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6.9 KiB
ReStructuredText
229 lines
6.9 KiB
ReStructuredText
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.. _cycle-terminology:
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======================
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LLVM Cycle Terminology
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======================
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.. contents::
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:local:
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Cycles
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======
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Cycles are a generalization of LLVM :ref:`loops <loop-terminology>`,
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defined recursively as follows [HavlakCycles]_:
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1. In a directed graph G, an *outermost cycle* is a maximal strongly
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connected region with at least one internal edge. (Informational
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note --- The requirement for at least one internal edge ensures
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that a single basic block is a cycle only if there is an edge that
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goes back to the same basic block.)
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2. A basic block in the cycle that can be reached from the entry of
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the function along a path that does not visit any other basic block
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in the cycle is called an *entry* of the cycle. A cycle can have
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multiple entries.
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3. In any depth-first search starting from the entry of the function,
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the first node of a cycle to be visited will be one of the entries.
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This entry is called the *header* of the cycle. (Informational note
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--- Thus, the header of the cycle is implementation-defined.)
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4. In any depth-first search starting from the entry, set of outermost
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cycles found in the CFG is the same. These are the *top-level
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cycles* that do not themselves have a parent.
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5. The cycles nested inside a cycle C with header H are the outermost
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cycles in the subgraph induced on the set of nodes (C - H). C is
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said to be the *parent* of these cycles, and each of these cycles
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is a *child* of C.
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Thus, cycles form an implementation-defined forest where each cycle C is
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the parent of any outermost cycles nested inside C. The tree closely
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follows the nesting of loops in the same function. The unique entry of
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a reducible cycle (an LLVM loop) L dominates all its other nodes, and
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is always chosen as the header of some cycle C regardless of the DFS
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tree used. This cycle C is a superset of the loop L. For an
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irreducible cycle, no one entry dominates the nodes of the cycle. One
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of the entries is chosen as header of the cycle, in an
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implementation-defined way.
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.. _cycle-irreducible:
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A cycle is *irreducible* if it has multiple entries and it is
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*reducible* otherwise.
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.. _cycle-parent-block:
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A cycle C is said to be the *parent* of a basic block B if B occurs in
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C but not in any child cycle of C. Then B is also said to be a *child*
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of cycle C.
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.. _cycle-sibling:
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A basic block or cycle X is a *sibling* of another basic block or
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cycle Y if they both have no parent or both have the same parent.
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Informational notes:
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- Non-header entry blocks of a cycle can be contained in child cycles.
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- If the CFG is reducible, the cycles are exactly the natural loops and
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every cycle has exactly one entry block.
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- Cycles are well-nested (by definition).
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- The entry blocks of a cycle are siblings in the dominator tree.
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.. [HavlakCycles] Paul Havlak, "Nesting of reducible and irreducible
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loops." ACM Transactions on Programming Languages
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and Systems (TOPLAS) 19.4 (1997): 557-567.
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.. _cycle-examples:
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Examples of Cycles
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==================
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Irreducible cycle enclosing natural loops
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-----------------------------------------
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.. Graphviz source; the indented blocks below form a comment.
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/// | |
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/// />A] [B<\
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/// | \ / |
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/// ^---C---^
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/// |
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strict digraph {
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{ rank=same; A B}
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Entry -> A
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Entry -> B
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A -> A
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A -> C
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B -> B
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B -> C
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C -> A
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C -> B
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C -> Exit
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}
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.. image:: cycle-1.png
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The self-loops of ``A`` and ``B`` give rise to two single-block
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natural loops. A possible hierarchy of cycles is::
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cycle: {A, B, C} entries: {A, B} header: A
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- cycle: {B, C} entries: {B, C} header: C
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- cycle: {B} entries: {B} header: B
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This hierarchy arises when DFS visits the blocks in the order ``A``,
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``C``, ``B`` (in preorder).
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Irreducible union of two natural loops
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--------------------------------------
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.. Graphviz source; the indented blocks below form a comment.
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/// | |
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/// A<->B
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/// ^ ^
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/// | |
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/// v v
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/// C D
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/// | |
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strict digraph {
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{ rank=same; A B}
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{ rank=same; C D}
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Entry -> A
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Entry -> B
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A -> B
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B -> A
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A -> C
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C -> A
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B -> D
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D -> B
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C -> Exit
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D -> Exit
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}
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.. image:: cycle-2.png
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There are two natural loops: ``{A, C}`` and ``{B, D}``. A possible
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hierarchy of cycles is::
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cycle: {A, B, C, D} entries: {A, B} header: A
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- cycle: {B, D} entries: {B} header: B
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Irreducible cycle without natural loops
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---------------------------------------
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.. Graphviz source; the indented blocks below form a comment.
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/// | |
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/// />A B<\
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/// | |\ /| |
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/// | | x | |
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/// | |/ \| |
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/// ^-C D-^
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/// | |
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///
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strict digraph {
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{ rank=same; A B}
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{ rank=same; C D}
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Entry -> A
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Entry -> B
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A -> C
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A -> D
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B -> C
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B -> D
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C -> A
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D -> B
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C -> Exit
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D -> Exit
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}
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.. image:: cycle-3.png
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This graph does not contain any natural loops --- the nodes ``A``,
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``B``, ``C`` and ``D`` are siblings in the dominator tree. A possible
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hierarchy of cycles is::
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cycle: {A, B, C, D} entries: {A, B} header: A
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- cycle: {B, D} entries: {B, D} header: D
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.. _cycle-closed-path:
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Closed Paths and Cycles
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=======================
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A *closed path* in a CFG is a connected sequence of nodes and edges in
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the CFG whose start and end points are the same.
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1. If a node D dominates one or more nodes in a closed path P and P
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does not contain D, then D dominates every node in P.
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**Proof:** Let U be a node in P that is dominated by D. If there
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was a node V in P not dominated by D, then U would be reachable
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from the function entry node via V without passing through D, which
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contradicts the fact that D dominates U.
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2. If a node D dominates one or more nodes in a closed path P and P
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does not contain D, then there exists a cycle C that contains P but
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not D.
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**Proof:** From the above property, D dominates all the nodes in P.
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For any nesting of cycles discovered by the implementation-defined
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DFS, consider the smallest cycle C which contains P. For the sake
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of contradiction, assume that D is in C. Then the header H of C
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cannot be in P, since the header of a cycle cannot be dominated by
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any other node in the cycle. Thus, P is in the set (C-H), and there
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must be a smaller cycle C' in C which also contains P, but that
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contradicts how we chose C.
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3. If a closed path P contains nodes U1 and U2 but not their
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dominators D1 and D2 respectively, then there exists a cycle C that
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contains U1 and U2 but neither of D1 and D2.
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**Proof:** From the above properties, each D1 and D2 separately
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dominate every node in P. There exists a cycle C1 (respectively,
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C2) that contains P but not D1 (respectively, D2). Either C1 and C2
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are the same cycle, or one of them is nested inside the other.
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Hence there is always a cycle that contains U1 and U2 but neither
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of D1 and D2.
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