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1 /*
2  * Copyright (C) 2017 The Android Open Source Project
3  *
4  * Licensed under the Apache License, Version 2.0 (the "License");
5  * you may not use this file except in compliance with the License.
6  * You may obtain a copy of the License at
7  *
8  *      http://www.apache.org/licenses/LICENSE-2.0
9  *
10  * Unless required by applicable law or agreed to in writing, software
11  * distributed under the License is distributed on an "AS IS" BASIS,
12  * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
13  * See the License for the specific language governing permissions and
14  * limitations under the License.
15  */
16 
17 #ifndef ART_COMPILER_OPTIMIZING_SUPERBLOCK_CLONER_H_
18 #define ART_COMPILER_OPTIMIZING_SUPERBLOCK_CLONER_H_
19 
20 #include "base/arena_bit_vector.h"
21 #include "base/arena_containers.h"
22 #include "base/bit_vector-inl.h"
23 #include "nodes.h"
24 
25 namespace art {
26 
27 class InductionVarRange;
28 
29 static const bool kSuperblockClonerLogging = false;
30 static const bool kSuperblockClonerVerify = false;
31 
32 // Represents an edge between two HBasicBlocks.
33 //
34 // Note: objects of this class are small - pass them by value.
35 class HEdge : public ArenaObject<kArenaAllocSuperblockCloner> {
36  public:
HEdge(HBasicBlock * from,HBasicBlock * to)37   HEdge(HBasicBlock* from, HBasicBlock* to) : from_(from->GetBlockId()), to_(to->GetBlockId()) {
38     DCHECK_NE(to_, kInvalidBlockId);
39     DCHECK_NE(from_, kInvalidBlockId);
40   }
HEdge(uint32_t from,uint32_t to)41   HEdge(uint32_t from, uint32_t to) : from_(from), to_(to) {
42     DCHECK_NE(to_, kInvalidBlockId);
43     DCHECK_NE(from_, kInvalidBlockId);
44   }
HEdge()45   HEdge() : from_(kInvalidBlockId), to_(kInvalidBlockId) {}
46 
GetFrom()47   uint32_t GetFrom() const { return from_; }
GetTo()48   uint32_t GetTo() const { return to_; }
49 
50   bool operator==(const HEdge& other) const {
51     return this->from_ == other.from_ && this->to_ == other.to_;
52   }
53 
54   bool operator!=(const HEdge& other) const { return !operator==(other); }
55   void Dump(std::ostream& stream) const;
56 
57   // Returns whether an edge represents a valid edge in CF graph: whether the from_ block
58   // has to_ block as a successor.
IsValid()59   bool IsValid() const { return from_ != kInvalidBlockId && to_ != kInvalidBlockId; }
60 
61  private:
62   // Predecessor block id.
63   uint32_t from_;
64   // Successor block id.
65   uint32_t to_;
66 };
67 
68 // Returns whether a HEdge edge corresponds to an existing edge in the graph.
IsEdgeValid(HEdge edge,HGraph * graph)69 inline bool IsEdgeValid(HEdge edge, HGraph* graph) {
70   if (!edge.IsValid()) {
71     return false;
72   }
73   uint32_t from = edge.GetFrom();
74   uint32_t to = edge.GetTo();
75   if (from >= graph->GetBlocks().size() || to >= graph->GetBlocks().size()) {
76     return false;
77   }
78 
79   HBasicBlock* block_from = graph->GetBlocks()[from];
80   HBasicBlock* block_to = graph->GetBlocks()[to];
81   if (block_from == nullptr || block_to == nullptr) {
82     return false;
83   }
84 
85   return block_from->HasSuccessor(block_to, 0);
86 }
87 
88 // SuperblockCloner provides a feature of cloning subgraphs in a smart, high level way without
89 // fine grain manipulation with IR; data flow and graph properties are resolved/adjusted
90 // automatically. The clone transformation is defined by specifying a set of basic blocks to copy
91 // and a set of rules how to treat edges, remap their successors. By using this approach such
92 // optimizations as Branch Target Expansion, Loop Peeling, Loop Unrolling, Loop Versioning can be
93 // implemented.
94 //
95 // The idea of the transformation is based on "Superblock cloning" technique described in the book
96 // "Engineering a Compiler. Second Edition", Keith D. Cooper, Linda Torczon, Rice University
97 // Houston, Texas. 2nd edition, Morgan Kaufmann. The original paper is "The Superblock: An Efective
98 // Technique for VLIW and Superscalar Compilation" by Hwu, W.M.W., Mahlke, S.A., Chen, W.Y. et al.
99 // J Supercomput (1993) 7: 229. doi:10.1007/BF01205185.
100 //
101 // There are two states of the IR graph: original graph (before the transformation) and
102 // copy graph (after).
103 //
104 // Before the transformation:
105 // Defining a set of basic block to copy (orig_bb_set) partitions all of the edges in the original
106 // graph into 4 categories/sets (use the following notation for edges: "(pred, succ)",
107 // where pred, succ - basic blocks):
108 //  - internal - pred, succ are members of ‘orig_bb_set’.
109 //  - outside  - pred, succ are not members of ‘orig_bb_set’.
110 //  - incoming - pred is not a member of ‘orig_bb_set’, succ is.
111 //  - outgoing - pred is a member of ‘orig_bb_set’, succ is not.
112 //
113 // Transformation:
114 //
115 // 1. Initial cloning:
116 //   1.1. For each ‘orig_block’ in orig_bb_set create a copy ‘copy_block’; these new blocks
117 //        form ‘copy_bb_set’.
118 //   1.2. For each edge (X, Y) from internal set create an edge (X_1, Y_1) where X_1, Y_1 are the
119 //        copies of X, Y basic blocks correspondingly; these new edges form ‘copy_internal’ edge
120 //        set.
121 //   1.3. For each edge (X, Y) from outgoing set create an edge (X_1, Y_1) where X_1, Y_1 are the
122 //        copies of X, Y basic blocks correspondingly; these new edges form ‘copy_outgoing’ edge
123 //        set.
124 // 2. Successors remapping.
125 //   2.1. 'remap_orig_internal’ - set of edges (X, Y) from ‘orig_bb_set’ whose successors should
126 //        be remapped to copy nodes: ((X, Y) will be transformed into (X, Y_1)).
127 //   2.2. ‘remap_copy_internal’ - set of edges (X_1, Y_1) from ‘copy_bb_set’ whose successors
128 //        should be remapped to copy nodes: (X_1, Y_1) will be transformed into (X_1, Y)).
129 //   2.3. 'remap_incoming’ - set of edges (X, Y) from the ‘incoming’ edge set in the original graph
130 //        whose successors should be remapped to copies nodes: ((X, Y) will be transformed into
131 //        (X, Y_1)).
132 // 3. Adjust control flow structures and relations (dominance, reverse post order, loops, etc).
133 // 4. Fix/resolve data flow.
134 // 5. Do cleanups (DCE, critical edges splitting, etc).
135 //
136 class SuperblockCloner : public ValueObject {
137  public:
138   // TODO: Investigate optimal types for the containers.
139   using HBasicBlockMap = ArenaSafeMap<HBasicBlock*, HBasicBlock*>;
140   using HInstructionMap = ArenaSafeMap<HInstruction*, HInstruction*>;
141   using HBasicBlockSet = ArenaBitVector;
142   using HEdgeSet = ArenaHashSet<HEdge>;
143 
144   SuperblockCloner(HGraph* graph,
145                    const HBasicBlockSet* orig_bb_set,
146                    HBasicBlockMap* bb_map,
147                    HInstructionMap* hir_map,
148                    InductionVarRange* induction_range);
149 
150   // Sets edge successor remapping info specified by corresponding edge sets.
151   void SetSuccessorRemappingInfo(const HEdgeSet* remap_orig_internal,
152                                  const HEdgeSet* remap_copy_internal,
153                                  const HEdgeSet* remap_incoming);
154 
155   // Returns whether the specified subgraph is copyable.
156   // TODO: Start from small range of graph patterns then extend it.
157   bool IsSubgraphClonable() const;
158 
159   // Returns whether selected subgraph satisfies the criteria for fast data flow resolution
160   // when iterative DF algorithm is not required and dominators/instructions inputs can be
161   // trivially adjusted.
162   //
163   // TODO: formally describe the criteria.
164   //
165   // Loop peeling, unrolling and versioning satisfy the criteria.
166   bool IsFastCase() const;
167 
168   // Runs the copy algorithm according to the description.
169   void Run();
170 
171   // Cleans up the graph after transformation: splits critical edges, recalculates control flow
172   // information (back-edges, dominators, loop info, etc), eliminates redundant phis.
173   void CleanUp();
174 
175   // Returns a clone of a basic block (orig_block).
176   //
177   //  - The copy block will have no successors/predecessors; they should be set up manually.
178   //  - For each instruction in the orig_block a copy is created and inserted into the copy block;
179   //    this correspondence is recorded in the map (old instruction, new instruction).
180   //  - Graph HIR is not valid after this transformation: all of the HIRs have their inputs the
181   //    same, as in the original block, PHIs do not reflect a correct correspondence between the
182   //    value and predecessors (as the copy block has no predecessors by now), etc.
183   HBasicBlock* CloneBasicBlock(const HBasicBlock* orig_block);
184 
185   // Creates a clone for each basic blocks in orig_bb_set adding corresponding entries into bb_map_
186   // and hir_map_.
187   void CloneBasicBlocks();
188 
GetInstrCopy(HInstruction * orig_instr)189   HInstruction* GetInstrCopy(HInstruction* orig_instr) const {
190     auto copy_input_iter = hir_map_->find(orig_instr);
191     DCHECK(copy_input_iter != hir_map_->end());
192     return copy_input_iter->second;
193   }
194 
GetBlockCopy(HBasicBlock * orig_block)195   HBasicBlock* GetBlockCopy(HBasicBlock* orig_block) const {
196     HBasicBlock* block = bb_map_->Get(orig_block);
197     DCHECK(block != nullptr);
198     return block;
199   }
200 
GetInstrOrig(HInstruction * copy_instr)201   HInstruction* GetInstrOrig(HInstruction* copy_instr) const {
202     for (auto it : *hir_map_) {
203       if (it.second == copy_instr) {
204         return it.first;
205       }
206     }
207     return nullptr;
208   }
209 
IsInOrigBBSet(uint32_t block_id)210   bool IsInOrigBBSet(uint32_t block_id) const {
211     return orig_bb_set_.IsBitSet(block_id);
212   }
213 
IsInOrigBBSet(const HBasicBlock * block)214   bool IsInOrigBBSet(const HBasicBlock* block) const {
215     return IsInOrigBBSet(block->GetBlockId());
216   }
217 
218   // Returns the area (the most outer loop) in the graph for which control flow (back edges, loops,
219   // dominators) needs to be adjusted.
GetRegionToBeAdjusted()220   HLoopInformation* GetRegionToBeAdjusted() const {
221     return outer_loop_;
222   }
223 
224  private:
225   // Fills the 'exits' vector with the subgraph exits.
226   void SearchForSubgraphExits(ArenaVector<HBasicBlock*>* exits) const;
227 
228   // Finds and records information about the area in the graph for which control flow (back edges,
229   // loops, dominators) needs to be adjusted.
230   void FindAndSetLocalAreaForAdjustments();
231 
232   // Remaps edges' successors according to the info specified in the edges sets.
233   //
234   // Only edge successors/predecessors and phis' input records (to have a correspondence between
235   // a phi input record (not value) and a block's predecessor) are adjusted at this stage: neither
236   // phis' nor instructions' inputs values are resolved.
237   void RemapEdgesSuccessors();
238 
239   // Adjusts control flow (back edges, loops, dominators) for the local area defined by
240   // FindAndSetLocalAreaForAdjustments.
241   void AdjustControlFlowInfo();
242 
243   // Resolves Data Flow - adjusts phis' and instructions' inputs in order to have a valid graph in
244   // the SSA form.
245   void ResolveDataFlow();
246 
247   //
248   // Helpers for live-outs processing and Subgraph-closed SSA.
249   //
250   //  - live-outs - values which are defined inside the subgraph and have uses outside.
251   //  - Subgraph-closed SSA - SSA form for which all the values defined inside the subgraph
252   //    have no outside uses except for the phi-nodes in the subgraph exits.
253   //
254   // Note: now if the subgraph has live-outs it is only clonable if it has a single exit; this
255   // makes the subgraph-closed SSA form construction much easier.
256   //
257   // TODO: Support subgraphs with live-outs and multiple exits.
258   //
259 
260   // For each live-out value 'val' in the region puts a record <val, val> into the map.
261   // Returns whether all of the instructions in the subgraph are clonable.
262   bool CollectLiveOutsAndCheckClonable(HInstructionMap* live_outs_) const;
263 
264   // Constructs Subgraph-closed SSA; precondition - a subgraph has a single exit.
265   //
266   // For each live-out 'val' in 'live_outs_' map inserts a HPhi 'phi' into the exit node, updates
267   // the record in the map to <val, phi> and replaces all outside uses with this phi.
268   void ConstructSubgraphClosedSSA();
269 
270   // Fixes the data flow for the live-out 'val' by adding a 'copy_val' input to the corresponding
271   // (<val, phi>) phi after the cloning is done.
272   void FixSubgraphClosedSSAAfterCloning();
273 
274   //
275   // Helpers for CloneBasicBlock.
276   //
277 
278   // Adjusts copy instruction's inputs: if the input of the original instruction is defined in the
279   // orig_bb_set, replaces it with a corresponding copy otherwise leaves it the same as original.
280   void ReplaceInputsWithCopies(HInstruction* copy_instr);
281 
282   // Recursively clones the environment for the copy instruction. If the input of the original
283   // environment is defined in the orig_bb_set, replaces it with a corresponding copy otherwise
284   // leaves it the same as original.
285   void DeepCloneEnvironmentWithRemapping(HInstruction* copy_instr, const HEnvironment* orig_env);
286 
287   //
288   // Helpers for RemapEdgesSuccessors.
289   //
290 
291   // Remaps incoming or original internal edge to its copy, adjusts the phi inputs in orig_succ and
292   // copy_succ.
293   void RemapOrigInternalOrIncomingEdge(HBasicBlock* orig_block, HBasicBlock* orig_succ);
294 
295   // Adds copy internal edge (from copy_block to copy_succ), updates phis in the copy_succ.
296   void AddCopyInternalEdge(HBasicBlock* orig_block, HBasicBlock* orig_succ);
297 
298   // Remaps copy internal edge to its origin, adjusts the phi inputs in orig_succ.
299   void RemapCopyInternalEdge(HBasicBlock* orig_block, HBasicBlock* orig_succ);
300 
301   // Checks whether the edges remapping info corresponds to the subgraph versioning case:
302   //  - none of the incoming edges are to be remapped (they are being duplicated).
303   //  - none of the internal edges are to be remapped.
304   bool IsRemapInfoForVersioning() const;
305 
306   // Processes incoming edges for subgraph versioning case: for each incoming edge (X, Y) adds
307   // an edge (X, Y_1) where Y_1 = Copy(Y) and add corresponding phi input to copy phi.
308   //
309   // Note: such node X will now have two successors, its unconditional branch instruction
310   // will be invalid and should be adjusted to some conditional branch by the client code.
311   void CopyIncomingEdgesForVersioning();
312 
313   //
314   // Local versions of control flow calculation/adjustment routines.
315   //
316 
317   void FindBackEdgesLocal(HBasicBlock* entry_block, ArenaBitVector* local_set);
318   void RecalculateBackEdgesInfo(ArenaBitVector* outer_loop_bb_set);
319   GraphAnalysisResult AnalyzeLoopsLocally(ArenaBitVector* outer_loop_bb_set);
320   void CleanUpControlFlow();
321 
322   //
323   // Helpers for ResolveDataFlow
324   //
325 
326   // Resolves the inputs of the phi.
327   void ResolvePhi(HPhi* phi);
328 
329   // Update induction range after when fixing SSA.
330   void UpdateInductionRangeInfoOf(
331       HInstruction* user, HInstruction* old_instruction, HInstruction* replacement);
332 
333   //
334   // Debug and logging methods.
335   //
336   void CheckInstructionInputsRemapping(HInstruction* orig_instr);
337   bool CheckRemappingInfoIsValid();
338   void VerifyGraph();
339   void DumpInputSets();
340 
GetBlockById(uint32_t block_id)341   HBasicBlock* GetBlockById(uint32_t block_id) const {
342     DCHECK(block_id < graph_->GetBlocks().size());
343     HBasicBlock* block = graph_->GetBlocks()[block_id];
344     DCHECK(block != nullptr);
345     return block;
346   }
347 
348   HGraph* const graph_;
349   ArenaAllocator* const arena_;
350 
351   // Set of basic block in the original graph to be copied.
352   HBasicBlockSet orig_bb_set_;
353 
354   // Sets of edges which require successors remapping.
355   const HEdgeSet* remap_orig_internal_;
356   const HEdgeSet* remap_copy_internal_;
357   const HEdgeSet* remap_incoming_;
358 
359   // Correspondence map for blocks: (original block, copy block).
360   HBasicBlockMap* bb_map_;
361   // Correspondence map for instructions: (original HInstruction, copy HInstruction).
362   HInstructionMap* hir_map_;
363   // As a result of cloning, the induction range analysis information can be invalidated
364   // and must be updated. If not null, the cloner updates it for changed instructions.
365   InductionVarRange* induction_range_;
366   // Area in the graph for which control flow (back edges, loops, dominators) needs to be adjusted.
367   HLoopInformation* outer_loop_;
368   HBasicBlockSet outer_loop_bb_set_;
369 
370   HInstructionMap live_outs_;
371 
372   ART_FRIEND_TEST(SuperblockClonerTest, AdjustControlFlowInfo);
373   ART_FRIEND_TEST(SuperblockClonerTest, IsGraphConnected);
374 
375   DISALLOW_COPY_AND_ASSIGN(SuperblockCloner);
376 };
377 
378 // Helper class to perform loop peeling/unrolling/versioning.
379 //
380 // This helper should be used when correspondence map between original and copied
381 // basic blocks/instructions are demanded.
382 class LoopClonerHelper : public ValueObject {
383  public:
LoopClonerHelper(HLoopInformation * info,SuperblockCloner::HBasicBlockMap * bb_map,SuperblockCloner::HInstructionMap * hir_map,InductionVarRange * induction_range)384   LoopClonerHelper(HLoopInformation* info,
385                    SuperblockCloner::HBasicBlockMap* bb_map,
386                    SuperblockCloner::HInstructionMap* hir_map,
387                    InductionVarRange* induction_range) :
388       loop_info_(info),
389       cloner_(info->GetHeader()->GetGraph(), &info->GetBlocks(), bb_map, hir_map, induction_range) {
390     // For now do transformations only for natural loops.
391     DCHECK(!info->IsIrreducible());
392   }
393 
394   // Returns whether the loop can be peeled/unrolled (static function).
395   static bool IsLoopClonable(HLoopInformation* loop_info);
396 
397   // Returns whether the loop can be peeled/unrolled.
IsLoopClonable()398   bool IsLoopClonable() const { return cloner_.IsSubgraphClonable(); }
399 
400   // Perform loop peeling.
401   //
402   // Control flow of an example (ignoring critical edges splitting).
403   //
404   //       Before                    After
405   //
406   //         |B|                      |B|
407   //          |                        |
408   //          v                        v
409   //         |1|                      |1|
410   //          |                        |
411   //          v                        v
412   //         |2|<-\                  |2A|
413   //         / \  /                   / \
414   //        v   v/                   /   v
415   //       |4|  |3|                 /   |3A|
416   //        |                      /     /
417   //        v                     |     v
418   //       |E|                     \   |2|<-\
419   //                                \ / \   /
420   //                                 v   v /
421   //                                |4|  |3|
422   //                                 |
423   //                                 v
424   //                                |E|
DoPeeling()425   HBasicBlock* DoPeeling() {
426     return DoLoopTransformationImpl(TransformationKind::kPeeling);
427   }
428 
429   // Perform loop unrolling.
430   //
431   // Control flow of an example (ignoring critical edges splitting).
432   //
433   //       Before                    After
434   //
435   //         |B|                      |B|
436   //          |                        |
437   //          v                        v
438   //         |1|                      |1|
439   //          |                        |
440   //          v                        v
441   //         |2|<-\                   |2A|<-\
442   //         / \  /                   / \    \
443   //        v   v/                   /   v    \
444   //       |4|  |3|                 /   |3A|   |
445   //        |                      /     /    /
446   //        v                     |     v    /
447   //       |E|                     \   |2|  /
448   //                                \ / \  /
449   //                                 v   v/
450   //                                |4| |3|
451   //                                 |
452   //                                 v
453   //                                |E|
DoUnrolling()454   HBasicBlock* DoUnrolling() {
455     return DoLoopTransformationImpl(TransformationKind::kUnrolling);
456   }
457 
458   // Perform loop versioning.
459   //
460   // Control flow of an example (ignoring critical edges splitting).
461   //
462   //       Before                    After
463   //
464   //         |B|                      |B|
465   //          |                        |
466   //          v                        v
467   //         |1|                      |1|_________
468   //          |                        |          |
469   //          v                        v          v
470   //         |2|<-\                   |2|<-\     |2A|<-\
471   //         / \  /                   / \  /     /  \  /
472   //        v   v/                   |   v/      |   v/
473   //        |   |3|                  |  |3|      | |3A|
474   //        |                        | __________|
475   //        |                        ||
476   //        v                        vv
477   //       |4|                       |4|
478   //        |                         |
479   //        v                         v
480   //       |E|                       |E|
DoVersioning()481   HBasicBlock* DoVersioning() {
482     return DoLoopTransformationImpl(TransformationKind::kVersioning);
483   }
484 
GetRegionToBeAdjusted()485   HLoopInformation* GetRegionToBeAdjusted() const { return cloner_.GetRegionToBeAdjusted(); }
486 
487  protected:
488   enum class TransformationKind {
489     kPeeling,
490     kUnrolling,
491     kVersioning,
492   };
493 
494   // Applies a specific loop transformation to the loop.
495   HBasicBlock* DoLoopTransformationImpl(TransformationKind transformation);
496 
497  private:
498   HLoopInformation* loop_info_;
499   SuperblockCloner cloner_;
500 
501   DISALLOW_COPY_AND_ASSIGN(LoopClonerHelper);
502 };
503 
504 // Helper class to perform loop peeling/unrolling/versioning.
505 //
506 // This helper should be used when there is no need to get correspondence information between
507 // original and copied basic blocks/instructions.
508 class LoopClonerSimpleHelper : public ValueObject {
509  public:
510   LoopClonerSimpleHelper(HLoopInformation* info, InductionVarRange* induction_range);
IsLoopClonable()511   bool IsLoopClonable() const { return helper_.IsLoopClonable(); }
DoPeeling()512   HBasicBlock* DoPeeling() { return helper_.DoPeeling(); }
DoUnrolling()513   HBasicBlock* DoUnrolling() { return helper_.DoUnrolling(); }
DoVersioning()514   HBasicBlock* DoVersioning() { return helper_.DoVersioning(); }
GetRegionToBeAdjusted()515   HLoopInformation* GetRegionToBeAdjusted() const { return helper_.GetRegionToBeAdjusted(); }
516 
GetBasicBlockMap()517   const SuperblockCloner::HBasicBlockMap* GetBasicBlockMap() const { return &bb_map_; }
GetInstructionMap()518   const SuperblockCloner::HInstructionMap* GetInstructionMap() const { return &hir_map_; }
519 
520  private:
521   SuperblockCloner::HBasicBlockMap bb_map_;
522   SuperblockCloner::HInstructionMap hir_map_;
523   LoopClonerHelper helper_;
524 
525   DISALLOW_COPY_AND_ASSIGN(LoopClonerSimpleHelper);
526 };
527 
528 // Collects edge remapping info for loop peeling/unrolling for the loop specified by loop info.
529 void CollectRemappingInfoForPeelUnroll(bool to_unroll,
530                                        HLoopInformation* loop_info,
531                                        SuperblockCloner::HEdgeSet* remap_orig_internal,
532                                        SuperblockCloner::HEdgeSet* remap_copy_internal,
533                                        SuperblockCloner::HEdgeSet* remap_incoming);
534 
535 // Returns whether blocks from 'work_set' are reachable from the rest of the graph.
536 //
537 // Returns whether such a set 'outer_entries' of basic blocks exists that:
538 // - each block from 'outer_entries' is not from 'work_set'.
539 // - each block from 'work_set' is reachable from at least one block from 'outer_entries'.
540 //
541 // After the function returns work_set contains only blocks from the original 'work_set'
542 // which are unreachable from the rest of the graph.
543 bool IsSubgraphConnected(SuperblockCloner::HBasicBlockSet* work_set, HGraph* graph);
544 
545 // Returns a common predecessor of loop1 and loop2 in the loop tree or nullptr if it is the whole
546 // graph.
547 HLoopInformation* FindCommonLoop(HLoopInformation* loop1, HLoopInformation* loop2);
548 }  // namespace art
549 
550 namespace std {
551 
552 template <>
553 struct hash<art::HEdge> {
554   size_t operator()(art::HEdge const& x) const noexcept  {
555     // Use Cantor pairing function as the hash function.
556     size_t a = x.GetFrom();
557     size_t b = x.GetTo();
558     return (a + b) * (a + b + 1) / 2 + b;
559   }
560 };
561 ostream& operator<<(ostream& os, const art::HEdge& e);
562 
563 }  // namespace std
564 
565 #endif  // ART_COMPILER_OPTIMIZING_SUPERBLOCK_CLONER_H_
566