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llvm-project/mlir/lib/Analysis/AffineStructures.cpp

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//===- AffineStructures.cpp - MLIR Affine Structures Class-------*- C++ -*-===//
//
// Copyright 2019 The MLIR Authors.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
// =============================================================================
//
// Structures for affine/polyhedral analysis of MLIR functions.
//
//===----------------------------------------------------------------------===//
#include "mlir/Analysis/AffineStructures.h"
#include "mlir/Analysis/AffineAnalysis.h"
#include "mlir/IR/AffineExprVisitor.h"
#include "mlir/IR/AffineMap.h"
#include "mlir/IR/BuiltinOps.h"
#include "mlir/IR/IntegerSet.h"
#include "mlir/IR/Statements.h"
#include "mlir/Support/MathExtras.h"
#include "llvm/ADT/DenseSet.h"
#include "llvm/Support/Debug.h"
#include "llvm/Support/raw_ostream.h"
#define DEBUG_TYPE "affine-structures"
using namespace mlir;
using namespace llvm;
namespace {
// Affine map composition terminology:
// *) current: refers to the target map of the composition operation. It is the
// map into which results from the 'input' map are forward substituted.
// *) input: refers to the map which is being forward substituted into the
// 'current' map.
// *) output: refers to the resulting affine map after composition.
// AffineMapCompositionUpdate encapsulates the state necessary to compose
// AffineExprs for two affine maps using AffineExprComposer (see below).
struct AffineMapCompositionUpdate {
using PositionMap = DenseMap<unsigned, unsigned>;
explicit AffineMapCompositionUpdate(ArrayRef<AffineExpr> inputResults)
: inputResults(inputResults), outputNumDims(0), outputNumSymbols(0) {}
// Map from 'curr' affine map dim position to 'output' affine map
// dim position.
PositionMap currDimMap;
// Map from dim position of 'curr' affine map to index into 'inputResults'.
PositionMap currDimToInputResultMap;
// Map from 'curr' affine map symbol position to 'output' affine map
// symbol position.
PositionMap currSymbolMap;
// Map from 'input' affine map dim position to 'output' affine map
// dim position.
PositionMap inputDimMap;
// Map from 'input' affine map symbol position to 'output' affine map
// symbol position.
PositionMap inputSymbolMap;
// Results of 'input' affine map.
ArrayRef<AffineExpr> inputResults;
// Number of dimension operands for 'output' affine map.
unsigned outputNumDims;
// Number of symbol operands for 'output' affine map.
unsigned outputNumSymbols;
};
// AffineExprComposer composes two AffineExprs as specified by 'mapUpdate'.
class AffineExprComposer {
public:
// Compose two AffineExprs using dimension and symbol position update maps,
// as well as input map result AffineExprs specified in 'mapUpdate'.
AffineExprComposer(const AffineMapCompositionUpdate &mapUpdate)
: mapUpdate(mapUpdate), walkingInputMap(false) {}
AffineExpr walk(AffineExpr expr) {
switch (expr.getKind()) {
case AffineExprKind::Add:
return walkBinExpr(
expr, [](AffineExpr lhs, AffineExpr rhs) { return lhs + rhs; });
case AffineExprKind::Mul:
return walkBinExpr(
expr, [](AffineExpr lhs, AffineExpr rhs) { return lhs * rhs; });
case AffineExprKind::Mod:
return walkBinExpr(
expr, [](AffineExpr lhs, AffineExpr rhs) { return lhs % rhs; });
case AffineExprKind::FloorDiv:
return walkBinExpr(expr, [](AffineExpr lhs, AffineExpr rhs) {
return lhs.floorDiv(rhs);
});
case AffineExprKind::CeilDiv:
return walkBinExpr(expr, [](AffineExpr lhs, AffineExpr rhs) {
return lhs.ceilDiv(rhs);
});
case AffineExprKind::Constant:
return expr;
case AffineExprKind::DimId: {
unsigned dimPosition = expr.cast<AffineDimExpr>().getPosition();
if (walkingInputMap) {
return getAffineDimExpr(mapUpdate.inputDimMap.lookup(dimPosition),
expr.getContext());
}
// Check if we are just mapping this dim to another position.
if (mapUpdate.currDimMap.count(dimPosition) > 0) {
assert(mapUpdate.currDimToInputResultMap.count(dimPosition) == 0);
return getAffineDimExpr(mapUpdate.currDimMap.lookup(dimPosition),
expr.getContext());
}
// We are substituting an input map result at 'dimPositon'
// Forward substitute currDimToInputResultMap[dimPosition] into this
// map.
AffineExprComposer composer(mapUpdate, /*walkingInputMap=*/true);
unsigned inputResultIndex =
mapUpdate.currDimToInputResultMap.lookup(dimPosition);
assert(inputResultIndex < mapUpdate.inputResults.size());
return composer.walk(mapUpdate.inputResults[inputResultIndex]);
}
case AffineExprKind::SymbolId:
unsigned symbolPosition = expr.cast<AffineSymbolExpr>().getPosition();
if (walkingInputMap) {
return getAffineSymbolExpr(
mapUpdate.inputSymbolMap.lookup(symbolPosition), expr.getContext());
}
return getAffineSymbolExpr(mapUpdate.currSymbolMap.lookup(symbolPosition),
expr.getContext());
}
}
private:
AffineExprComposer(const AffineMapCompositionUpdate &mapUpdate,
bool walkingInputMap)
: mapUpdate(mapUpdate), walkingInputMap(walkingInputMap) {}
AffineExpr walkBinExpr(AffineExpr expr,
std::function<AffineExpr(AffineExpr, AffineExpr)> op) {
auto binOpExpr = expr.cast<AffineBinaryOpExpr>();
return op(walk(binOpExpr.getLHS()), walk(binOpExpr.getRHS()));
}
// Map update specifies to dim and symbol postion maps, as well as the input
// result AffineExprs to forward subustitute into the input map.
const AffineMapCompositionUpdate &mapUpdate;
// True if we are walking an AffineExpr in the 'input' map, false if
// we are walking the 'input' map.
bool walkingInputMap;
};
} // end anonymous namespace
static void
forwardSubstituteMutableAffineMap(const AffineMapCompositionUpdate &mapUpdate,
MutableAffineMap *map) {
for (unsigned i = 0, e = map->getNumResults(); i < e; i++) {
AffineExprComposer composer(mapUpdate);
map->setResult(i, composer.walk(map->getResult(i)));
}
// TODO(andydavis) Evaluate if we need to update range sizes here.
map->setNumDims(mapUpdate.outputNumDims);
map->setNumSymbols(mapUpdate.outputNumSymbols);
}
//===----------------------------------------------------------------------===//
// MutableAffineMap.
//===----------------------------------------------------------------------===//
MutableAffineMap::MutableAffineMap(AffineMap map)
: numDims(map.getNumDims()), numSymbols(map.getNumSymbols()),
// A map always has at least 1 result by construction
context(map.getResult(0).getContext()) {
for (auto result : map.getResults())
results.push_back(result);
for (auto rangeSize : map.getRangeSizes())
results.push_back(rangeSize);
}
void MutableAffineMap::reset(AffineMap map) {
results.clear();
rangeSizes.clear();
numDims = map.getNumDims();
numSymbols = map.getNumSymbols();
// A map always has at least 1 result by construction
context = map.getResult(0).getContext();
for (auto result : map.getResults())
results.push_back(result);
for (auto rangeSize : map.getRangeSizes())
results.push_back(rangeSize);
}
bool MutableAffineMap::isMultipleOf(unsigned idx, int64_t factor) const {
if (results[idx].isMultipleOf(factor))
return true;
// TODO(bondhugula): use simplifyAffineExpr and FlatAffineConstraints to
// complete this (for a more powerful analysis).
return false;
}
// Simplifies the result affine expressions of this map. The expressions have to
// be pure for the simplification implemented.
void MutableAffineMap::simplify() {
// Simplify each of the results if possible.
// TODO(ntv): functional-style map
for (unsigned i = 0, e = getNumResults(); i < e; i++) {
results[i] = simplifyAffineExpr(getResult(i), numDims, numSymbols);
}
}
AffineMap MutableAffineMap::getAffineMap() const {
return AffineMap::get(numDims, numSymbols, results, rangeSizes);
}
MutableIntegerSet::MutableIntegerSet(IntegerSet set, MLIRContext *context)
: numDims(set.getNumDims()), numSymbols(set.getNumSymbols()),
context(context) {
// TODO(bondhugula)
}
// Universal set.
MutableIntegerSet::MutableIntegerSet(unsigned numDims, unsigned numSymbols,
MLIRContext *context)
: numDims(numDims), numSymbols(numSymbols), context(context) {}
//===----------------------------------------------------------------------===//
// AffineValueMap.
//===----------------------------------------------------------------------===//
AffineValueMap::AffineValueMap(const AffineApplyOp &op)
: map(op.getAffineMap()) {
for (auto *operand : op.getOperands())
operands.push_back(const_cast<Value *>(operand));
for (unsigned i = 0, e = op.getNumResults(); i < e; i++)
results.push_back(const_cast<Value *>(op.getResult(i)));
}
AffineValueMap::AffineValueMap(AffineMap map, ArrayRef<Value *> operands)
: map(map) {
for (Value *operand : operands) {
this->operands.push_back(operand);
}
}
void AffineValueMap::reset(AffineMap map, ArrayRef<Value *> operands) {
this->operands.clear();
this->results.clear();
this->map.reset(map);
for (Value *operand : operands) {
this->operands.push_back(operand);
}
}
void AffineValueMap::forwardSubstitute(const AffineApplyOp &inputOp) {
SmallVector<bool, 4> inputResultsToSubstitute(inputOp.getNumResults());
for (unsigned i = 0, e = inputOp.getNumResults(); i < e; i++)
inputResultsToSubstitute[i] = true;
forwardSubstitute(inputOp, inputResultsToSubstitute);
}
void AffineValueMap::forwardSubstituteSingle(const AffineApplyOp &inputOp,
unsigned inputResultIndex) {
SmallVector<bool, 4> inputResultsToSubstitute(inputOp.getNumResults(), false);
inputResultsToSubstitute[inputResultIndex] = true;
forwardSubstitute(inputOp, inputResultsToSubstitute);
}
// Returns true and sets 'indexOfMatch' if 'valueToMatch' is found in
// 'valuesToSearch' beginning at 'indexStart'. Returns false otherwise.
static bool findIndex(Value *valueToMatch, ArrayRef<Value *> valuesToSearch,
unsigned indexStart, unsigned *indexOfMatch) {
unsigned size = valuesToSearch.size();
for (unsigned i = indexStart; i < size; ++i) {
if (valueToMatch == valuesToSearch[i]) {
*indexOfMatch = i;
return true;
}
}
return false;
}
// AffineValueMap forward substitution composes results from the affine map
// associated with 'inputOp', with the map it currently represents. This is
// accomplished by updating its MutableAffineMap and operand list to represent
// a new 'output' map which is the composition of the 'current' and 'input'
// maps (see "Affine map composition terminology" above for details).
//
// Affine map forward substitution is comprised of the following steps:
// *) Compute input affine map result indices used by the current map.
// *) Gather all dim and symbol positions from all AffineExpr input results
// computed in previous step.
// *) Build output operand list:
// *) Add curr map dim operands:
// *) If curr dim operand is being forward substituted by result of input
// map, store mapping from curr postion to input result index.
// *) Else add curr dim operand to output operand list.
// *) Add input map dim operands:
// *) If input map dim operand is used (step 2), add to output operand
// list (scanning current list for dups before updating mapping).
// *) Add curr map dim symbols.
// *) Add input map dim symbols (if used from step 2), dedup if needed.
// *) Update operands and forward substitute new dim and symbol mappings
// into MutableAffineMap 'map'.
//
// TODO(andydavis) Move this to a function which can be shared with
// forwardSubstitute(const AffineValueMap &inputMap).
void AffineValueMap::forwardSubstitute(
const AffineApplyOp &inputOp, ArrayRef<bool> inputResultsToSubstitute) {
unsigned currNumDims = map.getNumDims();
unsigned inputNumResults = inputOp.getNumResults();
// Gather result indices from 'inputOp' used by current map.
DenseSet<unsigned> inputResultsUsed;
DenseMap<unsigned, unsigned> currOperandToInputResult;
for (unsigned i = 0; i < currNumDims; ++i) {
for (unsigned j = 0; j < inputNumResults; ++j) {
if (!inputResultsToSubstitute[j])
continue;
if (operands[i] == const_cast<Value *>(inputOp.getResult(j))) {
currOperandToInputResult[i] = j;
inputResultsUsed.insert(j);
}
}
}
// Return if there were no uses of 'inputOp' results in 'operands'.
if (inputResultsUsed.empty()) {
return;
}
class AffineExprPositionGatherer
: public AffineExprVisitor<AffineExprPositionGatherer> {
public:
unsigned numDims;
DenseSet<unsigned> *positions;
AffineExprPositionGatherer(unsigned numDims, DenseSet<unsigned> *positions)
: numDims(numDims), positions(positions) {}
void visitDimExpr(AffineDimExpr expr) {
positions->insert(expr.getPosition());
}
void visitSymbolExpr(AffineSymbolExpr expr) {
positions->insert(numDims + expr.getPosition());
}
};
// Gather dim and symbol positions from 'inputOp' on which
// 'inputResultsUsed' depend.
AffineMap inputMap = inputOp.getAffineMap();
unsigned inputNumDims = inputMap.getNumDims();
DenseSet<unsigned> inputPositionsUsed;
AffineExprPositionGatherer gatherer(inputNumDims, &inputPositionsUsed);
for (unsigned i = 0; i < inputNumResults; ++i) {
if (inputResultsUsed.count(i) == 0)
continue;
gatherer.walkPostOrder(inputMap.getResult(i));
}
// Build new output operands list and map update.
SmallVector<Value *, 4> outputOperands;
unsigned outputOperandPosition = 0;
AffineMapCompositionUpdate mapUpdate(inputOp.getAffineMap().getResults());
// Add dim operands from current map.
for (unsigned i = 0; i < currNumDims; ++i) {
if (currOperandToInputResult.count(i) > 0) {
mapUpdate.currDimToInputResultMap[i] = currOperandToInputResult[i];
} else {
mapUpdate.currDimMap[i] = outputOperandPosition++;
outputOperands.push_back(operands[i]);
}
}
// Add dim operands from input map.
for (unsigned i = 0; i < inputNumDims; ++i) {
// Skip input dim operands that we won't use.
if (inputPositionsUsed.count(i) == 0)
continue;
// Check if input operand has a dup in current operand list.
auto *inputOperand = const_cast<Value *>(inputOp.getOperand(i));
unsigned outputIndex;
if (findIndex(inputOperand, outputOperands, /*indexStart=*/0,
&outputIndex)) {
mapUpdate.inputDimMap[i] = outputIndex;
} else {
mapUpdate.inputDimMap[i] = outputOperandPosition++;
outputOperands.push_back(inputOperand);
}
}
// Done adding dimension operands, so store new output num dims.
unsigned outputNumDims = outputOperandPosition;
// Add symbol operands from current map.
unsigned currNumOperands = operands.size();
for (unsigned i = currNumDims; i < currNumOperands; ++i) {
unsigned currSymbolPosition = i - currNumDims;
unsigned outputSymbolPosition = outputOperandPosition - outputNumDims;
mapUpdate.currSymbolMap[currSymbolPosition] = outputSymbolPosition;
outputOperands.push_back(operands[i]);
++outputOperandPosition;
}
// Add symbol operands from input map.
unsigned inputNumOperands = inputOp.getNumOperands();
for (unsigned i = inputNumDims; i < inputNumOperands; ++i) {
// Skip input symbol operands that we won't use.
if (inputPositionsUsed.count(i) == 0)
continue;
unsigned inputSymbolPosition = i - inputNumDims;
// Check if input operand has a dup in current operand list.
auto *inputOperand = const_cast<Value *>(inputOp.getOperand(i));
// Find output operand index of 'inputOperand' dup.
unsigned outputIndex;
// Start at index 'outputNumDims' so that only symbol operands are searched.
if (findIndex(inputOperand, outputOperands, /*indexStart=*/outputNumDims,
&outputIndex)) {
unsigned outputSymbolPosition = outputIndex - outputNumDims;
mapUpdate.inputSymbolMap[inputSymbolPosition] = outputSymbolPosition;
} else {
unsigned outputSymbolPosition = outputOperandPosition - outputNumDims;
mapUpdate.inputSymbolMap[inputSymbolPosition] = outputSymbolPosition;
outputOperands.push_back(inputOperand);
++outputOperandPosition;
}
}
// Set output number of dimension and symbol operands.
mapUpdate.outputNumDims = outputNumDims;
mapUpdate.outputNumSymbols = outputOperands.size() - outputNumDims;
// Update 'operands' with new 'outputOperands'.
operands.swap(outputOperands);
// Forward substitute 'mapUpdate' into 'map'.
forwardSubstituteMutableAffineMap(mapUpdate, &map);
}
inline bool AffineValueMap::isMultipleOf(unsigned idx, int64_t factor) const {
return map.isMultipleOf(idx, factor);
}
/// This method uses the invariant that operands are always positionally aligned
/// with the AffineDimExpr in the underlying AffineMap.
bool AffineValueMap::isFunctionOf(unsigned idx, Value *value) const {
unsigned index;
findIndex(value, operands, /*indexStart=*/0, &index);
auto expr = const_cast<AffineValueMap *>(this)->getAffineMap().getResult(idx);
// TODO(ntv): this is better implemented on a flattened representation.
// At least for now it is conservative.
return expr.isFunctionOfDim(index);
}
Value *AffineValueMap::getOperand(unsigned i) const {
return static_cast<Value *>(operands[i]);
}
ArrayRef<Value *> AffineValueMap::getOperands() const {
return ArrayRef<Value *>(operands);
}
AffineMap AffineValueMap::getAffineMap() const { return map.getAffineMap(); }
AffineValueMap::~AffineValueMap() {}
//===----------------------------------------------------------------------===//
// FlatAffineConstraints.
//===----------------------------------------------------------------------===//
// Copy constructor.
FlatAffineConstraints::FlatAffineConstraints(
const FlatAffineConstraints &other) {
numReservedCols = other.numReservedCols;
numDims = other.getNumDimIds();
numSymbols = other.getNumSymbolIds();
numIds = other.getNumIds();
auto otherIds = other.getIds();
ids.reserve(numReservedCols);
ids.insert(ids.end(), otherIds.begin(), otherIds.end());
unsigned numReservedEqualities = other.getNumReservedEqualities();
unsigned numReservedInequalities = other.getNumReservedInequalities();
equalities.reserve(numReservedEqualities * numReservedCols);
inequalities.reserve(numReservedInequalities * numReservedCols);
for (unsigned r = 0, e = other.getNumInequalities(); r < e; r++) {
addInequality(other.getInequality(r));
}
for (unsigned r = 0, e = other.getNumEqualities(); r < e; r++) {
addEquality(other.getEquality(r));
}
}
// Clones this object.
std::unique_ptr<FlatAffineConstraints> FlatAffineConstraints::clone() const {
return std::make_unique<FlatAffineConstraints>(*this);
}
// Construct from an IntegerSet.
FlatAffineConstraints::FlatAffineConstraints(IntegerSet set)
: numReservedCols(set.getNumOperands() + 1),
numIds(set.getNumDims() + set.getNumSymbols()), numDims(set.getNumDims()),
numSymbols(set.getNumSymbols()) {
equalities.reserve(set.getNumEqualities() * numReservedCols);
inequalities.reserve(set.getNumInequalities() * numReservedCols);
ids.resize(numIds, None);
// Flatten expressions and add them to the constraint system.
std::vector<SmallVector<int64_t, 8>> flatExprs;
FlatAffineConstraints localVarCst;
if (!getFlattenedAffineExprs(set, &flatExprs, &localVarCst)) {
assert(false && "flattening unimplemented for semi-affine integer sets");
return;
}
assert(flatExprs.size() == set.getNumConstraints());
for (unsigned l = 0, e = localVarCst.getNumLocalIds(); l < e; l++) {
addLocalId(getNumLocalIds());
}
for (unsigned i = 0, e = flatExprs.size(); i < e; ++i) {
const auto &flatExpr = flatExprs[i];
assert(flatExpr.size() == getNumCols());
if (set.getEqFlags()[i]) {
addEquality(flatExpr);
} else {
addInequality(flatExpr);
}
}
// Add the other constraints involving local id's from flattening.
append(localVarCst);
}
void FlatAffineConstraints::reset(unsigned numReservedInequalities,
unsigned numReservedEqualities,
unsigned newNumReservedCols,
unsigned newNumDims, unsigned newNumSymbols,
unsigned newNumLocals,
ArrayRef<Value *> idArgs) {
assert(newNumReservedCols >= newNumDims + newNumSymbols + newNumLocals + 1 &&
"minimum 1 column");
numReservedCols = newNumReservedCols;
numDims = newNumDims;
numSymbols = newNumSymbols;
numIds = numDims + numSymbols + newNumLocals;
equalities.clear();
inequalities.clear();
if (numReservedEqualities >= 1)
equalities.reserve(newNumReservedCols * numReservedEqualities);
if (numReservedInequalities >= 1)
inequalities.reserve(newNumReservedCols * numReservedInequalities);
ids.clear();
if (idArgs.empty()) {
ids.resize(numIds, None);
} else {
ids.reserve(idArgs.size());
ids.insert(ids.end(), idArgs.begin(), idArgs.end());
}
}
void FlatAffineConstraints::reset(unsigned newNumDims, unsigned newNumSymbols,
unsigned newNumLocals,
ArrayRef<Value *> idArgs) {
reset(0, 0, newNumDims + newNumSymbols + newNumLocals + 1, newNumDims,
newNumSymbols, newNumLocals, idArgs);
}
void FlatAffineConstraints::append(const FlatAffineConstraints &other) {
assert(other.getNumCols() == getNumCols());
assert(other.getNumDimIds() == getNumDimIds());
assert(other.getNumSymbolIds() == getNumSymbolIds());
inequalities.reserve(inequalities.size() +
other.getNumInequalities() * numReservedCols);
equalities.reserve(equalities.size() +
other.getNumEqualities() * numReservedCols);
for (unsigned r = 0, e = other.getNumInequalities(); r < e; r++) {
addInequality(other.getInequality(r));
}
for (unsigned r = 0, e = other.getNumEqualities(); r < e; r++) {
addEquality(other.getEquality(r));
}
}
void FlatAffineConstraints::addLocalId(unsigned pos) {
addId(IdKind::Local, pos);
}
void FlatAffineConstraints::addDimId(unsigned pos, Value *id) {
addId(IdKind::Dimension, pos, id);
}
void FlatAffineConstraints::addSymbolId(unsigned pos, Value *id) {
addId(IdKind::Symbol, pos, id);
}
/// Adds a dimensional identifier. The added column is initialized to
/// zero.
void FlatAffineConstraints::addId(IdKind kind, unsigned pos, Value *id) {
if (kind == IdKind::Dimension) {
assert(pos <= getNumDimIds());
} else if (kind == IdKind::Symbol) {
assert(pos <= getNumSymbolIds());
} else {
assert(pos <= getNumLocalIds());
}
unsigned oldNumReservedCols = numReservedCols;
// Check if a resize is necessary.
if (getNumCols() + 1 > numReservedCols) {
equalities.resize(getNumEqualities() * (getNumCols() + 1));
inequalities.resize(getNumInequalities() * (getNumCols() + 1));
numReservedCols++;
}
unsigned absolutePos;
if (kind == IdKind::Dimension) {
absolutePos = pos;
numDims++;
} else if (kind == IdKind::Symbol) {
absolutePos = pos + getNumDimIds();
numSymbols++;
} else {
absolutePos = pos + getNumDimIds() + getNumSymbolIds();
}
numIds++;
// Note that getNumCols() now will already return the new size, which will be
// at least one.
int numInequalities = static_cast<int>(getNumInequalities());
int numEqualities = static_cast<int>(getNumEqualities());
int numCols = static_cast<int>(getNumCols());
for (int r = numInequalities - 1; r >= 0; r--) {
for (int c = numCols - 2; c >= 0; c--) {
if (c < absolutePos)
atIneq(r, c) = inequalities[r * oldNumReservedCols + c];
else
atIneq(r, c + 1) = inequalities[r * oldNumReservedCols + c];
}
atIneq(r, absolutePos) = 0;
}
for (int r = numEqualities - 1; r >= 0; r--) {
for (int c = numCols - 2; c >= 0; c--) {
// All values in column absolutePositions < absolutePos have the same
// coordinates in the 2-d view of the coefficient buffer.
if (c < absolutePos)
atEq(r, c) = equalities[r * oldNumReservedCols + c];
else
// Those at absolutePosition >= absolutePos, get a shifted
// absolutePosition.
atEq(r, c + 1) = equalities[r * oldNumReservedCols + c];
}
// Initialize added dimension to zero.
atEq(r, absolutePos) = 0;
}
// If an 'id' is provided, insert it; otherwise use None.
if (id) {
ids.insert(ids.begin() + absolutePos, id);
} else {
ids.insert(ids.begin() + absolutePos, None);
}
assert(ids.size() == getNumIds());
}
// This routine may add additional local variables if the flattened expression
// corresponding to the map has such variables due to the presence of
// mod's, ceildiv's, and floordiv's.
bool FlatAffineConstraints::composeMap(AffineValueMap *vMap) {
// Assert if the map and this constraint set aren't associated with the same
// identifiers in the same order.
assert(vMap->getNumDims() <= getNumDimIds());
assert(vMap->getNumSymbols() <= getNumSymbolIds());
for (unsigned i = 0, e = vMap->getNumDims(); i < e; i++) {
assert(ids[i].hasValue());
assert(vMap->getOperand(i) == ids[i].getValue());
}
for (unsigned i = 0, e = vMap->getNumSymbols(); i < e; i++) {
assert(ids[numDims + i].hasValue());
assert(vMap->getOperand(vMap->getNumDims() + i) ==
ids[numDims + i].getValue());
}
std::vector<SmallVector<int64_t, 8>> flatExprs;
FlatAffineConstraints cst;
if (!getFlattenedAffineExprs(vMap->getAffineMap(), &flatExprs, &cst)) {
LLVM_DEBUG(llvm::dbgs()
<< "composition unimplemented for semi-affine maps");
return false;
}
assert(flatExprs.size() == vMap->getNumResults());
// Make the value map and the flat affine cst dimensions compatible.
// A lot of this code will be refactored/cleaned up.
// TODO(bondhugula): the next ~20 lines of code is pretty UGLY. This needs
// to be factored out into an FlatAffineConstraints::alignAndMerge().
for (unsigned l = 0, e = cst.getNumLocalIds(); l < e; l++) {
addLocalId(0);
}
for (unsigned t = 0, e = vMap->getNumResults(); t < e; t++) {
// TODO: Consider using a batched version to add a range of IDs.
addDimId(0);
cst.addDimId(0);
}
assert(cst.getNumDimIds() <= getNumDimIds());
for (unsigned t = 0, e = getNumDimIds() - cst.getNumDimIds(); t < e; t++) {
// Dimensions that are in 'this' but not in vMap/cst are added at the end.
cst.addDimId(cst.getNumDimIds());
}
assert(cst.getNumSymbolIds() <= getNumSymbolIds());
for (unsigned t = 0, e = getNumSymbolIds() - cst.getNumSymbolIds(); t < e;
t++) {
// Dimensions that are in 'this' but not in vMap/cst are added at the end.
cst.addSymbolId(cst.getNumSymbolIds());
}
assert(cst.getNumLocalIds() <= getNumLocalIds());
for (unsigned t = 0, e = getNumLocalIds() - cst.getNumLocalIds(); t < e;
t++) {
cst.addLocalId(cst.getNumLocalIds());
}
/// Finally, append cst to this constraint set.
append(cst);
// We add one equality for each result connecting the result dim of the map to
// the other identifiers.
// For eg: if the expression is 16*i0 + i1, and this is the r^th
// iteration/result of the value map, we are adding the equality:
// d_r - 16*i0 - i1 = 0. Hence, when flattening say (i0 + 1, i0 + 8*i2), we
// add two equalities overall: d_0 - i0 - 1 == 0, d1 - i0 - 8*i2 == 0.
for (unsigned r = 0, e = flatExprs.size(); r < e; r++) {
const auto &flatExpr = flatExprs[r];
// eqToAdd is the equality corresponding to the flattened affine expression.
SmallVector<int64_t, 8> eqToAdd(getNumCols(), 0);
// Set the coefficient for this result to one.
eqToAdd[r] = 1;
assert(flatExpr.size() >= vMap->getNumOperands() + 1);
// Dims and symbols.
for (unsigned i = 0, e = vMap->getNumOperands(); i < e; i++) {
unsigned loc;
bool ret = findId(*vMap->getOperand(i), &loc);
assert(ret && "value map's id can't be found");
(void)ret;
// We need to negate 'eq[r]' since the newly added dimension is going to
// be set to this one.
eqToAdd[loc] = -flatExpr[i];
}
// Local vars common to eq and cst are at the beginning.
int j = getNumDimIds() + getNumSymbolIds();
int end = flatExpr.size() - 1;
for (int i = vMap->getNumOperands(); i < end; i++, j++) {
eqToAdd[j] = -flatExpr[i];
}
// Constant term.
eqToAdd[getNumCols() - 1] = -flatExpr[flatExpr.size() - 1];
// Add the equality connecting the result of the map to this constraint set.
addEquality(eqToAdd);
}
return true;
}
// Searches for a constraint with a non-zero coefficient at 'colIdx' in
// equality (isEq=true) or inequality (isEq=false) constraints.
// Returns true and sets row found in search in 'rowIdx'.
// Returns false otherwise.
static bool
findConstraintWithNonZeroAt(const FlatAffineConstraints &constraints,
unsigned colIdx, bool isEq, unsigned &rowIdx) {
auto at = [&](unsigned rowIdx) -> int64_t {
return isEq ? constraints.atEq(rowIdx, colIdx)
: constraints.atIneq(rowIdx, colIdx);
};
unsigned e =
isEq ? constraints.getNumEqualities() : constraints.getNumInequalities();
for (rowIdx = 0; rowIdx < e; ++rowIdx) {
if (at(rowIdx) != 0) {
return true;
}
}
return false;
}
// Normalizes the coefficient values across all columns in 'rowIDx' by their
// GCD in equality or inequality contraints as specified by 'isEq'.
template <bool isEq>
static void normalizeConstraintByGCD(FlatAffineConstraints *constraints,
unsigned rowIdx) {
auto at = [&](unsigned colIdx) -> int64_t {
return isEq ? constraints->atEq(rowIdx, colIdx)
: constraints->atIneq(rowIdx, colIdx);
};
uint64_t gcd = std::abs(at(0));
for (unsigned j = 1, e = constraints->getNumCols(); j < e; ++j) {
gcd = llvm::GreatestCommonDivisor64(gcd, std::abs(at(j)));
}
if (gcd > 0 && gcd != 1) {
for (unsigned j = 0, e = constraints->getNumCols(); j < e; ++j) {
int64_t v = at(j) / static_cast<int64_t>(gcd);
isEq ? constraints->atEq(rowIdx, j) = v
: constraints->atIneq(rowIdx, j) = v;
}
}
}
void FlatAffineConstraints::normalizeConstraintsByGCD() {
for (unsigned i = 0, e = getNumEqualities(); i < e; ++i) {
normalizeConstraintByGCD</*isEq=*/true>(this, i);
}
for (unsigned i = 0, e = getNumInequalities(); i < e; ++i) {
normalizeConstraintByGCD</*isEq=*/false>(this, i);
}
}
bool FlatAffineConstraints::hasConsistentState() const {
if (inequalities.size() != getNumInequalities() * numReservedCols)
return false;
if (equalities.size() != getNumEqualities() * numReservedCols)
return false;
if (ids.size() != getNumIds())
return false;
// Catches errors where numDims, numSymbols, numIds aren't consistent.
if (numDims > numIds || numSymbols > numIds || numDims + numSymbols > numIds)
return false;
return true;
}
/// Checks all rows of equality/inequality constraints for trivial
/// contradictions (for example: 1 == 0, 0 >= 1), which may have surfaced
/// after elimination. Returns 'true' if an invalid constraint is found;
/// 'false' otherwise.
bool FlatAffineConstraints::hasInvalidConstraint() const {
assert(hasConsistentState());
auto check = [&](bool isEq) -> bool {
unsigned numCols = getNumCols();
unsigned numRows = isEq ? getNumEqualities() : getNumInequalities();
for (unsigned i = 0, e = numRows; i < e; ++i) {
unsigned j;
for (j = 0; j < numCols - 1; ++j) {
int64_t v = isEq ? atEq(i, j) : atIneq(i, j);
// Skip rows with non-zero variable coefficients.
if (v != 0)
break;
}
if (j < numCols - 1) {
continue;
}
// Check validity of constant term at 'numCols - 1' w.r.t 'isEq'.
// Example invalid constraints include: '1 == 0' or '-1 >= 0'
int64_t v = isEq ? atEq(i, numCols - 1) : atIneq(i, numCols - 1);
if ((isEq && v != 0) || (!isEq && v < 0)) {
return true;
}
}
return false;
};
if (check(/*isEq=*/true))
return true;
return check(/*isEq=*/false);
}
// Eliminate identifier from constraint at 'rowIdx' based on coefficient at
// pivotRow, pivotCol. Columns in range [elimColStart, pivotCol) will not be
// updated as they have already been eliminated.
static void eliminateFromConstraint(FlatAffineConstraints *constraints,
unsigned rowIdx, unsigned pivotRow,
unsigned pivotCol, unsigned elimColStart,
bool isEq) {
// Skip if equality 'rowIdx' if same as 'pivotRow'.
if (isEq && rowIdx == pivotRow)
return;
auto at = [&](unsigned i, unsigned j) -> int64_t {
return isEq ? constraints->atEq(i, j) : constraints->atIneq(i, j);
};
int64_t leadCoeff = at(rowIdx, pivotCol);
// Skip if leading coefficient at 'rowIdx' is already zero.
if (leadCoeff == 0)
return;
int64_t pivotCoeff = constraints->atEq(pivotRow, pivotCol);
int64_t sign = (leadCoeff * pivotCoeff > 0) ? -1 : 1;
int64_t lcm = mlir::lcm(pivotCoeff, leadCoeff);
int64_t pivotMultiplier = sign * (lcm / std::abs(pivotCoeff));
int64_t rowMultiplier = lcm / std::abs(leadCoeff);
unsigned numCols = constraints->getNumCols();
for (unsigned j = 0; j < numCols; ++j) {
// Skip updating column 'j' if it was just eliminated.
if (j >= elimColStart && j < pivotCol)
continue;
int64_t v = pivotMultiplier * constraints->atEq(pivotRow, j) +
rowMultiplier * at(rowIdx, j);
isEq ? constraints->atEq(rowIdx, j) = v
: constraints->atIneq(rowIdx, j) = v;
}
}
// Remove coefficients in column range [colStart, colLimit) in place.
// This removes in data in the specified column range, and copies any
// remaining valid data into place.
static void shiftColumnsToLeft(FlatAffineConstraints *constraints,
unsigned colStart, unsigned colLimit,
bool isEq) {
assert(colStart >= 0 && colLimit <= constraints->getNumIds());
if (colLimit <= colStart)
return;
unsigned numCols = constraints->getNumCols();
unsigned numRows = isEq ? constraints->getNumEqualities()
: constraints->getNumInequalities();
unsigned numToEliminate = colLimit - colStart;
for (unsigned r = 0, e = numRows; r < e; ++r) {
for (unsigned c = colLimit; c < numCols; ++c) {
if (isEq) {
constraints->atEq(r, c - numToEliminate) = constraints->atEq(r, c);
} else {
constraints->atIneq(r, c - numToEliminate) = constraints->atIneq(r, c);
}
}
}
}
// Removes identifiers in column range [idStart, idLimit), and copies any
// remaining valid data into place, and updates member variables.
void FlatAffineConstraints::removeIdRange(unsigned idStart, unsigned idLimit) {
assert(idLimit < getNumCols());
// TODO(andydavis) Make 'removeIdRange' a lambda called from here.
// Remove eliminated identifiers from equalities.
shiftColumnsToLeft(this, idStart, idLimit, /*isEq=*/true);
// Remove eliminated identifiers from inequalities.
shiftColumnsToLeft(this, idStart, idLimit, /*isEq=*/false);
// Update members numDims, numSymbols and numIds.
unsigned numDimsEliminated = 0;
if (idStart < numDims) {
numDimsEliminated = std::min(numDims, idLimit) - idStart;
}
unsigned numColsEliminated = idLimit - idStart;
unsigned numSymbolsEliminated =
std::min(numSymbols, numColsEliminated - numDimsEliminated);
numDims -= numDimsEliminated;
numSymbols -= numSymbolsEliminated;
numIds = numIds - numColsEliminated;
ids.erase(ids.begin() + idStart, ids.begin() + idLimit);
// No resize necessary. numReservedCols remains the same.
}
// Checks for emptiness of the set by eliminating identifiers successively and
// using the GCD test (on all equality constraints) and checking for trivially
// invalid constraints. Returns 'true' if the constraint system is found to be
// empty; false otherwise.
bool FlatAffineConstraints::isEmpty() const {
if (isEmptyByGCDTest() || hasInvalidConstraint())
return true;
auto tmpCst = clone();
for (unsigned i = 0, e = tmpCst->getNumIds(); i < e; i++) {
// We check emptiness through trivial checks after eliminating each ID to
// detect emptiness early. Since the checks isEmptyByGCDTest() and
// hasInvalidConstraint() are linear time and single sweep on the constraint
// buffer, this appears reasonable - but can optimize in the future.
if (tmpCst->gaussianEliminateId(0)) {
if (tmpCst->hasInvalidConstraint() || tmpCst->isEmptyByGCDTest())
return true;
} else {
tmpCst->FourierMotzkinEliminate(0);
// If the variable couldn't be eliminated by Gaussian, FM wouldn't have
// modified the equalities in any way. So no need to again run GCD test.
// Check for trivial invalid constraints.
if (tmpCst->hasInvalidConstraint())
return true;
}
}
return false;
}
// Runs the GCD test on all equality constraints. Returns 'true' if this test
// fails on any equality. Returns 'false' otherwise.
// This test can be used to disprove the existence of a solution. If it returns
// true, no integer solution to the equality constraints can exist.
//
// GCD test definition:
//
// The equality constraint:
//
// c_1*x_1 + c_2*x_2 + ... + c_n*x_n = c_0
//
// has an integer solution iff:
//
// GCD of c_1, c_2, ..., c_n divides c_0.
//
bool FlatAffineConstraints::isEmptyByGCDTest() const {
assert(hasConsistentState());
unsigned numCols = getNumCols();
for (unsigned i = 0, e = getNumEqualities(); i < e; ++i) {
uint64_t gcd = std::abs(atEq(i, 0));
for (unsigned j = 1; j < numCols - 1; ++j) {
gcd = llvm::GreatestCommonDivisor64(gcd, std::abs(atEq(i, j)));
}
int64_t v = std::abs(atEq(i, numCols - 1));
if (gcd > 0 && (v % gcd != 0)) {
return true;
}
}
return false;
}
/// Tightens inequalities given that we are dealing with integer spaces. This is
/// analogous to the GCD test but applied to inequalities. The constant term can
/// be reduced to the preceding multiple of the GCD of the coefficients, i.e.,
/// 64*i - 100 >= 0 => 64*i - 128 >= 0 (since 'i' is an integer). This is a
/// fast method - linear in the number of coefficients.
// Example on how this affects practical cases: consider the scenario:
// 64*i >= 100, j = 64*i; without a tightening, elimination of i would yield
// j >= 100 instead of the tighter (exact) j >= 128.
void FlatAffineConstraints::GCDTightenInequalities() {
unsigned numCols = getNumCols();
for (unsigned i = 0, e = getNumInequalities(); i < e; ++i) {
uint64_t gcd = std::abs(atIneq(i, 0));
for (unsigned j = 1; j < numCols - 1; ++j) {
gcd = llvm::GreatestCommonDivisor64(gcd, std::abs(atIneq(i, j)));
}
if (gcd > 0) {
int64_t gcdI = static_cast<int64_t>(gcd);
atIneq(i, numCols - 1) =
gcdI * mlir::floorDiv(atIneq(i, numCols - 1), gcdI);
}
}
}
// Eliminates all identifer variables in column range [posStart, posLimit).
// Returns the number of variables eliminated.
unsigned FlatAffineConstraints::gaussianEliminateIds(unsigned posStart,
unsigned posLimit) {
// Return if identifier positions to eliminate are out of range.
assert(posStart >= 0 && posLimit <= numIds);
assert(hasConsistentState());
if (posStart >= posLimit)
return 0;
GCDTightenInequalities();
unsigned pivotCol = 0;
for (pivotCol = posStart; pivotCol < posLimit; ++pivotCol) {
// Find a row which has a non-zero coefficient in column 'j'.
unsigned pivotRow;
if (!findConstraintWithNonZeroAt(*this, pivotCol, /*isEq=*/true,
pivotRow)) {
// No pivot row in equalities with non-zero at 'pivotCol'.
if (!findConstraintWithNonZeroAt(*this, pivotCol, /*isEq=*/false,
pivotRow)) {
// If inequalities are also non-zero in 'pivotCol', it can be
// eliminated.
continue;
}
break;
}
// Eliminate identifier at 'pivotCol' from each equality row.
for (unsigned i = 0, e = getNumEqualities(); i < e; ++i) {
eliminateFromConstraint(this, i, pivotRow, pivotCol, posStart,
/*isEq=*/true);
normalizeConstraintByGCD</*isEq=*/true>(this, i);
}
// Eliminate identifier at 'pivotCol' from each inequality row.
for (unsigned i = 0, e = getNumInequalities(); i < e; ++i) {
eliminateFromConstraint(this, i, pivotRow, pivotCol, posStart,
/*isEq=*/false);
normalizeConstraintByGCD</*isEq=*/false>(this, i);
}
removeEquality(pivotRow);
}
// Update position limit based on number eliminated.
posLimit = pivotCol;
// Remove eliminated columns from all constraints.
removeIdRange(posStart, posLimit);
return posLimit - posStart;
}
// Returns an AffineMap which represents 'pos' in equality constraint 'idx',
// as a function of dim and symbols identifers in all other positions.
// TODO(andydavis) Add local variable support to this function.
AffineMap FlatAffineConstraints::toAffineMapFromEq(
unsigned idx, unsigned pos, MLIRContext *context,
SmallVectorImpl<unsigned> *nonZeroDimIds,
SmallVectorImpl<unsigned> *nonZeroSymbolIds) {
assert(getNumLocalIds() == 0);
assert(idx < getNumEqualities());
int64_t v = atEq(idx, pos);
// Return if coefficient at (idx, pos) is zero or does not divide constant.
if (v == 0 || (atEq(idx, getNumIds()) % v != 0))
return AffineMap::Null();
// Check that coefficient at 'pos' divides all other coefficient in row 'idx'.
for (unsigned j = 0, e = getNumIds(); j < e; ++j) {
if (j != pos && (atEq(idx, j) % v != 0))
return AffineMap::Null();
}
// Build AffineExpr solving for identifier 'pos' in terms of all others.
auto expr = getAffineConstantExpr(0, context);
unsigned mapNumDims = 0;
unsigned mapNumSymbols = 0;
for (unsigned j = 0, e = getNumIds(); j < e; ++j) {
if (j == pos)
continue;
int64_t c = atEq(idx, j);
if (c == 0)
continue;
// Divide 'c' by 'v' from 'pos' for which we are solving.
c /= v;
if (j < numDims) {
expr = expr + getAffineDimExpr(mapNumDims++, context) * c;
nonZeroDimIds->push_back(j);
} else {
expr =
expr + getAffineSymbolExpr(mapNumDims + mapNumSymbols++, context) * c;
nonZeroSymbolIds->push_back(j);
}
expr = expr * (-1);
}
// Add constant term to AffineExpr.
int64_t c = atEq(idx, getNumIds());
if (c > 0) {
expr = expr + (c / v) * (-1);
}
return AffineMap::get(mapNumDims, mapNumSymbols, {expr}, {});
}
void FlatAffineConstraints::addEquality(ArrayRef<int64_t> eq) {
assert(eq.size() == getNumCols());
unsigned offset = equalities.size();
equalities.resize(equalities.size() + numReservedCols);
std::copy(eq.begin(), eq.end(), equalities.begin() + offset);
}
void FlatAffineConstraints::addInequality(ArrayRef<int64_t> inEq) {
assert(inEq.size() == getNumCols());
unsigned offset = inequalities.size();
inequalities.resize(inequalities.size() + numReservedCols);
std::copy(inEq.begin(), inEq.end(), inequalities.begin() + offset);
}
void FlatAffineConstraints::addConstantLowerBound(unsigned pos, int64_t lb) {
assert(pos < getNumCols());
unsigned offset = inequalities.size();
inequalities.resize(inequalities.size() + numReservedCols);
std::fill(inequalities.begin() + offset,
inequalities.begin() + offset + getNumCols(), 0);
inequalities[offset + pos] = 1;
inequalities[offset + getNumCols() - 1] = -lb;
}
void FlatAffineConstraints::addConstantUpperBound(unsigned pos, int64_t ub) {
assert(pos < getNumCols());
unsigned offset = inequalities.size();
inequalities.resize(inequalities.size() + numReservedCols);
std::fill(inequalities.begin() + offset,
inequalities.begin() + offset + getNumCols(), 0);
inequalities[offset + pos] = -1;
inequalities[offset + getNumCols() - 1] = ub;
}
void FlatAffineConstraints::addConstantLowerBound(ArrayRef<int64_t> expr,
int64_t lb) {
assert(expr.size() == getNumCols());
unsigned offset = inequalities.size();
inequalities.resize(inequalities.size() + numReservedCols);
std::fill(inequalities.begin() + offset,
inequalities.begin() + offset + getNumCols(), 0);
std::copy(expr.begin(), expr.end(), inequalities.begin() + offset);
inequalities[offset + getNumCols() - 1] += -lb;
}
void FlatAffineConstraints::addConstantUpperBound(ArrayRef<int64_t> expr,
int64_t ub) {
assert(expr.size() == getNumCols());
unsigned offset = inequalities.size();
inequalities.resize(inequalities.size() + numReservedCols);
std::fill(inequalities.begin() + offset,
inequalities.begin() + offset + getNumCols(), 0);
for (unsigned i = 0, e = getNumCols(); i < e; i++) {
inequalities[offset + i] = -expr[i];
}
inequalities[offset + getNumCols() - 1] += ub;
}
void FlatAffineConstraints::addLowerBound(ArrayRef<int64_t> expr,
ArrayRef<int64_t> lb) {
assert(expr.size() == getNumCols());
assert(lb.size() == getNumCols());
unsigned offset = inequalities.size();
inequalities.resize(inequalities.size() + numReservedCols);
std::fill(inequalities.begin() + offset,
inequalities.begin() + offset + getNumCols(), 0);
for (unsigned i = 0, e = getNumCols(); i < e; i++) {
inequalities[offset + i] = expr[i] - lb[i];
}
}
void FlatAffineConstraints::addUpperBound(ArrayRef<int64_t> expr,
ArrayRef<int64_t> ub) {
assert(expr.size() == getNumCols());
assert(ub.size() == getNumCols());
unsigned offset = inequalities.size();
inequalities.resize(inequalities.size() + numReservedCols);
std::fill(inequalities.begin() + offset,
inequalities.begin() + offset + getNumCols(), 0);
for (unsigned i = 0, e = getNumCols(); i < e; i++) {
inequalities[offset + i] = ub[i] - expr[i];
}
}
bool FlatAffineConstraints::findId(const Value &id, unsigned *pos) const {
unsigned i = 0;
for (const auto &mayBeId : ids) {
if (mayBeId.hasValue() && mayBeId.getValue() == &id) {
*pos = i;
return true;
}
i++;
}
return false;
}
void FlatAffineConstraints::setDimSymbolSeparation(unsigned newSymbolCount) {
assert(newSymbolCount <= numDims + numSymbols &&
"invalid separation position");
numDims = numDims + numSymbols - newSymbolCount;
numSymbols = newSymbolCount;
}
bool FlatAffineConstraints::addForStmtDomain(const ForStmt &forStmt) {
unsigned pos;
// Pre-condition for this method.
if (!findId(forStmt, &pos)) {
assert(0 && "Value not found");
return false;
}
if (forStmt.getStep() != 1)
LLVM_DEBUG(llvm::dbgs()
<< "Domain conservative: non-unit stride not handled\n");
// Adds a lower or upper bound when the bounds aren't constant.
auto addLowerOrUpperBound = [&](bool lower) -> bool {
auto operands = lower ? forStmt.getLowerBoundOperands()
: forStmt.getUpperBoundOperands();
for (const auto &operand : operands) {
unsigned loc;
if (!findId(*operand, &loc)) {
if (operand->isValidSymbol()) {
addSymbolId(getNumSymbolIds(), const_cast<Value *>(operand));
loc = getNumDimIds() + getNumSymbolIds() - 1;
// Check if the symbol is a constant.
if (auto *opStmt = operand->getDefiningStmt()) {
if (auto constOp = opStmt->dyn_cast<ConstantIndexOp>()) {
setIdToConstant(*operand, constOp->getValue());
}
}
} else {
addDimId(getNumDimIds(), const_cast<Value *>(operand));
loc = getNumDimIds() - 1;
}
}
}
// Record positions of the operands in the constraint system.
SmallVector<unsigned, 8> positions;
for (const auto &operand : operands) {
unsigned loc;
if (!findId(*operand, &loc))
assert(0 && "expected to be found");
positions.push_back(loc);
}
auto boundMap =
lower ? forStmt.getLowerBoundMap() : forStmt.getUpperBoundMap();
FlatAffineConstraints localVarCst;
std::vector<SmallVector<int64_t, 8>> flatExprs;
if (!getFlattenedAffineExprs(boundMap, &flatExprs, &localVarCst)) {
LLVM_DEBUG(llvm::dbgs() << "semi-affine expressions not yet supported\n");
return false;
}
if (localVarCst.getNumLocalIds() > 0) {
LLVM_DEBUG(llvm::dbgs()
<< "loop bounds with mod/floordiv expr's not yet supported\n");
return false;
}
for (const auto &flatExpr : flatExprs) {
SmallVector<int64_t, 4> ineq(getNumCols(), 0);
ineq[pos] = lower ? 1 : -1;
for (unsigned j = 0, e = boundMap.getNumInputs(); j < e; j++) {
ineq[positions[j]] = lower ? -flatExpr[j] : flatExpr[j];
}
// Constant term.
ineq[getNumCols() - 1] =
lower ? -flatExpr[flatExpr.size() - 1]
// Upper bound in flattenedExpr is an exclusive one.
: flatExpr[flatExpr.size() - 1] - 1;
addInequality(ineq);
}
return true;
};
if (forStmt.hasConstantLowerBound()) {
addConstantLowerBound(pos, forStmt.getConstantLowerBound());
} else {
// Non-constant lower bound case.
if (!addLowerOrUpperBound(/*lower=*/true))
return false;
}
if (forStmt.hasConstantUpperBound()) {
addConstantUpperBound(pos, forStmt.getConstantUpperBound() - 1);
return true;
}
// Non-constant upper bound case.
return addLowerOrUpperBound(/*lower=*/false);
}
/// Sets the specified identifer to a constant value.
void FlatAffineConstraints::setIdToConstant(unsigned pos, int64_t val) {
unsigned offset = equalities.size();
equalities.resize(equalities.size() + numReservedCols);
std::fill(equalities.begin() + offset,
equalities.begin() + offset + getNumCols(), 0);
equalities[offset + pos] = 1;
equalities[offset + getNumCols() - 1] = -val;
}
/// Sets the specified identifer to a constant value; asserts if the id is not
/// found.
void FlatAffineConstraints::setIdToConstant(const Value &id, int64_t val) {
unsigned pos;
if (!findId(id, &pos))
// This is a pre-condition for this method.
assert(0 && "id not found");
setIdToConstant(pos, val);
}
void FlatAffineConstraints::removeEquality(unsigned pos) {
unsigned numEqualities = getNumEqualities();
assert(pos < numEqualities);
unsigned outputIndex = pos * numReservedCols;
unsigned inputIndex = (pos + 1) * numReservedCols;
unsigned numElemsToCopy = (numEqualities - pos - 1) * numReservedCols;
std::copy(equalities.begin() + inputIndex,
equalities.begin() + inputIndex + numElemsToCopy,
equalities.begin() + outputIndex);
equalities.resize(equalities.size() - numReservedCols);
}
/// Finds an equality that equates the specified identifier to a constant.
/// Returns the position of the equality row. If 'symbolic' is set to true,
/// symbols are also treated like a constant, i.e., an affine function of the
/// symbols is also treated like a constant.
static int findEqualityToConstant(const FlatAffineConstraints &cst,
unsigned pos, bool symbolic = false) {
assert(pos < cst.getNumIds() && "invalid position");
for (unsigned r = 0, e = cst.getNumEqualities(); r < e; r++) {
int64_t v = cst.atEq(r, pos);
if (v * v != 1)
continue;
unsigned c;
unsigned f = symbolic ? cst.getNumDimIds() : cst.getNumIds();
// This checks for zeros in all positions other than 'pos' in [0, f)
for (c = 0; c < f; c++) {
if (c == pos)
continue;
if (cst.atEq(r, c) != 0) {
// Dependent on another identifier.
break;
}
}
if (c == f)
// Equality is free of other identifiers.
return r;
}
return -1;
}
void FlatAffineConstraints::setAndEliminate(unsigned pos, int64_t constVal) {
assert(pos < getNumIds() && "invalid position");
for (unsigned r = 0, e = getNumInequalities(); r < e; r++) {
atIneq(r, getNumCols() - 1) += atIneq(r, pos) * constVal;
}
for (unsigned r = 0, e = getNumEqualities(); r < e; r++) {
atEq(r, getNumCols() - 1) += atEq(r, pos) * constVal;
}
removeId(pos);
}
bool FlatAffineConstraints::constantFoldId(unsigned pos) {
assert(pos < getNumIds() && "invalid position");
int rowIdx;
if ((rowIdx = findEqualityToConstant(*this, pos)) == -1)
return false;
// atEq(rowIdx, pos) is either -1 or 1.
assert(atEq(rowIdx, pos) * atEq(rowIdx, pos) == 1);
int64_t constVal = atEq(rowIdx, getNumCols() - 1) / -atEq(rowIdx, pos);
setAndEliminate(pos, constVal);
return true;
}
void FlatAffineConstraints::constantFoldIdRange(unsigned pos, unsigned num) {
for (unsigned s = pos, t = pos, e = pos + num; s < e; s++) {
if (!constantFoldId(t))
t++;
}
}
/// Returns the extent (upper bound - lower bound) of the specified
/// identifier if it is found to be a constant; returns None if it's not a
/// constant. This methods treats symbolic identifiers specially, i.e.,
/// it looks for constant differences between affine expressions involving
/// only the symbolic identifiers. See comments at function definition for
/// example. 'lb', if provided, is set to the lower bound associated with the
/// constant difference. Note that 'lb' is purely symbolic and thus will contain
/// the coefficients of the symbolic identifiers and the constant coefficient.
// Egs: 0 <= i <= 15, return 16.
// s0 + 2 <= i <= s0 + 17, returns 16. (s0 has to be a symbol)
// i + s0 + 16 <= d0 <= i + s0 + 31, returns 16.
Optional<int64_t> FlatAffineConstraints::getConstantBoundOnDimSize(
unsigned pos, SmallVectorImpl<int64_t> *lb) const {
assert(pos < getNumDimIds() && "Invalid position");
assert(getNumLocalIds() == 0);
// TODO(bondhugula): eliminate all remaining dimensional identifiers (other
// than the one at 'pos' to make this more powerful. Not needed for
// hyper-rectangular spaces.
// Find an equality for 'pos'^th identifier that equates it to some function
// of the symbolic identifiers (+ constant).
int eqRow = findEqualityToConstant(*this, pos, /*symbolic=*/true);
if (eqRow != -1) {
// This identifier can only take a single value.
if (lb) {
// Set lb to the symbolic value.
lb->resize(getNumSymbolIds() + 1);
for (unsigned c = 0, f = getNumSymbolIds() + 1; c < f; c++) {
int64_t v = atEq(eqRow, pos);
// atEq(eqRow, pos) is either -1 or 1.
assert(v * v == 1);
(*lb)[c] = v < 0 ? atEq(eqRow, getNumDimIds() + c) / -v
: -atEq(eqRow, getNumDimIds() + c) / v;
}
}
return 1;
}
// Check if the identifier appears at all in any of the inequalities.
unsigned r, e;
for (r = 0, e = getNumInequalities(); r < e; r++) {
if (atIneq(r, pos) != 0)
break;
}
if (r == e) {
// If it doesn't appear, just remove the column and return.
// TODO(andydavis,bondhugula): refactor removeColumns to use it from here.
return None;
}
// Positions of constraints that are lower/upper bounds on the variable.
SmallVector<unsigned, 4> lbIndices, ubIndices;
// Gather all lower bounds and upper bounds of the variable. Since the
// canonical form c_1*x_1 + c_2*x_2 + ... + c_0 >= 0, a constraint is a lower
// bound for x_i if c_i >= 1, and an upper bound if c_i <= -1.
for (unsigned r = 0, e = getNumInequalities(); r < e; r++) {
if (atIneq(r, pos) >= 1)
// Lower bound.
lbIndices.push_back(r);
else if (atIneq(r, pos) <= -1)
// Upper bound.
ubIndices.push_back(r);
}
// TODO(bondhugula): eliminate other dimensional identifiers to make this more
// powerful. Not needed for hyper-rectangular iteration spaces.
Optional<int64_t> minDiff = None;
unsigned minLbPosition;
for (auto ubPos : ubIndices) {
for (auto lbPos : lbIndices) {
// Look for a lower bound and an upper bound that only differ by a
// constant, i.e., pairs of the form 0 <= c_pos - f(c_i's) <= diffConst.
// For example, if ii is the pos^th variable, we are looking for
// constraints like ii >= i, ii <= ii + 50, 50 being the difference. The
// minimum among all such constant differences is kept since that's the
// constant bounding the extent of the pos^th variable.
unsigned j, e;
for (j = 0, e = getNumCols() - 1; j < e; j++)
if (atIneq(ubPos, j) != -atIneq(lbPos, j)) {
break;
}
if (j < getNumCols() - 1)
continue;
int64_t mayDiff =
atIneq(ubPos, getNumCols() - 1) + atIneq(lbPos, getNumCols() - 1) + 1;
if (minDiff == None || mayDiff < minDiff) {
minDiff = mayDiff;
minLbPosition = lbPos;
}
}
}
if (lb && minDiff.hasValue()) {
// Set lb to the symbolic lower bound.
lb->resize(getNumSymbolIds() + 1);
for (unsigned c = 0, e = getNumSymbolIds() + 1; c < e; c++) {
(*lb)[c] = -atIneq(minLbPosition, getNumDimIds() + c);
}
}
return minDiff;
}
// A simple (naive and conservative) check for hyper-rectangularlity.
bool FlatAffineConstraints::isHyperRectangular(unsigned pos,
unsigned num) const {
assert(pos < getNumCols() - 1);
// Check for two non-zero coefficients in the range [pos, pos + sum).
for (unsigned r = 0, e = getNumInequalities(); r < e; r++) {
unsigned sum = 0;
for (unsigned c = pos; c < pos + num; c++) {
if (atIneq(r, c) != 0)
sum++;
}
if (sum > 1)
return false;
}
for (unsigned r = 0, e = getNumEqualities(); r < e; r++) {
unsigned sum = 0;
for (unsigned c = pos; c < pos + num; c++) {
if (atEq(r, c) != 0)
sum++;
}
if (sum > 1)
return false;
}
return true;
}
void FlatAffineConstraints::print(raw_ostream &os) const {
assert(hasConsistentState());
os << "\nConstraints (" << getNumDimIds() << " dims, " << getNumSymbolIds()
<< " symbols, " << getNumLocalIds() << " locals), (" << getNumConstraints()
<< " constraints)\n";
os << "(";
for (unsigned i = 0, e = getNumIds(); i < e; i++) {
if (ids[i] == None)
os << "None ";
else
os << "Value ";
}
os << " const)\n";
for (unsigned i = 0, e = getNumEqualities(); i < e; ++i) {
for (unsigned j = 0, f = getNumCols(); j < f; ++j) {
os << atEq(i, j) << " ";
}
os << "= 0\n";
}
for (unsigned i = 0, e = getNumInequalities(); i < e; ++i) {
for (unsigned j = 0, f = getNumCols(); j < f; ++j) {
os << atIneq(i, j) << " ";
}
os << ">= 0\n";
}
os << '\n';
}
void FlatAffineConstraints::dump() const { print(llvm::errs()); }
/// Removes duplicate constraints and trivially true constraints: a constraint
/// of the form <non-negative constant> >= 0 is considered a trivially true
/// constraint.
// Uses a DenseSet to hash and detect duplicates followed by a linear scan to
// remove duplicates in place.
void FlatAffineConstraints::removeTrivialRedundancy() {
DenseSet<ArrayRef<int64_t>> rowSet;
// Check if constraint is of the form <non-negative-constant> >= 0.
auto isTriviallyValid = [&](unsigned r) -> bool {
for (unsigned c = 0, e = getNumCols() - 1; c < e; c++) {
if (atIneq(r, c) != 0)
return false;
}
return atIneq(r, getNumCols() - 1) >= 0;
};
// Detect and mark redundant constraints.
std::vector<bool> redunIneq(getNumInequalities(), false);
for (unsigned r = 0, e = getNumInequalities(); r < e; r++) {
int64_t *rowStart = inequalities.data() + numReservedCols * r;
auto row = ArrayRef<int64_t>(rowStart, getNumCols());
if (isTriviallyValid(r) || !rowSet.insert(row).second) {
redunIneq[r] = true;
}
}
auto copyRow = [&](unsigned src, unsigned dest) {
if (src == dest)
return;
for (unsigned c = 0, e = getNumCols(); c < e; c++) {
atIneq(dest, c) = atIneq(src, c);
}
};
// Scan to get rid of all rows marked redundant, in-place.
unsigned pos = 0;
for (unsigned r = 0, e = getNumInequalities(); r < e; r++) {
if (!redunIneq[r])
copyRow(r, pos++);
}
inequalities.resize(numReservedCols * pos);
// TODO(bondhugula): consider doing this for equalities as well, but probably
// not worth the savings.
}
void FlatAffineConstraints::clearAndCopyFrom(
const FlatAffineConstraints &other) {
FlatAffineConstraints copy(other);
std::swap(*this, copy);
assert(copy.getNumIds() == copy.getIds().size());
}
void FlatAffineConstraints::removeId(unsigned pos) {
removeIdRange(pos, pos + 1);
}
static std::pair<unsigned, unsigned>
getNewNumDimsSymbols(unsigned pos, const FlatAffineConstraints &cst) {
unsigned numDims = cst.getNumDimIds();
unsigned numSymbols = cst.getNumSymbolIds();
unsigned newNumDims, newNumSymbols;
if (pos < numDims) {
newNumDims = numDims - 1;
newNumSymbols = numSymbols;
} else if (pos < numDims + numSymbols) {
assert(numSymbols >= 1);
newNumDims = numDims;
newNumSymbols = numSymbols - 1;
} else {
newNumDims = numDims;
newNumSymbols = numSymbols;
}
return {newNumDims, newNumSymbols};
}
/// Eliminates identifier at the specified position using Fourier-Motzkin
/// variable elimination. This technique is exact for rational spaces but
/// conservative (in "rare" cases) for integer spaces. The operation corresponds
/// to a projection operation yielding the (convex) set of integer points
/// contained in the rational shadow of the set. An emptiness test that relies
/// on this method will guarantee emptiness, i.e., it disproves the existence of
/// a solution if it says it's empty.
/// If a non-null isResultIntegerExact is passed, it is set to true if the
/// result is also integer exact. If it's set to false, the obtained solution
/// *may* not be exact, i.e., it may contain integer points that do not have an
/// integer pre-image in the original set.
///
/// Eg:
/// j >= 0, j <= i + 1
/// i >= 0, i <= N + 1
/// Eliminating i yields,
/// j >= 0, 0 <= N + 1, j - 1 <= N + 1
///
/// If darkShadow = true, this method computes the dark shadow on elimination;
/// the dark shadow is a convex integer subset of the exact integer shadow. A
/// non-empty dark shadow proves the existence of an integer solution. The
/// elimination in such a case could however be an under-approximation, and thus
/// should not be used for scanning sets or used by itself for dependence
/// checking.
///
/// Eg: 2-d set, * represents grid points, 'o' represents a point in the set.
/// ^
/// |
/// | * * * * o o
/// i | * * o o o o
/// | o * * * * *
/// --------------->
/// j ->
///
/// Eliminating i from this system (projecting on the j dimension):
/// rational shadow / integer light shadow: 1 <= j <= 6
/// dark shadow: 3 <= j <= 6
/// exact integer shadow: j = 1 \union 3 <= j <= 6
/// holes/splinters: j = 2
///
/// darkShadow = false, isResultIntegerExact = nullptr are default values.
// TODO(bondhugula): a slight modification to yield dark shadow version of FM
// (tightened), which can prove the existence of a solution if there is one.
void FlatAffineConstraints::FourierMotzkinEliminate(
unsigned pos, bool darkShadow, bool *isResultIntegerExact) {
LLVM_DEBUG(llvm::dbgs() << "FM input (eliminate pos " << pos << "):\n");
LLVM_DEBUG(dump());
assert(pos < getNumIds() && "invalid position");
assert(hasConsistentState());
// Check if this identifier can be eliminated through a substitution.
for (unsigned r = 0, e = getNumEqualities(); r < e; r++) {
if (atEq(r, pos) != 0) {
// Use Gaussian elimination here (since we have an equality).
bool ret = gaussianEliminateId(pos);
(void)ret;
assert(ret && "Gaussian elimination guaranteed to succeed");
LLVM_DEBUG(llvm::dbgs() << "FM output:\n");
LLVM_DEBUG(dump());
return;
}
}
// A fast linear time tightening.
GCDTightenInequalities();
// Check if the identifier appears at all in any of the inequalities.
unsigned r, e;
for (r = 0, e = getNumInequalities(); r < e; r++) {
if (atIneq(r, pos) != 0)
break;
}
if (r == getNumInequalities()) {
// If it doesn't appear, just remove the column and return.
// TODO(andydavis,bondhugula): refactor removeColumns to use it from here.
removeId(pos);
LLVM_DEBUG(llvm::dbgs() << "FM output:\n");
LLVM_DEBUG(dump());
return;
}
// Positions of constraints that are lower bounds on the variable.
SmallVector<unsigned, 4> lbIndices;
// Positions of constraints that are lower bounds on the variable.
SmallVector<unsigned, 4> ubIndices;
// Positions of constraints that do not involve the variable.
std::vector<unsigned> nbIndices;
nbIndices.reserve(getNumInequalities());
// Gather all lower bounds and upper bounds of the variable. Since the
// canonical form c_1*x_1 + c_2*x_2 + ... + c_0 >= 0, a constraint is a lower
// bound for x_i if c_i >= 1, and an upper bound if c_i <= -1.
for (unsigned r = 0, e = getNumInequalities(); r < e; r++) {
if (atIneq(r, pos) == 0) {
// Id does not appear in bound.
nbIndices.push_back(r);
} else if (atIneq(r, pos) >= 1) {
// Lower bound.
lbIndices.push_back(r);
} else {
// Upper bound.
ubIndices.push_back(r);
}
}
// Set the number of dimensions, symbols in the resulting system.
const auto &dimsSymbols = getNewNumDimsSymbols(pos, *this);
unsigned newNumDims = dimsSymbols.first;
unsigned newNumSymbols = dimsSymbols.second;
SmallVector<Optional<Value *>, 8> newIds;
newIds.reserve(numIds - 1);
newIds.insert(newIds.end(), ids.begin(), ids.begin() + pos);
newIds.insert(newIds.end(), ids.begin() + pos + 1, ids.end());
/// Create the new system which has one identifier less.
FlatAffineConstraints newFac(
lbIndices.size() * ubIndices.size() + nbIndices.size(),
getNumEqualities(), getNumCols() - 1, newNumDims, newNumSymbols,
/*numLocals=*/getNumIds() - 1 - newNumDims - newNumSymbols, newIds);
assert(newFac.getIds().size() == newFac.getNumIds());
// This will be used to check if the elimination was integer exact.
unsigned lcmProducts = 1;
// Let x be the variable we are eliminating.
// For each lower bound, lb <= c_l*x, and each upper bound c_u*x <= ub, (note
// that c_l, c_u >= 1) we have:
// lb*lcm(c_l, c_u)/c_l <= lcm(c_l, c_u)*x <= ub*lcm(c_l, c_u)/c_u
// We thus generate a constraint:
// lcm(c_l, c_u)/c_l*lb <= lcm(c_l, c_u)/c_u*ub.
// Note if c_l = c_u = 1, all integer points captured by the resulting
// constraint correspond to integer points in the original system (i.e., they
// have integer pre-images). Hence, if the lcm's are all 1, the elimination is
// integer exact.
for (auto ubPos : ubIndices) {
for (auto lbPos : lbIndices) {
SmallVector<int64_t, 4> ineq;
ineq.reserve(newFac.getNumCols());
int64_t lbCoeff = atIneq(lbPos, pos);
// Note that in the comments above, ubCoeff is the negation of the
// coefficient in the canonical form as the view taken here is that of the
// term being moved to the other size of '>='.
int64_t ubCoeff = -atIneq(ubPos, pos);
// TODO(bondhugula): refactor this loop to avoid all branches inside.
for (unsigned l = 0, e = getNumCols(); l < e; l++) {
if (l == pos)
continue;
assert(lbCoeff >= 1 && ubCoeff >= 1 && "bounds wrongly identified");
int64_t lcm = mlir::lcm(lbCoeff, ubCoeff);
ineq.push_back(atIneq(ubPos, l) * (lcm / ubCoeff) +
atIneq(lbPos, l) * (lcm / lbCoeff));
lcmProducts *= lcm;
}
if (darkShadow) {
// The dark shadow is a convex subset of the exact integer shadow. If
// there is a point here, it proves the existence of a solution.
ineq[ineq.size() - 1] += lbCoeff * ubCoeff - lbCoeff - ubCoeff + 1;
}
// TODO: we need to have a way to add inequalities in-place in
// FlatAffineConstraints instead of creating and copying over.
newFac.addInequality(ineq);
}
}
if (lcmProducts == 1 && isResultIntegerExact)
*isResultIntegerExact = 1;
// Copy over the constraints not involving this variable.
for (auto nbPos : nbIndices) {
SmallVector<int64_t, 4> ineq;
ineq.reserve(getNumCols() - 1);
for (unsigned l = 0, e = getNumCols(); l < e; l++) {
if (l == pos)
continue;
ineq.push_back(atIneq(nbPos, l));
}
newFac.addInequality(ineq);
}
assert(newFac.getNumConstraints() ==
lbIndices.size() * ubIndices.size() + nbIndices.size());
// Copy over the equalities.
for (unsigned r = 0, e = getNumEqualities(); r < e; r++) {
SmallVector<int64_t, 4> eq;
eq.reserve(newFac.getNumCols());
for (unsigned l = 0, e = getNumCols(); l < e; l++) {
if (l == pos)
continue;
eq.push_back(atEq(r, l));
}
newFac.addEquality(eq);
}
newFac.removeTrivialRedundancy();
clearAndCopyFrom(newFac);
LLVM_DEBUG(llvm::dbgs() << "FM output:\n");
LLVM_DEBUG(dump());
}
/// Returns the position of the identifier that has the minimum <number of lower
/// bounds> times <number of upper bounds> from the specified range of
/// identifiers [start, end). It is often best to eliminate in the increasing
/// order of these counts when doing Fourier-Motzkin elimination since FM adds
/// that many new constraints.
static unsigned getBestIdToEliminate(const FlatAffineConstraints &cst,
unsigned start, unsigned end) {
assert(start < cst.getNumIds() && end < cst.getNumIds() + 1);
auto getProductOfNumLowerUpperBounds = [&](unsigned pos) {
unsigned numLb = 0;
unsigned numUb = 0;
for (unsigned r = 0, e = cst.getNumInequalities(); r < e; r++) {
if (cst.atIneq(r, pos) > 0) {
++numLb;
} else if (cst.atIneq(r, pos) < 0) {
++numUb;
}
}
return numLb * numUb;
};
unsigned minLoc = start;
unsigned min = getProductOfNumLowerUpperBounds(start);
for (unsigned c = start + 1; c < end; c++) {
unsigned numLbUbProduct = getProductOfNumLowerUpperBounds(c);
if (numLbUbProduct < min) {
min = numLbUbProduct;
minLoc = c;
}
}
return minLoc;
}
void FlatAffineConstraints::projectOut(unsigned pos, unsigned num) {
if (num == 0)
return;
// 'pos' can be at most getNumCols() - 2.
assert(pos <= getNumCols() - 2 && "invalid position");
assert(pos + num < getNumCols() && "invalid range");
// Eliminate as many identifiers as possible using Gaussian elimination.
unsigned currentPos = pos;
unsigned numToEliminate = num;
unsigned numGaussianEliminated = 0;
do {
unsigned curNumEliminated =
gaussianEliminateIds(currentPos, currentPos + numToEliminate);
if (curNumEliminated == 0) {
++currentPos;
--numToEliminate;
} else {
numToEliminate -= curNumEliminated;
}
numGaussianEliminated += curNumEliminated;
} while (numToEliminate != 0);
// Eliminate the remaining using Fourier-Motzkin.
for (unsigned i = 0; i < num - numGaussianEliminated; i++) {
unsigned elimId = getBestIdToEliminate(*this, pos, getNumIds());
FourierMotzkinEliminate(elimId);
}
// Fast/trivial simplifications.
GCDTightenInequalities();
// Normalize constraints after tightening since the latter impacts this, but
// not the other way round.
normalizeConstraintsByGCD();
}
void FlatAffineConstraints::projectOut(Value *id) {
unsigned pos;
bool ret = findId(*id, &pos);
assert(ret);
(void)ret;
FourierMotzkinEliminate(pos);
}
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