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 "base/macros.h"
24 #include "nodes.h"
25 
26 namespace art HIDDEN {
27 
28 class InductionVarRange;
29 
30 static const bool kSuperblockClonerLogging = false;
31 static const bool kSuperblockClonerVerify = false;
32 
33 // Represents an edge between two HBasicBlocks.
34 //
35 // Note: objects of this class are small - pass them by value.
36 class HEdge : public ArenaObject<kArenaAllocSuperblockCloner> {
37  public:
HEdge(HBasicBlock * from,HBasicBlock * to)38   HEdge(HBasicBlock* from, HBasicBlock* to) : from_(from->GetBlockId()), to_(to->GetBlockId()) {
39     DCHECK_NE(to_, kInvalidBlockId);
40     DCHECK_NE(from_, kInvalidBlockId);
41   }
HEdge(uint32_t from,uint32_t to)42   HEdge(uint32_t from, uint32_t to) : from_(from), to_(to) {
43     DCHECK_NE(to_, kInvalidBlockId);
44     DCHECK_NE(from_, kInvalidBlockId);
45   }
HEdge()46   HEdge() : from_(kInvalidBlockId), to_(kInvalidBlockId) {}
47 
GetFrom()48   uint32_t GetFrom() const { return from_; }
GetTo()49   uint32_t GetTo() const { return to_; }
50 
51   bool operator==(const HEdge& other) const {
52     return this->from_ == other.from_ && this->to_ == other.to_;
53   }
54 
55   bool operator!=(const HEdge& other) const { return !operator==(other); }
56   void Dump(std::ostream& stream) const;
57 
58   // Returns whether an edge represents a valid edge in CF graph: whether the from_ block
59   // has to_ block as a successor.
IsValid()60   bool IsValid() const { return from_ != kInvalidBlockId && to_ != kInvalidBlockId; }
61 
62  private:
63   // Predecessor block id.
64   uint32_t from_;
65   // Successor block id.
66   uint32_t to_;
67 };
68 
69 // Returns whether a HEdge edge corresponds to an existing edge in the graph.
IsEdgeValid(HEdge edge,HGraph * graph)70 inline bool IsEdgeValid(HEdge edge, HGraph* graph) {
71   if (!edge.IsValid()) {
72     return false;
73   }
74   uint32_t from = edge.GetFrom();
75   uint32_t to = edge.GetTo();
76   if (from >= graph->GetBlocks().size() || to >= graph->GetBlocks().size()) {
77     return false;
78   }
79 
80   HBasicBlock* block_from = graph->GetBlocks()[from];
81   HBasicBlock* block_to = graph->GetBlocks()[to];
82   if (block_from == nullptr || block_to == nullptr) {
83     return false;
84   }
85 
86   return block_from->HasSuccessor(block_to, 0);
87 }
88 
89 // SuperblockCloner provides a feature of cloning subgraphs in a smart, high level way without
90 // fine grain manipulation with IR; data flow and graph properties are resolved/adjusted
91 // automatically. The clone transformation is defined by specifying a set of basic blocks to copy
92 // and a set of rules how to treat edges, remap their successors. By using this approach such
93 // optimizations as Branch Target Expansion, Loop Peeling, Loop Unrolling, Loop Versioning can be
94 // implemented.
95 //
96 // The idea of the transformation is based on "Superblock cloning" technique described in the book
97 // "Engineering a Compiler. Second Edition", Keith D. Cooper, Linda Torczon, Rice University
98 // Houston, Texas. 2nd edition, Morgan Kaufmann. The original paper is "The Superblock: An Efective
99 // Technique for VLIW and Superscalar Compilation" by Hwu, W.M.W., Mahlke, S.A., Chen, W.Y. et al.
100 // J Supercomput (1993) 7: 229. doi:10.1007/BF01205185.
101 //
102 // There are two states of the IR graph: original graph (before the transformation) and
103 // copy graph (after).
104 //
105 // Before the transformation:
106 // Defining a set of basic block to copy (orig_bb_set) partitions all of the edges in the original
107 // graph into 4 categories/sets (use the following notation for edges: "(pred, succ)",
108 // where pred, succ - basic blocks):
109 //  - internal - pred, succ are members of ‘orig_bb_set’.
110 //  - outside  - pred, succ are not members of ‘orig_bb_set’.
111 //  - incoming - pred is not a member of ‘orig_bb_set’, succ is.
112 //  - outgoing - pred is a member of ‘orig_bb_set’, succ is not.
113 //
114 // Transformation:
115 //
116 // 1. Initial cloning:
117 //   1.1. For each ‘orig_block’ in orig_bb_set create a copy ‘copy_block’; these new blocks
118 //        form ‘copy_bb_set’.
119 //   1.2. For each edge (X, Y) from internal set create an edge (X_1, Y_1) where X_1, Y_1 are the
120 //        copies of X, Y basic blocks correspondingly; these new edges form ‘copy_internal’ edge
121 //        set.
122 //   1.3. For each edge (X, Y) from outgoing set create an edge (X_1, Y_1) where X_1, Y_1 are the
123 //        copies of X, Y basic blocks correspondingly; these new edges form ‘copy_outgoing’ edge
124 //        set.
125 // 2. Successors remapping.
126 //   2.1. 'remap_orig_internal’ - set of edges (X, Y) from ‘orig_bb_set’ whose successors should
127 //        be remapped to copy nodes: ((X, Y) will be transformed into (X, Y_1)).
128 //   2.2. ‘remap_copy_internal’ - set of edges (X_1, Y_1) from ‘copy_bb_set’ whose successors
129 //        should be remapped to copy nodes: (X_1, Y_1) will be transformed into (X_1, Y)).
130 //   2.3. 'remap_incoming’ - set of edges (X, Y) from the ‘incoming’ edge set in the original graph
131 //        whose successors should be remapped to copies nodes: ((X, Y) will be transformed into
132 //        (X, Y_1)).
133 // 3. Adjust control flow structures and relations (dominance, reverse post order, loops, etc).
134 // 4. Fix/resolve data flow.
135 // 5. Do cleanups (DCE, critical edges splitting, etc).
136 //
137 class SuperblockCloner : public ValueObject {
138  public:
139   // TODO: Investigate optimal types for the containers.
140   using HBasicBlockMap = ArenaSafeMap<HBasicBlock*, HBasicBlock*>;
141   using HInstructionMap = ArenaSafeMap<HInstruction*, HInstruction*>;
142   using HBasicBlockSet = ArenaBitVector;
143   using HEdgeSet = ArenaHashSet<HEdge>;
144 
145   SuperblockCloner(HGraph* graph,
146                    const HBasicBlockSet* orig_bb_set,
147                    HBasicBlockMap* bb_map,
148                    HInstructionMap* hir_map,
149                    InductionVarRange* induction_range);
150 
151   // Sets edge successor remapping info specified by corresponding edge sets.
152   void SetSuccessorRemappingInfo(const HEdgeSet* remap_orig_internal,
153                                  const HEdgeSet* remap_copy_internal,
154                                  const HEdgeSet* remap_incoming);
155 
156   // Returns whether the specified subgraph is copyable.
157   // TODO: Start from small range of graph patterns then extend it.
158   bool IsSubgraphClonable() const;
159 
160   // Returns whether selected subgraph satisfies the criteria for fast data flow resolution
161   // when iterative DF algorithm is not required and dominators/instructions inputs can be
162   // trivially adjusted.
163   //
164   // TODO: formally describe the criteria.
165   //
166   // Loop peeling, unrolling and versioning satisfy the criteria.
167   bool IsFastCase() const;
168 
169   // Runs the copy algorithm according to the description.
170   void Run();
171 
172   // Cleans up the graph after transformation: splits critical edges, recalculates control flow
173   // information (back-edges, dominators, loop info, etc), eliminates redundant phis.
174   void CleanUp();
175 
176   // Returns a clone of a basic block (orig_block).
177   //
178   //  - The copy block will have no successors/predecessors; they should be set up manually.
179   //  - For each instruction in the orig_block a copy is created and inserted into the copy block;
180   //    this correspondence is recorded in the map (old instruction, new instruction).
181   //  - Graph HIR is not valid after this transformation: all of the HIRs have their inputs the
182   //    same, as in the original block, PHIs do not reflect a correct correspondence between the
183   //    value and predecessors (as the copy block has no predecessors by now), etc.
184   HBasicBlock* CloneBasicBlock(const HBasicBlock* orig_block);
185 
186   // Creates a clone for each basic blocks in orig_bb_set adding corresponding entries into bb_map_
187   // and hir_map_.
188   void CloneBasicBlocks();
189 
GetInstrCopy(HInstruction * orig_instr)190   HInstruction* GetInstrCopy(HInstruction* orig_instr) const {
191     auto copy_input_iter = hir_map_->find(orig_instr);
192     DCHECK(copy_input_iter != hir_map_->end());
193     return copy_input_iter->second;
194   }
195 
GetBlockCopy(HBasicBlock * orig_block)196   HBasicBlock* GetBlockCopy(HBasicBlock* orig_block) const {
197     HBasicBlock* block = bb_map_->Get(orig_block);
198     DCHECK(block != nullptr);
199     return block;
200   }
201 
GetInstrOrig(HInstruction * copy_instr)202   HInstruction* GetInstrOrig(HInstruction* copy_instr) const {
203     for (auto it : *hir_map_) {
204       if (it.second == copy_instr) {
205         return it.first;
206       }
207     }
208     return nullptr;
209   }
210 
IsInOrigBBSet(uint32_t block_id)211   bool IsInOrigBBSet(uint32_t block_id) const {
212     return orig_bb_set_.IsBitSet(block_id);
213   }
214 
IsInOrigBBSet(const HBasicBlock * block)215   bool IsInOrigBBSet(const HBasicBlock* block) const {
216     return IsInOrigBBSet(block->GetBlockId());
217   }
218 
219   // Returns the area (the most outer loop) in the graph for which control flow (back edges, loops,
220   // dominators) needs to be adjusted.
GetRegionToBeAdjusted()221   HLoopInformation* GetRegionToBeAdjusted() const {
222     return outer_loop_;
223   }
224 
225  private:
226   // Fills the 'exits' vector with the subgraph exits.
227   void SearchForSubgraphExits(ArenaVector<HBasicBlock*>* exits) const;
228 
229   // Finds and records information about the area in the graph for which control flow (back edges,
230   // loops, dominators) needs to be adjusted.
231   void FindAndSetLocalAreaForAdjustments();
232 
233   // Remaps edges' successors according to the info specified in the edges sets.
234   //
235   // Only edge successors/predecessors and phis' input records (to have a correspondence between
236   // a phi input record (not value) and a block's predecessor) are adjusted at this stage: neither
237   // phis' nor instructions' inputs values are resolved.
238   void RemapEdgesSuccessors();
239 
240   // Adjusts control flow (back edges, loops, dominators) for the local area defined by
241   // FindAndSetLocalAreaForAdjustments.
242   void AdjustControlFlowInfo();
243 
244   // Resolves Data Flow - adjusts phis' and instructions' inputs in order to have a valid graph in
245   // the SSA form.
246   void ResolveDataFlow();
247 
248   //
249   // Helpers for live-outs processing and Subgraph-closed SSA.
250   //
251   //  - live-outs - values which are defined inside the subgraph and have uses outside.
252   //  - Subgraph-closed SSA - SSA form for which all the values defined inside the subgraph
253   //    have no outside uses except for the phi-nodes in the subgraph exits.
254   //
255   // Note: now if the subgraph has live-outs it is only clonable if it has a single exit; this
256   // makes the subgraph-closed SSA form construction much easier.
257   //
258   // TODO: Support subgraphs with live-outs and multiple exits.
259   //
260 
261   // For each live-out value 'val' in the region puts a record <val, val> into the map.
262   // Returns whether all of the instructions in the subgraph are clonable.
263   bool CollectLiveOutsAndCheckClonable(HInstructionMap* live_outs_) const;
264 
265   // Constructs Subgraph-closed SSA; precondition - a subgraph has a single exit.
266   //
267   // For each live-out 'val' in 'live_outs_' map inserts a HPhi 'phi' into the exit node, updates
268   // the record in the map to <val, phi> and replaces all outside uses with this phi.
269   void ConstructSubgraphClosedSSA();
270 
271   // Fixes the data flow for the live-out 'val' by adding a 'copy_val' input to the corresponding
272   // (<val, phi>) phi after the cloning is done.
273   void FixSubgraphClosedSSAAfterCloning();
274 
275   //
276   // Helpers for CloneBasicBlock.
277   //
278 
279   // Adjusts copy instruction's inputs: if the input of the original instruction is defined in the
280   // orig_bb_set, replaces it with a corresponding copy otherwise leaves it the same as original.
281   void ReplaceInputsWithCopies(HInstruction* copy_instr);
282 
283   // Recursively clones the environment for the copy instruction. If the input of the original
284   // environment is defined in the orig_bb_set, replaces it with a corresponding copy otherwise
285   // leaves it the same as original.
286   void DeepCloneEnvironmentWithRemapping(HInstruction* copy_instr, const HEnvironment* orig_env);
287 
288   //
289   // Helpers for RemapEdgesSuccessors.
290   //
291 
292   // Remaps incoming or original internal edge to its copy, adjusts the phi inputs in orig_succ and
293   // copy_succ.
294   void RemapOrigInternalOrIncomingEdge(HBasicBlock* orig_block, HBasicBlock* orig_succ);
295 
296   // Adds copy internal edge (from copy_block to copy_succ), updates phis in the copy_succ.
297   void AddCopyInternalEdge(HBasicBlock* orig_block, HBasicBlock* orig_succ);
298 
299   // Remaps copy internal edge to its origin, adjusts the phi inputs in orig_succ.
300   void RemapCopyInternalEdge(HBasicBlock* orig_block, HBasicBlock* orig_succ);
301 
302   // Checks whether the edges remapping info corresponds to the subgraph versioning case:
303   //  - none of the incoming edges are to be remapped (they are being duplicated).
304   //  - none of the internal edges are to be remapped.
305   bool IsRemapInfoForVersioning() const;
306 
307   // Processes incoming edges for subgraph versioning case: for each incoming edge (X, Y) adds
308   // an edge (X, Y_1) where Y_1 = Copy(Y) and add corresponding phi input to copy phi.
309   //
310   // Note: such node X will now have two successors, its unconditional branch instruction
311   // will be invalid and should be adjusted to some conditional branch by the client code.
312   void CopyIncomingEdgesForVersioning();
313 
314   //
315   // Local versions of control flow calculation/adjustment routines.
316   //
317 
318   void FindBackEdgesLocal(HBasicBlock* entry_block, ArenaBitVector* local_set);
319   void RecalculateBackEdgesInfo(ArenaBitVector* outer_loop_bb_set);
320   GraphAnalysisResult AnalyzeLoopsLocally(ArenaBitVector* outer_loop_bb_set);
321   void CleanUpControlFlow();
322 
323   //
324   // Helpers for ResolveDataFlow
325   //
326 
327   // Resolves the inputs of the phi.
328   void ResolvePhi(HPhi* phi);
329 
330   // Update induction range after when fixing SSA.
331   void UpdateInductionRangeInfoOf(
332       HInstruction* user, HInstruction* old_instruction, HInstruction* replacement);
333 
334   //
335   // Debug and logging methods.
336   //
337   void CheckInstructionInputsRemapping(HInstruction* orig_instr);
338   bool CheckRemappingInfoIsValid();
339   void VerifyGraph();
340   void DumpInputSets();
341 
GetBlockById(uint32_t block_id)342   HBasicBlock* GetBlockById(uint32_t block_id) const {
343     DCHECK(block_id < graph_->GetBlocks().size());
344     HBasicBlock* block = graph_->GetBlocks()[block_id];
345     DCHECK(block != nullptr);
346     return block;
347   }
348 
349   HGraph* const graph_;
350   ArenaAllocator* const arena_;
351 
352   // Set of basic block in the original graph to be copied.
353   HBasicBlockSet orig_bb_set_;
354 
355   // Sets of edges which require successors remapping.
356   const HEdgeSet* remap_orig_internal_;
357   const HEdgeSet* remap_copy_internal_;
358   const HEdgeSet* remap_incoming_;
359 
360   // Correspondence map for blocks: (original block, copy block).
361   HBasicBlockMap* bb_map_;
362   // Correspondence map for instructions: (original HInstruction, copy HInstruction).
363   HInstructionMap* hir_map_;
364   // As a result of cloning, the induction range analysis information can be invalidated
365   // and must be updated. If not null, the cloner updates it for changed instructions.
366   InductionVarRange* induction_range_;
367   // Area in the graph for which control flow (back edges, loops, dominators) needs to be adjusted.
368   HLoopInformation* outer_loop_;
369   HBasicBlockSet outer_loop_bb_set_;
370 
371   HInstructionMap live_outs_;
372 
373   ART_FRIEND_TEST(SuperblockClonerTest, AdjustControlFlowInfo);
374   ART_FRIEND_TEST(SuperblockClonerTest, IsGraphConnected);
375 
376   DISALLOW_COPY_AND_ASSIGN(SuperblockCloner);
377 };
378 
379 // Helper class to perform loop peeling/unrolling/versioning.
380 //
381 // This helper should be used when correspondence map between original and copied
382 // basic blocks/instructions are demanded.
383 class LoopClonerHelper : public ValueObject {
384  public:
LoopClonerHelper(HLoopInformation * info,SuperblockCloner::HBasicBlockMap * bb_map,SuperblockCloner::HInstructionMap * hir_map,InductionVarRange * induction_range)385   LoopClonerHelper(HLoopInformation* info,
386                    SuperblockCloner::HBasicBlockMap* bb_map,
387                    SuperblockCloner::HInstructionMap* hir_map,
388                    InductionVarRange* induction_range) :
389       loop_info_(info),
390       cloner_(info->GetHeader()->GetGraph(), &info->GetBlocks(), bb_map, hir_map, induction_range) {
391     // For now do transformations only for natural loops.
392     DCHECK(!info->IsIrreducible());
393   }
394 
395   // Returns whether the loop can be peeled/unrolled (static function).
396   static bool IsLoopClonable(HLoopInformation* loop_info);
397 
398   // Returns whether the loop can be peeled/unrolled.
IsLoopClonable()399   bool IsLoopClonable() const { return cloner_.IsSubgraphClonable(); }
400 
401   // Perform loop peeling.
402   //
403   // Control flow of an example (ignoring critical edges splitting).
404   //
405   //       Before                    After
406   //
407   //         |B|                      |B|
408   //          |                        |
409   //          v                        v
410   //         |1|                      |1|
411   //          |                        |
412   //          v                        v
413   //         |2|<-\                  |2A|
414   //         / \  /                   / \
415   //        v   v/                   /   v
416   //       |4|  |3|                 /   |3A|
417   //        |                      /     /
418   //        v                     |     v
419   //       |E|                     \   |2|<-\
420   //                                \ / \   /
421   //                                 v   v /
422   //                                |4|  |3|
423   //                                 |
424   //                                 v
425   //                                |E|
DoPeeling()426   HBasicBlock* DoPeeling() {
427     return DoLoopTransformationImpl(TransformationKind::kPeeling);
428   }
429 
430   // Perform loop unrolling.
431   //
432   // Control flow of an example (ignoring critical edges splitting).
433   //
434   //       Before                    After
435   //
436   //         |B|                      |B|
437   //          |                        |
438   //          v                        v
439   //         |1|                      |1|
440   //          |                        |
441   //          v                        v
442   //         |2|<-\                   |2A|<-\
443   //         / \  /                   / \    \
444   //        v   v/                   /   v    \
445   //       |4|  |3|                 /   |3A|   |
446   //        |                      /     /    /
447   //        v                     |     v    /
448   //       |E|                     \   |2|  /
449   //                                \ / \  /
450   //                                 v   v/
451   //                                |4| |3|
452   //                                 |
453   //                                 v
454   //                                |E|
DoUnrolling()455   HBasicBlock* DoUnrolling() {
456     return DoLoopTransformationImpl(TransformationKind::kUnrolling);
457   }
458 
459   // Perform loop versioning.
460   //
461   // Control flow of an example (ignoring critical edges splitting).
462   //
463   //       Before                    After
464   //
465   //         |B|                      |B|
466   //          |                        |
467   //          v                        v
468   //         |1|                      |1|_________
469   //          |                        |          |
470   //          v                        v          v
471   //         |2|<-\                   |2|<-\     |2A|<-\
472   //         / \  /                   / \  /     /  \  /
473   //        v   v/                   |   v/      |   v/
474   //        |   |3|                  |  |3|      | |3A|
475   //        |                        | __________|
476   //        |                        ||
477   //        v                        vv
478   //       |4|                       |4|
479   //        |                         |
480   //        v                         v
481   //       |E|                       |E|
DoVersioning()482   HBasicBlock* DoVersioning() {
483     return DoLoopTransformationImpl(TransformationKind::kVersioning);
484   }
485 
GetRegionToBeAdjusted()486   HLoopInformation* GetRegionToBeAdjusted() const { return cloner_.GetRegionToBeAdjusted(); }
487 
488  protected:
489   enum class TransformationKind {
490     kPeeling,
491     kUnrolling,
492     kVersioning,
493   };
494 
495   // Applies a specific loop transformation to the loop.
496   HBasicBlock* DoLoopTransformationImpl(TransformationKind transformation);
497 
498  private:
499   HLoopInformation* loop_info_;
500   SuperblockCloner cloner_;
501 
502   DISALLOW_COPY_AND_ASSIGN(LoopClonerHelper);
503 };
504 
505 // Helper class to perform loop peeling/unrolling/versioning.
506 //
507 // This helper should be used when there is no need to get correspondence information between
508 // original and copied basic blocks/instructions.
509 class LoopClonerSimpleHelper : public ValueObject {
510  public:
511   LoopClonerSimpleHelper(HLoopInformation* info, InductionVarRange* induction_range);
IsLoopClonable()512   bool IsLoopClonable() const { return helper_.IsLoopClonable(); }
DoPeeling()513   HBasicBlock* DoPeeling() { return helper_.DoPeeling(); }
DoUnrolling()514   HBasicBlock* DoUnrolling() { return helper_.DoUnrolling(); }
DoVersioning()515   HBasicBlock* DoVersioning() { return helper_.DoVersioning(); }
GetRegionToBeAdjusted()516   HLoopInformation* GetRegionToBeAdjusted() const { return helper_.GetRegionToBeAdjusted(); }
517 
GetBasicBlockMap()518   const SuperblockCloner::HBasicBlockMap* GetBasicBlockMap() const { return &bb_map_; }
GetInstructionMap()519   const SuperblockCloner::HInstructionMap* GetInstructionMap() const { return &hir_map_; }
520 
521  private:
522   SuperblockCloner::HBasicBlockMap bb_map_;
523   SuperblockCloner::HInstructionMap hir_map_;
524   LoopClonerHelper helper_;
525 
526   DISALLOW_COPY_AND_ASSIGN(LoopClonerSimpleHelper);
527 };
528 
529 // Collects edge remapping info for loop peeling/unrolling for the loop specified by loop info.
530 void CollectRemappingInfoForPeelUnroll(bool to_unroll,
531                                        HLoopInformation* loop_info,
532                                        SuperblockCloner::HEdgeSet* remap_orig_internal,
533                                        SuperblockCloner::HEdgeSet* remap_copy_internal,
534                                        SuperblockCloner::HEdgeSet* remap_incoming);
535 
536 // Returns whether blocks from 'work_set' are reachable from the rest of the graph.
537 //
538 // Returns whether such a set 'outer_entries' of basic blocks exists that:
539 // - each block from 'outer_entries' is not from 'work_set'.
540 // - each block from 'work_set' is reachable from at least one block from 'outer_entries'.
541 //
542 // After the function returns work_set contains only blocks from the original 'work_set'
543 // which are unreachable from the rest of the graph.
544 bool IsSubgraphConnected(SuperblockCloner::HBasicBlockSet* work_set, HGraph* graph);
545 
546 // Returns a common predecessor of loop1 and loop2 in the loop tree or nullptr if it is the whole
547 // graph.
548 HLoopInformation* FindCommonLoop(HLoopInformation* loop1, HLoopInformation* loop2);
549 }  // namespace art
550 
551 namespace std {
552 
553 template <>
554 struct hash<art::HEdge> {
555   size_t operator()(art::HEdge const& x) const noexcept  {
556     // Use Cantor pairing function as the hash function.
557     size_t a = x.GetFrom();
558     size_t b = x.GetTo();
559     return (a + b) * (a + b + 1) / 2 + b;
560   }
561 };
562 ostream& operator<<(ostream& os, const art::HEdge& e);
563 
564 }  // namespace std
565 
566 #endif  // ART_COMPILER_OPTIMIZING_SUPERBLOCK_CLONER_H_
567