergence-maximal: | |

Maximal Convergence | |

------------------- | |

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, with the assumption that threads converge at the header on every | |

"iteration" of the cycle. Informally, two threads execute the same iteration of | |

a cycle if they both previously executed the cycle header the same number of | |

times after they entered that cycle. In general, this needs to account for the | |

iterations of parent cycles as well. | |

**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:: | |

Cycle headers may not be unique to a given CFG if it is irreducible. Each | |

cycle hierarchy for the same CFG results in a different maximal | |

converged-with relation. | |

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: | |

.. table:: | |

:align: left | |

+----------+--------+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+ | |

| | | 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``. | |

.. _convergence-cycle-headers: | |

Dependence on Cycles Headers | |

---------------------------- | |

Contradictions in *convergence-before* 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 :ref:`future | |

work<convergence-note-convergence>`. 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: | |

1. The thread directly entered cycle ``Ck`` without having executed | |

any of the headers ``H1`` to ``Hk``. | |

2. 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: | |

1. When two threads enter a top-level cycle ``C1``, they execute | |

converged dynamic instances of every node that is a :ref:`child | |

<cycle-parent-block>` of ``C1``. | |

2. 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: | |

1. If neither thread executed any header from ``H1`` to ``Hk``, then | |

``X1`` and ``X2`` are converged. | |

2. 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. | |

3. 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:** | |

.. figure:: convergence-both-diverged-nested.png | |

:name: convergence-both-diverged-nested | |

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. | |

.. table:: | |

:align: left | |

+---------+-------+-----+-----+-----+-----+-----+-----+-----+------+ | |

| | 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. | |

.. _convergence-uniformity: | |

Uniformity | |

========== | |

1. The output of two converged dynamic instances is uniform if and | |

only if it compares equal for those two dynamic instances. | |

2. 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: | |

1. The semantics of the instruction may specify the output to be | |

uniform. | |

2. Otherwise, the output is divergent if the static instance is not | |

:ref:`m-converged <convergence-m-converged>`. | |

3. Otherwise, if the static instance is m-converged: | |

1. 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``: | |

a. Both instances choose the same output from converged | |

dynamic instances, and, | |

b. That output is uniform for all threads in ``S``. | |

2. Otherwise, the output is uniform if and only if the input | |

operands are uniform for all threads in ``S``. | |

Divergent Cycle Exits | |

--------------------- | |

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 :ref:`diverged entry | |

<convergence-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. | |

.. _uniformity-analysis: | |

Static Uniformity Analysis | |

========================== | |

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: | |

.. _convergence-m-converged: | |

**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 every cycle that contains ``X`` satisfies the following necessary | |

conditions: | |

1. Every divergent branch inside the cycle satisfies the | |

:ref:`diverged entry criterion<convergence-diverged-entry>`, and, | |

2. There are no :ref:`diverged paths reaching the | |

cycle<convergence-diverged-outside>` from a divergent branch | |

outside it. | |

.. note:: | |

A reducible cycle :ref:`trivially satisfies | |

<convergence-reducible-cycle>` the above conditions. In particular, | |

if the whole CFG is reducible, then all nodes in the CFG are | |

m-converged. | |

The uniformity of each output of a static instance | |

is determined using the criteria | |

:ref:`described earlier <convergence-uniformity>`. 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. | |

Divergent Branch inside a Cycle | |

------------------------------- | |

.. figure:: convergence-divergent-inside.png | |

:name: convergence-divergent-inside | |

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: | |

1. In an implementation that detects a single cycle ``C`` with header | |

``P``, convergence inside the cycle is determined by ``P``. | |

2. In an implementation that detects two nested cycles with headers | |

``R`` and ``S``, convergence inside those cycles is determined by | |

their respective headers. | |

.. _convergence-diverged-entry: | |

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``. | |

.. figure:: convergence-closed-path.png | |

:name: convergence-closed-path | |

Consider the closed path ``P -> Q -> R -> S`` in the above figure. | |

``P`` and ``R`` are :ref:`entries to the closed | |

path<cycle-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 :ref:`child cycle<cycle-definition>` ``C'`` | |

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. | |

.. table:: | |

:align: left | |

+---------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+ | |

| | 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``. | |

.. table:: | |

:align: left | |

+---------+-------+-----+-----+-----+-----+-----+-----+-----+-----+------+ | |

| | 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 | |

:ref:`every closed path passes through the header of some | |

cycle<cycle-closed-path-header>`, 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. | |

.. _convergence-diverged-outside: | |

Diverged Paths reaching a Cycle | |

------------------------------- | |

.. figure:: convergence-divergent-outside.png | |

:name: convergence-divergent-outside | |

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. | |

.. table:: | |

:align: left | |

+---------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+ | |

| | 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. | |

.. table:: | |

:align: left | |

+---------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+ | |

| | 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. | |

.. _convergence-reducible-cycle: | |

Reducible Cycle | |

--------------- | |

If ``C`` is a reducible cycle with header ``H``, then in any DFS, | |

``H`` :ref:`must be the header of some cycle<cycle-reducible-headers>` | |

``C'`` that contains ``C``. Independent of the DFS, there is no entry | |

to the subgraph ``C`` other than ``H`` itself. Thus, we have the | |

following: | |

1. The diverged entry criterion is trivially satisfied for a divergent | |

branch and its join, where both are inside subgraph ``C``. | |

2. 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``. | |

Controlled Convergence | |

====================== | |

:ref:`Convergence control tokens <dynamic_instances_and_convergence_tokens>` | |

provide an explicit semantics for determining which threads are converged at a | |

given point in the program. The impact of this is incorporated in a | |

:ref:`controlled maximal converged-with <controlled_maximal_converged_with>` | |

relation over dynamic instances and a :ref:`controlled m-converged | |

<controlled_m_converged>` property of static instances. The :ref:`uniformity | |

analysis <uniformity-analysis>` implemented in LLVM includes this for targets | |

that support convergence control tokens. |