Google Git
Sign in
llvm / llvm-project / refs/tags/llvmorg-11.0.1 / . / mlir / lib / Analysis / AffineStructures.cpp
blob: c09a9af45712a18579c3ee5c6d1e3d42749b92e1 [file] [log] [blame]
//===- AffineStructures.cpp - MLIR Affine Structures Class-----------------===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//
//
// Structures for affine/polyhedral analysis of affine dialect ops.
//
//===----------------------------------------------------------------------===//
#include "mlir/Analysis/AffineStructures.h"
#include "mlir/Analysis/Presburger/Simplex.h"
#include "mlir/Dialect/Affine/IR/AffineOps.h"
#include "mlir/Dialect/Affine/IR/AffineValueMap.h"
#include "mlir/Dialect/StandardOps/IR/Ops.h"
#include "mlir/IR/AffineExprVisitor.h"
#include "mlir/IR/IntegerSet.h"
#include "mlir/Support/LLVM.h"
#include "mlir/Support/MathExtras.h"
#include "llvm/ADT/SmallPtrSet.h"
#include "llvm/Support/Debug.h"
#include "llvm/Support/raw_ostream.h"
#define DEBUG_TYPE "affine-structures"
using namespace mlir;
using llvm::SmallDenseMap;
using llvm::SmallDenseSet;
namespace {
// See comments for SimpleAffineExprFlattener.
// An AffineExprFlattener extends a SimpleAffineExprFlattener by recording
// constraint information associated with mod's, floordiv's, and ceildiv's
// in FlatAffineConstraints 'localVarCst'.
struct AffineExprFlattener : public SimpleAffineExprFlattener {
public:
// Constraints connecting newly introduced local variables (for mod's and
// div's) to existing (dimensional and symbolic) ones. These are always
// inequalities.
FlatAffineConstraints localVarCst;
AffineExprFlattener(unsigned nDims, unsigned nSymbols, MLIRContext *ctx)
: SimpleAffineExprFlattener(nDims, nSymbols) {
localVarCst.reset(nDims, nSymbols, /*numLocals=*/0);
}
private:
// Add a local identifier (needed to flatten a mod, floordiv, ceildiv expr).
// The local identifier added is always a floordiv of a pure add/mul affine
// function of other identifiers, coefficients of which are specified in
// `dividend' and with respect to the positive constant `divisor'. localExpr
// is the simplified tree expression (AffineExpr) corresponding to the
// quantifier.
void addLocalFloorDivId(ArrayRef<int64_t> dividend, int64_t divisor,
AffineExpr localExpr) override {
SimpleAffineExprFlattener::addLocalFloorDivId(dividend, divisor, localExpr);
// Update localVarCst.
localVarCst.addLocalFloorDiv(dividend, divisor);
}
};
} // end anonymous namespace
// Flattens the expressions in map. Returns failure if 'expr' was unable to be
// flattened (i.e., semi-affine expressions not handled yet).
static LogicalResult
getFlattenedAffineExprs(ArrayRef<AffineExpr> exprs, unsigned numDims,
unsigned numSymbols,
std::vector<SmallVector<int64_t, 8>> *flattenedExprs,
FlatAffineConstraints *localVarCst) {
if (exprs.empty()) {
localVarCst->reset(numDims, numSymbols);
return success();
}
AffineExprFlattener flattener(numDims, numSymbols, exprs[0].getContext());
// Use the same flattener to simplify each expression successively. This way
// local identifiers / expressions are shared.
for (auto expr : exprs) {
if (!expr.isPureAffine())
return failure();
flattener.walkPostOrder(expr);
}
assert(flattener.operandExprStack.size() == exprs.size());
flattenedExprs->clear();
flattenedExprs->assign(flattener.operandExprStack.begin(),
flattener.operandExprStack.end());
if (localVarCst)
localVarCst->clearAndCopyFrom(flattener.localVarCst);
return success();
}
// Flattens 'expr' into 'flattenedExpr'. Returns failure if 'expr' was unable to
// be flattened (semi-affine expressions not handled yet).
LogicalResult
mlir::getFlattenedAffineExpr(AffineExpr expr, unsigned numDims,
unsigned numSymbols,
SmallVectorImpl<int64_t> *flattenedExpr,
FlatAffineConstraints *localVarCst) {
std::vector<SmallVector<int64_t, 8>> flattenedExprs;
LogicalResult ret = ::getFlattenedAffineExprs({expr}, numDims, numSymbols,
&flattenedExprs, localVarCst);
*flattenedExpr = flattenedExprs[0];
return ret;
}
/// Flattens the expressions in map. Returns failure if 'expr' was unable to be
/// flattened (i.e., semi-affine expressions not handled yet).
LogicalResult mlir::getFlattenedAffineExprs(
AffineMap map, std::vector<SmallVector<int64_t, 8>> *flattenedExprs,
FlatAffineConstraints *localVarCst) {
if (map.getNumResults() == 0) {
localVarCst->reset(map.getNumDims(), map.getNumSymbols());
return success();
}
return ::getFlattenedAffineExprs(map.getResults(), map.getNumDims(),
map.getNumSymbols(), flattenedExprs,
localVarCst);
}
LogicalResult mlir::getFlattenedAffineExprs(
IntegerSet set, std::vector<SmallVector<int64_t, 8>> *flattenedExprs,
FlatAffineConstraints *localVarCst) {
if (set.getNumConstraints() == 0) {
localVarCst->reset(set.getNumDims(), set.getNumSymbols());
return success();
}
return ::getFlattenedAffineExprs(set.getConstraints(), set.getNumDims(),
set.getNumSymbols(), flattenedExprs,
localVarCst);
}
//===----------------------------------------------------------------------===//
// 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.append(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.getNumInputs() + 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 (failed(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;
assert(idArgs.empty() || idArgs.size() == numIds);
clearConstraints();
if (numReservedEqualities >= 1)
equalities.reserve(newNumReservedCols * numReservedEqualities);
if (numReservedInequalities >= 1)
inequalities.reserve(newNumReservedCols * numReservedInequalities);
if (idArgs.empty()) {
ids.resize(numIds, None);
} else {
ids.assign(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++;
}
int 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());
}
/// Checks if two constraint systems are in the same space, i.e., if they are
/// associated with the same set of identifiers, appearing in the same order.
static bool areIdsAligned(const FlatAffineConstraints &A,
const FlatAffineConstraints &B) {
return A.getNumDimIds() == B.getNumDimIds() &&
A.getNumSymbolIds() == B.getNumSymbolIds() &&
A.getNumIds() == B.getNumIds() && A.getIds().equals(B.getIds());
}
/// Calls areIdsAligned to check if two constraint systems have the same set
/// of identifiers in the same order.
bool FlatAffineConstraints::areIdsAlignedWithOther(
const FlatAffineConstraints &other) {
return areIdsAligned(*this, other);
}
/// Checks if the SSA values associated with `cst''s identifiers are unique.
static bool LLVM_ATTRIBUTE_UNUSED
areIdsUnique(const FlatAffineConstraints &cst) {
SmallPtrSet<Value, 8> uniqueIds;
for (auto id : cst.getIds()) {
if (id.hasValue() && !uniqueIds.insert(id.getValue()).second)
return false;
}
return true;
}
// Swap the posA^th identifier with the posB^th identifier.
static void swapId(FlatAffineConstraints *A, unsigned posA, unsigned posB) {
assert(posA < A->getNumIds() && "invalid position A");
assert(posB < A->getNumIds() && "invalid position B");
if (posA == posB)
return;
for (unsigned r = 0, e = A->getNumInequalities(); r < e; r++) {
std::swap(A->atIneq(r, posA), A->atIneq(r, posB));
}
for (unsigned r = 0, e = A->getNumEqualities(); r < e; r++) {
std::swap(A->atEq(r, posA), A->atEq(r, posB));
}
std::swap(A->getId(posA), A->getId(posB));
}
/// Merge and align the identifiers of A and B starting at 'offset', so that
/// both constraint systems get the union of the contained identifiers that is
/// dimension-wise and symbol-wise unique; both constraint systems are updated
/// so that they have the union of all identifiers, with A's original
/// identifiers appearing first followed by any of B's identifiers that didn't
/// appear in A. Local identifiers of each system are by design separate/local
/// and are placed one after other (A's followed by B's).
// Eg: Input: A has ((%i %j) [%M %N]) and B has (%k, %j) [%P, %N, %M])
// Output: both A, B have (%i, %j, %k) [%M, %N, %P]
//
static void mergeAndAlignIds(unsigned offset, FlatAffineConstraints *A,
FlatAffineConstraints *B) {
assert(offset <= A->getNumDimIds() && offset <= B->getNumDimIds());
// A merge/align isn't meaningful if a cst's ids aren't distinct.
assert(areIdsUnique(*A) && "A's id values aren't unique");
assert(areIdsUnique(*B) && "B's id values aren't unique");
assert(std::all_of(A->getIds().begin() + offset,
A->getIds().begin() + A->getNumDimAndSymbolIds(),
[](Optional<Value> id) { return id.hasValue(); }));
assert(std::all_of(B->getIds().begin() + offset,
B->getIds().begin() + B->getNumDimAndSymbolIds(),
[](Optional<Value> id) { return id.hasValue(); }));
// Place local id's of A after local id's of B.
for (unsigned l = 0, e = A->getNumLocalIds(); l < e; l++) {
B->addLocalId(0);
}
for (unsigned t = 0, e = B->getNumLocalIds() - A->getNumLocalIds(); t < e;
t++) {
A->addLocalId(A->getNumLocalIds());
}
SmallVector<Value, 4> aDimValues, aSymValues;
A->getIdValues(offset, A->getNumDimIds(), &aDimValues);
A->getIdValues(A->getNumDimIds(), A->getNumDimAndSymbolIds(), &aSymValues);
{
// Merge dims from A into B.
unsigned d = offset;
for (auto aDimValue : aDimValues) {
unsigned loc;
if (B->findId(aDimValue, &loc)) {
assert(loc >= offset && "A's dim appears in B's aligned range");
assert(loc < B->getNumDimIds() &&
"A's dim appears in B's non-dim position");
swapId(B, d, loc);
} else {
B->addDimId(d);
B->setIdValue(d, aDimValue);
}
d++;
}
// Dimensions that are in B, but not in A, are added at the end.
for (unsigned t = A->getNumDimIds(), e = B->getNumDimIds(); t < e; t++) {
A->addDimId(A->getNumDimIds());
A->setIdValue(A->getNumDimIds() - 1, B->getIdValue(t));
}
}
{
// Merge symbols: merge A's symbols into B first.
unsigned s = B->getNumDimIds();
for (auto aSymValue : aSymValues) {
unsigned loc;
if (B->findId(aSymValue, &loc)) {
assert(loc >= B->getNumDimIds() && loc < B->getNumDimAndSymbolIds() &&
"A's symbol appears in B's non-symbol position");
swapId(B, s, loc);
} else {
B->addSymbolId(s - B->getNumDimIds());
B->setIdValue(s, aSymValue);
}
s++;
}
// Symbols that are in B, but not in A, are added at the end.
for (unsigned t = A->getNumDimAndSymbolIds(),
e = B->getNumDimAndSymbolIds();
t < e; t++) {
A->addSymbolId(A->getNumSymbolIds());
A->setIdValue(A->getNumDimAndSymbolIds() - 1, B->getIdValue(t));
}
}
assert(areIdsAligned(*A, *B) && "IDs expected to be aligned");
}
// Call 'mergeAndAlignIds' to align constraint systems of 'this' and 'other'.
void FlatAffineConstraints::mergeAndAlignIdsWithOther(
unsigned offset, FlatAffineConstraints *other) {
mergeAndAlignIds(offset, this, other);
}
// This routine may add additional local variables if the flattened expression
// corresponding to the map has such variables due to mod's, ceildiv's, and
// floordiv's in it.
LogicalResult FlatAffineConstraints::composeMap(const AffineValueMap *vMap) {
std::vector<SmallVector<int64_t, 8>> flatExprs;
FlatAffineConstraints localCst;
if (failed(getFlattenedAffineExprs(vMap->getAffineMap(), &flatExprs,
&localCst))) {
LLVM_DEBUG(llvm::dbgs()
<< "composition unimplemented for semi-affine maps\n");
return failure();
}
assert(flatExprs.size() == vMap->getNumResults());
// Add localCst information.
if (localCst.getNumLocalIds() > 0) {
localCst.setIdValues(0, /*end=*/localCst.getNumDimAndSymbolIds(),
/*values=*/vMap->getOperands());
// Align localCst and this.
mergeAndAlignIds(/*offset=*/0, &localCst, this);
// Finally, append localCst to this constraint set.
append(localCst);
}
// Add dimensions corresponding to the map's results.
for (unsigned t = 0, e = vMap->getNumResults(); t < e; t++) {
// TODO: Consider using a batched version to add a range of IDs.
addDimId(0);
}
// 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];
assert(flatExpr.size() >= vMap->getNumOperands() + 1);
// 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;
// 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;
// Negate 'eq[r]' since the newly added dimension will be set to this one.
eqToAdd[loc] = -flatExpr[i];
}
// Local vars common to eq and localCst are at the beginning.
unsigned j = getNumDimIds() + getNumSymbolIds();
unsigned end = flatExpr.size() - 1;
for (unsigned 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 success();
}
// Similar to composeMap except that no Value's need be associated with the
// constraint system nor are they looked at -- since the dimensions and
// symbols of 'other' are expected to correspond 1:1 to 'this' system. It
// is thus not convenient to share code with composeMap.
LogicalResult FlatAffineConstraints::composeMatchingMap(AffineMap other) {
assert(other.getNumDims() == getNumDimIds() && "dim mismatch");
assert(other.getNumSymbols() == getNumSymbolIds() && "symbol mismatch");
std::vector<SmallVector<int64_t, 8>> flatExprs;
FlatAffineConstraints localCst;
if (failed(getFlattenedAffineExprs(other, &flatExprs, &localCst))) {
LLVM_DEBUG(llvm::dbgs()
<< "composition unimplemented for semi-affine maps\n");
return failure();
}
assert(flatExprs.size() == other.getNumResults());
// Add localCst information.
if (localCst.getNumLocalIds() > 0) {
// Place local id's of A after local id's of B.
for (unsigned l = 0, e = localCst.getNumLocalIds(); l < e; l++) {
addLocalId(0);
}
// Finally, append localCst to this constraint set.
append(localCst);
}
// Add dimensions corresponding to the map's results.
for (unsigned t = 0, e = other.getNumResults(); t < e; t++) {
addDimId(0);
}
// 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];
assert(flatExpr.size() >= other.getNumInputs() + 1);
// 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;
// Dims and symbols.
for (unsigned i = 0, f = other.getNumInputs(); i < f; i++) {
// Negate 'eq[r]' since the newly added dimension will be set to this one.
eqToAdd[e + i] = -flatExpr[i];
}
// Local vars common to eq and localCst are at the beginning.
unsigned j = getNumDimIds() + getNumSymbolIds();
unsigned end = flatExpr.size() - 1;
for (unsigned i = other.getNumInputs(); 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 success();
}
// Turn a dimension into a symbol.
static void turnDimIntoSymbol(FlatAffineConstraints *cst, Value id) {
unsigned pos;
if (cst->findId(id, &pos) && pos < cst->getNumDimIds()) {
swapId(cst, pos, cst->getNumDimIds() - 1);
cst->setDimSymbolSeparation(cst->getNumSymbolIds() + 1);
}
}
// Turn a symbol into a dimension.
static void turnSymbolIntoDim(FlatAffineConstraints *cst, Value id) {
unsigned pos;
if (cst->findId(id, &pos) && pos >= cst->getNumDimIds() &&
pos < cst->getNumDimAndSymbolIds()) {
swapId(cst, pos, cst->getNumDimIds());
cst->setDimSymbolSeparation(cst->getNumSymbolIds() - 1);
}
}
// Changes all symbol identifiers which are loop IVs to dim identifiers.
void FlatAffineConstraints::convertLoopIVSymbolsToDims() {
// Gather all symbols which are loop IVs.
SmallVector<Value, 4> loopIVs;
for (unsigned i = getNumDimIds(), e = getNumDimAndSymbolIds(); i < e; i++) {
if (ids[i].hasValue() && getForInductionVarOwner(ids[i].getValue()))
loopIVs.push_back(ids[i].getValue());
}
// Turn each symbol in 'loopIVs' into a dim identifier.
for (auto iv : loopIVs) {
turnSymbolIntoDim(this, iv);
}
}
void FlatAffineConstraints::addInductionVarOrTerminalSymbol(Value id) {
if (containsId(id))
return;
// Caller is expected to fully compose map/operands if necessary.
assert((isTopLevelValue(id) || isForInductionVar(id)) &&
"non-terminal symbol / loop IV expected");
// Outer loop IVs could be used in forOp's bounds.
if (auto loop = getForInductionVarOwner(id)) {
addDimId(getNumDimIds(), id);
if (failed(this->addAffineForOpDomain(loop)))
LLVM_DEBUG(
loop.emitWarning("failed to add domain info to constraint system"));
return;
}
// Add top level symbol.
addSymbolId(getNumSymbolIds(), id);
// Check if the symbol is a constant.
if (auto constOp = id.getDefiningOp<ConstantIndexOp>())
setIdToConstant(id, constOp.getValue());
}
LogicalResult FlatAffineConstraints::addAffineForOpDomain(AffineForOp forOp) {
unsigned pos;
// Pre-condition for this method.
if (!findId(forOp.getInductionVar(), &pos)) {
assert(false && "Value not found");
return failure();
}
int64_t step = forOp.getStep();
if (step != 1) {
if (!forOp.hasConstantLowerBound())
forOp.emitWarning("domain conservatively approximated");
else {
// Add constraints for the stride.
// (iv - lb) % step = 0 can be written as:
// (iv - lb) - step * q = 0 where q = (iv - lb) / step.
// Add local variable 'q' and add the above equality.
// The first constraint is q = (iv - lb) floordiv step
SmallVector<int64_t, 8> dividend(getNumCols(), 0);
int64_t lb = forOp.getConstantLowerBound();
dividend[pos] = 1;
dividend.back() -= lb;
addLocalFloorDiv(dividend, step);
// Second constraint: (iv - lb) - step * q = 0.
SmallVector<int64_t, 8> eq(getNumCols(), 0);
eq[pos] = 1;
eq.back() -= lb;
// For the local var just added above.
eq[getNumCols() - 2] = -step;
addEquality(eq);
}
}
if (forOp.hasConstantLowerBound()) {
addConstantLowerBound(pos, forOp.getConstantLowerBound());
} else {
// Non-constant lower bound case.
if (failed(addLowerOrUpperBound(pos, forOp.getLowerBoundMap(),
forOp.getLowerBoundOperands(),
/*eq=*/false, /*lower=*/true)))
return failure();
}
if (forOp.hasConstantUpperBound()) {
addConstantUpperBound(pos, forOp.getConstantUpperBound() - 1);
return success();
}
// Non-constant upper bound case.
return addLowerOrUpperBound(pos, forOp.getUpperBoundMap(),
forOp.getUpperBoundOperands(),
/*eq=*/false, /*lower=*/false);
}
// 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 &cst,
unsigned colIdx, bool isEq,
unsigned *rowIdx) {
assert(colIdx < cst.getNumCols() && "position out of bounds");
auto at = [&](unsigned rowIdx) -> int64_t {
return isEq ? cst.atEq(rowIdx, colIdx) : cst.atIneq(rowIdx, colIdx);
};
unsigned e = isEq ? cst.getNumEqualities() : cst.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 constraints 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(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() && "invalid id limit");
if (idStart >= idLimit)
return;
// We are going to be removing one or more identifiers from the range.
assert(idStart < numIds && "invalid idStart position");
// TODO: 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;
unsigned numLocalsEliminated = 0;
unsigned numColsEliminated = idLimit - idStart;
if (idStart < numDims) {
numDimsEliminated = std::min(numDims, idLimit) - idStart;
}
// Check how many local id's were removed. Note that our identifier order is
// [dims, symbols, locals]. Local id start at position numDims + numSymbols.
if (idLimit > numDims + numSymbols) {
numLocalsEliminated = std::min(
idLimit - std::max(idStart, numDims + numSymbols), getNumLocalIds());
}
unsigned numSymbolsEliminated =
numColsEliminated - numDimsEliminated - numLocalsEliminated;
numDims -= numDimsEliminated;
numSymbols -= numSymbolsEliminated;
numIds = numIds - numColsEliminated;
ids.erase(ids.begin() + idStart, ids.begin() + idLimit);
// No resize necessary. numReservedCols remains the same.
}
/// 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;
}
// 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;
// First, eliminate as many identifiers as possible using Gaussian
// elimination.
FlatAffineConstraints tmpCst(*this);
unsigned currentPos = 0;
while (currentPos < tmpCst.getNumIds()) {
tmpCst.gaussianEliminateIds(currentPos, tmpCst.getNumIds());
++currentPos;
// 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.hasInvalidConstraint() || tmpCst.isEmptyByGCDTest())
return true;
}
// Eliminate the remaining using FM.
for (unsigned i = 0, e = tmpCst.getNumIds(); i < e; i++) {
tmpCst.FourierMotzkinEliminate(
getBestIdToEliminate(tmpCst, 0, tmpCst.getNumIds()));
// Check for a constraint explosion. This rarely happens in practice, but
// this check exists as a safeguard against improperly constructed
// constraint systems or artificially created arbitrarily complex systems
// that aren't the intended use case for FlatAffineConstraints. This is
// needed since FM has a worst case exponential complexity in theory.
if (tmpCst.getNumConstraints() >= kExplosionFactor * getNumIds()) {
LLVM_DEBUG(llvm::dbgs() << "FM constraint explosion detected\n");
return false;
}
// 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;
}
// First, try the GCD test heuristic.
//
// If that doesn't find the set empty, check if the set is unbounded. If it is,
// we cannot use the GBR algorithm and we conservatively return false.
//
// If the set is bounded, we use the complete emptiness check for this case
// provided by Simplex::findIntegerSample(), which gives a definitive answer.
bool FlatAffineConstraints::isIntegerEmpty() const {
if (isEmptyByGCDTest())
return true;
Simplex simplex(*this);
if (simplex.isUnbounded())
return false;
return !simplex.findIntegerSample().hasValue();
}
Optional<SmallVector<int64_t, 8>>
FlatAffineConstraints::findIntegerSample() const {
return Simplex(*this).findIntegerSample();
}
/// 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 && gcd != 1) {
int64_t gcdI = static_cast<int64_t>(gcd);
// Tighten the constant term and normalize the constraint by the GCD.
atIneq(i, numCols - 1) = mlir::floorDiv(atIneq(i, numCols - 1), gcdI);
for (unsigned j = 0, e = numCols - 1; j < e; ++j)
atIneq(i, j) /= gcdI;
}
}
}
// Eliminates all identifier 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(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);
GCDTightenInequalities();
}
// Update position limit based on number eliminated.
posLimit = pivotCol;
// Remove eliminated columns from all constraints.
removeIdRange(posStart, posLimit);
return posLimit - posStart;
}
// Detect the identifier at 'pos' (say id_r) as modulo of another identifier
// (say id_n) w.r.t a constant. When this happens, another identifier (say id_q)
// could be detected as the floordiv of n. For eg:
// id_n - 4*id_q - id_r = 0, 0 <= id_r <= 3 <=>
// id_r = id_n mod 4, id_q = id_n floordiv 4.
// lbConst and ubConst are the constant lower and upper bounds for 'pos' -
// pre-detected at the caller.
static bool detectAsMod(const FlatAffineConstraints &cst, unsigned pos,
int64_t lbConst, int64_t ubConst,
SmallVectorImpl<AffineExpr> *memo) {
assert(pos < cst.getNumIds() && "invalid position");
// Check if 0 <= id_r <= divisor - 1 and if id_r is equal to
// id_n - divisor * id_q. If these are true, then id_n becomes the dividend
// and id_q the quotient when dividing id_n by the divisor.
if (lbConst != 0 || ubConst < 1)
return false;
int64_t divisor = ubConst + 1;
// Now check for: id_r = id_n - divisor * id_q. As an example, we
// are looking r = d - 4q, i.e., either r - d + 4q = 0 or -r + d - 4q = 0.
unsigned seenQuotient = 0, seenDividend = 0;
int quotientPos = -1, dividendPos = -1;
for (unsigned r = 0, e = cst.getNumEqualities(); r < e; r++) {
// id_n should have coeff 1 or -1.
if (std::abs(cst.atEq(r, pos)) != 1)
continue;
// constant term should be 0.
if (cst.atEq(r, cst.getNumCols() - 1) != 0)
continue;
unsigned c, f;
int quotientSign = 1, dividendSign = 1;
for (c = 0, f = cst.getNumDimAndSymbolIds(); c < f; c++) {
if (c == pos)
continue;
// The coefficient of the quotient should be +/-divisor.
// TODO: could be extended to detect an affine function for the quotient
// (i.e., the coeff could be a non-zero multiple of divisor).
int64_t v = cst.atEq(r, c) * cst.atEq(r, pos);
if (v == divisor || v == -divisor) {
seenQuotient++;
quotientPos = c;
quotientSign = v > 0 ? 1 : -1;
}
// The coefficient of the dividend should be +/-1.
// TODO: could be extended to detect an affine function of the other
// identifiers as the dividend.
else if (v == -1 || v == 1) {
seenDividend++;
dividendPos = c;
dividendSign = v < 0 ? 1 : -1;
} else if (cst.atEq(r, c) != 0) {
// Cannot be inferred as a mod since the constraint has a coefficient
// for an identifier that's neither a unit nor the divisor (see TODOs
// above).
break;
}
}
if (c < f)
// Cannot be inferred as a mod since the constraint has a coefficient for
// an identifier that's neither a unit nor the divisor (see TODOs above).
continue;
// We are looking for exactly one identifier as the dividend.
if (seenDividend == 1 && seenQuotient >= 1) {
if (!(*memo)[dividendPos])
return false;
// Successfully detected a mod.
(*memo)[pos] = (*memo)[dividendPos] % divisor * dividendSign;
auto ub = cst.getConstantUpperBound(dividendPos);
if (ub.hasValue() && ub.getValue() < divisor)
// The mod can be optimized away.
(*memo)[pos] = (*memo)[dividendPos] * dividendSign;
else
(*memo)[pos] = (*memo)[dividendPos] % divisor * dividendSign;
if (seenQuotient == 1 && !(*memo)[quotientPos])
// Successfully detected a floordiv as well.
(*memo)[quotientPos] =
(*memo)[dividendPos].floorDiv(divisor) * quotientSign;
return true;
}
}
return false;
}
/// Gather all lower and upper bounds of the identifier at `pos`, and
/// optionally any equalities on it. In addition, the bounds are to be
/// independent of identifiers in position range [`offset`, `offset` + `num`).
void FlatAffineConstraints::getLowerAndUpperBoundIndices(
unsigned pos, SmallVectorImpl<unsigned> *lbIndices,
SmallVectorImpl<unsigned> *ubIndices, SmallVectorImpl<unsigned> *eqIndices,
unsigned offset, unsigned num) const {
assert(pos < getNumIds() && "invalid position");
assert(offset + num < getNumCols() && "invalid range");
// Checks for a constraint that has a non-zero coeff for the identifiers in
// the position range [offset, offset + num) while ignoring `pos`.
auto containsConstraintDependentOnRange = [&](unsigned r, bool isEq) {
unsigned c, f;
auto cst = isEq ? getEquality(r) : getInequality(r);
for (c = offset, f = offset + num; c < f; ++c) {
if (c == pos)
continue;
if (cst[c] != 0)
break;
}
return c < f;
};
// 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++) {
// The bounds are to be independent of [offset, offset + num) columns.
if (containsConstraintDependentOnRange(r, /*isEq=*/false))
continue;
if (atIneq(r, pos) >= 1) {
// Lower bound.
lbIndices->push_back(r);
} else if (atIneq(r, pos) <= -1) {
// Upper bound.
ubIndices->push_back(r);
}
}
// An equality is both a lower and upper bound. Record any equalities
// involving the pos^th identifier.
if (!eqIndices)
return;
for (unsigned r = 0, e = getNumEqualities(); r < e; r++) {
if (atEq(r, pos) == 0)
continue;
if (containsConstraintDependentOnRange(r, /*isEq=*/true))
continue;
eqIndices->push_back(r);
}
}
/// Check if the pos^th identifier can be expressed as a floordiv of an affine
/// function of other identifiers (where the divisor is a positive constant)
/// given the initial set of expressions in `exprs`. If it can be, the
/// corresponding position in `exprs` is set as the detected affine expr. For
/// eg: 4q <= i + j <= 4q + 3 <=> q = (i + j) floordiv 4. An equality can
/// also yield a floordiv: eg. 4q = i + j <=> q = (i + j) floordiv 4. 32q + 28
/// <= i <= 32q + 31 => q = i floordiv 32.
static bool detectAsFloorDiv(const FlatAffineConstraints &cst, unsigned pos,
MLIRContext *context,
SmallVectorImpl<AffineExpr> &exprs) {
assert(pos < cst.getNumIds() && "invalid position");
SmallVector<unsigned, 4> lbIndices, ubIndices;
cst.getLowerAndUpperBoundIndices(pos, &lbIndices, &ubIndices);
// Check if any lower bound, upper bound pair is of the form:
// divisor * id >= expr - (divisor - 1) <-- Lower bound for 'id'
// divisor * id <= expr <-- Upper bound for 'id'
// Then, 'id' is equivalent to 'expr floordiv divisor'. (where divisor > 1).
//
// For example, if -32*k + 16*i + j >= 0
// 32*k - 16*i - j + 31 >= 0 <=>
// k = ( 16*i + j ) floordiv 32
unsigned seenDividends = 0;
for (auto ubPos : ubIndices) {
for (auto lbPos : lbIndices) {
// Check if the lower bound's constant term is divisor - 1. The
// 'divisor' here is cst.atIneq(lbPos, pos) and we already know that it's
// positive (since cst.Ineq(lbPos, ...) is a lower bound expr for 'pos'.
int64_t divisor = cst.atIneq(lbPos, pos);
int64_t lbConstTerm = cst.atIneq(lbPos, cst.getNumCols() - 1);
if (lbConstTerm != divisor - 1)
continue;
// Check if upper bound's constant term is 0.
if (cst.atIneq(ubPos, cst.getNumCols() - 1) != 0)
continue;
// For the remaining part, check if the lower bound expr's coeff's are
// negations of corresponding upper bound ones'.
unsigned c, f;
for (c = 0, f = cst.getNumCols() - 1; c < f; c++) {
if (cst.atIneq(lbPos, c) != -cst.atIneq(ubPos, c))
break;
if (c != pos && cst.atIneq(lbPos, c) != 0)
seenDividends++;
}
// Lb coeff's aren't negative of ub coeff's (for the non constant term
// part).
if (c < f)
continue;
if (seenDividends >= 1) {
// Construct the dividend expression.
auto dividendExpr = getAffineConstantExpr(0, context);
unsigned c, f;
for (c = 0, f = cst.getNumCols() - 1; c < f; c++) {
if (c == pos)
continue;
int64_t ubVal = cst.atIneq(ubPos, c);
if (ubVal == 0)
continue;
if (!exprs[c])
break;
dividendExpr = dividendExpr + ubVal * exprs[c];
}
// Expression can't be constructed as it depends on a yet unknown
// identifier.
// TODO: Visit/compute the identifiers in an order so that this doesn't
// happen. More complex but much more efficient.
if (c < f)
continue;
// Successfully detected the floordiv.
exprs[pos] = dividendExpr.floorDiv(divisor);
return true;
}
}
}
return false;
}
// Fills an inequality row with the value 'val'.
static inline void fillInequality(FlatAffineConstraints *cst, unsigned r,
int64_t val) {
for (unsigned c = 0, f = cst->getNumCols(); c < f; c++) {
cst->atIneq(r, c) = val;
}
}
// Negates an inequality.
static inline void negateInequality(FlatAffineConstraints *cst, unsigned r) {
for (unsigned c = 0, f = cst->getNumCols(); c < f; c++) {
cst->atIneq(r, c) = -cst->atIneq(r, c);
}
}
// A more complex check to eliminate redundant inequalities. Uses FourierMotzkin
// to check if a constraint is redundant.
void FlatAffineConstraints::removeRedundantInequalities() {
SmallVector<bool, 32> redun(getNumInequalities(), false);
// To check if an inequality is redundant, we replace the inequality by its
// complement (for eg., i - 1 >= 0 by i <= 0), and check if the resulting
// system is empty. If it is, the inequality is redundant.
FlatAffineConstraints tmpCst(*this);
for (unsigned r = 0, e = getNumInequalities(); r < e; r++) {
// Change the inequality to its complement.
negateInequality(&tmpCst, r);
tmpCst.atIneq(r, tmpCst.getNumCols() - 1)--;
if (tmpCst.isEmpty()) {
redun[r] = true;
// Zero fill the redundant inequality.
fillInequality(this, r, /*val=*/0);
fillInequality(&tmpCst, r, /*val=*/0);
} else {
// Reverse the change (to avoid recreating tmpCst each time).
tmpCst.atIneq(r, tmpCst.getNumCols() - 1)++;
negateInequality(&tmpCst, r);
}
}
// Scan to get rid of all rows marked redundant, in-place.
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);
}
};
unsigned pos = 0;
for (unsigned r = 0, e = getNumInequalities(); r < e; r++) {
if (!redun[r])
copyRow(r, pos++);
}
inequalities.resize(numReservedCols * pos);
}
std::pair<AffineMap, AffineMap> FlatAffineConstraints::getLowerAndUpperBound(
unsigned pos, unsigned offset, unsigned num, unsigned symStartPos,
ArrayRef<AffineExpr> localExprs, MLIRContext *context) const {
assert(pos + offset < getNumDimIds() && "invalid dim start pos");
assert(symStartPos >= (pos + offset) && "invalid sym start pos");
assert(getNumLocalIds() == localExprs.size() &&
"incorrect local exprs count");
SmallVector<unsigned, 4> lbIndices, ubIndices, eqIndices;
getLowerAndUpperBoundIndices(pos + offset, &lbIndices, &ubIndices, &eqIndices,
offset, num);
/// Add to 'b' from 'a' in set [0, offset) U [offset + num, symbStartPos).
auto addCoeffs = [&](ArrayRef<int64_t> a, SmallVectorImpl<int64_t> &b) {
b.clear();
for (unsigned i = 0, e = a.size(); i < e; ++i) {
if (i < offset || i >= offset + num)
b.push_back(a[i]);
}
};
SmallVector<int64_t, 8> lb, ub;
SmallVector<AffineExpr, 4> lbExprs;
unsigned dimCount = symStartPos - num;
unsigned symCount = getNumDimAndSymbolIds() - symStartPos;
lbExprs.reserve(lbIndices.size() + eqIndices.size());
// Lower bound expressions.
for (auto idx : lbIndices) {
auto ineq = getInequality(idx);
// Extract the lower bound (in terms of other coeff's + const), i.e., if
// i - j + 1 >= 0 is the constraint, 'pos' is for i the lower bound is j
// - 1.
addCoeffs(ineq, lb);
std::transform(lb.begin(), lb.end(), lb.begin(), std::negate<int64_t>());
auto expr =
getAffineExprFromFlatForm(lb, dimCount, symCount, localExprs, context);
// expr ceildiv divisor is (expr + divisor - 1) floordiv divisor
int64_t divisor = std::abs(ineq[pos + offset]);
expr = (expr + divisor - 1).floorDiv(divisor);
lbExprs.push_back(expr);
}
SmallVector<AffineExpr, 4> ubExprs;
ubExprs.reserve(ubIndices.size() + eqIndices.size());
// Upper bound expressions.
for (auto idx : ubIndices) {
auto ineq = getInequality(idx);
// Extract the upper bound (in terms of other coeff's + const).
addCoeffs(ineq, ub);
auto expr =
getAffineExprFromFlatForm(ub, dimCount, symCount, localExprs, context);
expr = expr.floorDiv(std::abs(ineq[pos + offset]));
// Upper bound is exclusive.
ubExprs.push_back(expr + 1);
}
// Equalities. It's both a lower and a upper bound.
SmallVector<int64_t, 4> b;
for (auto idx : eqIndices) {
auto eq = getEquality(idx);
addCoeffs(eq, b);
if (eq[pos + offset] > 0)
std::transform(b.begin(), b.end(), b.begin(), std::negate<int64_t>());
// Extract the upper bound (in terms of other coeff's + const).
auto expr =
getAffineExprFromFlatForm(b, dimCount, symCount, localExprs, context);
expr = expr.floorDiv(std::abs(eq[pos + offset]));
// Upper bound is exclusive.
ubExprs.push_back(expr + 1);
// Lower bound.
expr =
getAffineExprFromFlatForm(b, dimCount, symCount, localExprs, context);
expr = expr.ceilDiv(std::abs(eq[pos + offset]));
lbExprs.push_back(expr);
}
auto lbMap = AffineMap::get(dimCount, symCount, lbExprs, context);
auto ubMap = AffineMap::get(dimCount, symCount, ubExprs, context);
return {lbMap, ubMap};
}
/// Computes the lower and upper bounds of the first 'num' dimensional
/// identifiers (starting at 'offset') as affine maps of the remaining
/// identifiers (dimensional and symbolic identifiers). Local identifiers are
/// themselves explicitly computed as affine functions of other identifiers in
/// this process if needed.
void FlatAffineConstraints::getSliceBounds(unsigned offset, unsigned num,
MLIRContext *context,
SmallVectorImpl<AffineMap> *lbMaps,
SmallVectorImpl<AffineMap> *ubMaps) {
assert(num < getNumDimIds() && "invalid range");
// Basic simplification.
normalizeConstraintsByGCD();
LLVM_DEBUG(llvm::dbgs() << "getSliceBounds for first " << num
<< " identifiers\n");
LLVM_DEBUG(dump());
// Record computed/detected identifiers.
SmallVector<AffineExpr, 8> memo(getNumIds());
// Initialize dimensional and symbolic identifiers.
for (unsigned i = 0, e = getNumDimIds(); i < e; i++) {
if (i < offset)
memo[i] = getAffineDimExpr(i, context);
else if (i >= offset + num)
memo[i] = getAffineDimExpr(i - num, context);
}
for (unsigned i = getNumDimIds(), e = getNumDimAndSymbolIds(); i < e; i++)
memo[i] = getAffineSymbolExpr(i - getNumDimIds(), context);
bool changed;
do {
changed = false;
// Identify yet unknown identifiers as constants or mod's / floordiv's of
// other identifiers if possible.
for (unsigned pos = 0; pos < getNumIds(); pos++) {
if (memo[pos])
continue;
auto lbConst = getConstantLowerBound(pos);
auto ubConst = getConstantUpperBound(pos);
if (lbConst.hasValue() && ubConst.hasValue()) {
// Detect equality to a constant.
if (lbConst.getValue() == ubConst.getValue()) {
memo[pos] = getAffineConstantExpr(lbConst.getValue(), context);
changed = true;
continue;
}
// Detect an identifier as modulo of another identifier w.r.t a
// constant.
if (detectAsMod(*this, pos, lbConst.getValue(), ubConst.getValue(),
&memo)) {
changed = true;
continue;
}
}
// Detect an identifier as a floordiv of an affine function of other
// identifiers (divisor is a positive constant).
if (detectAsFloorDiv(*this, pos, context, memo)) {
changed = true;
continue;
}
// Detect an identifier as an expression of other identifiers.
unsigned idx;
if (!findConstraintWithNonZeroAt(*this, pos, /*isEq=*/true, &idx)) {
continue;
}
// Build AffineExpr solving for identifier 'pos' in terms of all others.
auto expr = getAffineConstantExpr(0, context);
unsigned j, e;
for (j = 0, e = getNumIds(); j < e; ++j) {
if (j == pos)
continue;
int64_t c = atEq(idx, j);
if (c == 0)
continue;
// If any of the involved IDs hasn't been found yet, we can't proceed.
if (!memo[j])
break;
expr = expr + memo[j] * c;
}
if (j < e)
// Can't construct expression as it depends on a yet uncomputed
// identifier.
continue;
// Add constant term to AffineExpr.
expr = expr + atEq(idx, getNumIds());
int64_t vPos = atEq(idx, pos);
assert(vPos != 0 && "expected non-zero here");
if (vPos > 0)
expr = (-expr).floorDiv(vPos);
else
// vPos < 0.
expr = expr.floorDiv(-vPos);
// Successfully constructed expression.
memo[pos] = expr;
changed = true;
}
// This loop is guaranteed to reach a fixed point - since once an
// identifier's explicit form is computed (in memo[pos]), it's not updated
// again.
} while (changed);
// Set the lower and upper bound maps for all the identifiers that were
// computed as affine expressions of the rest as the "detected expr" and
// "detected expr + 1" respectively; set the undetected ones to null.
Optional<FlatAffineConstraints> tmpClone;
for (unsigned pos = 0; pos < num; pos++) {
unsigned numMapDims = getNumDimIds() - num;
unsigned numMapSymbols = getNumSymbolIds();
AffineExpr expr = memo[pos + offset];
if (expr)
expr = simplifyAffineExpr(expr, numMapDims, numMapSymbols);
AffineMap &lbMap = (*lbMaps)[pos];
AffineMap &ubMap = (*ubMaps)[pos];
if (expr) {
lbMap = AffineMap::get(numMapDims, numMapSymbols, expr);
ubMap = AffineMap::get(numMapDims, numMapSymbols, expr + 1);
} else {
// TODO: Whenever there are local identifiers in the dependence
// constraints, we'll conservatively over-approximate, since we don't
// always explicitly compute them above (in the while loop).
if (getNumLocalIds() == 0) {
// Work on a copy so that we don't update this constraint system.
if (!tmpClone) {
tmpClone.emplace(FlatAffineConstraints(*this));
// Removing redundant inequalities is necessary so that we don't get
// redundant loop bounds.
tmpClone->removeRedundantInequalities();
}
std::tie(lbMap, ubMap) = tmpClone->getLowerAndUpperBound(
pos, offset, num, getNumDimIds(), /*localExprs=*/{}, context);
}
// If the above fails, we'll just use the constant lower bound and the
// constant upper bound (if they exist) as the slice bounds.
// TODO: being conservative for the moment in cases that
// lead to multiple bounds - until getConstDifference in LoopFusion.cpp is
// fixed (b/126426796).
if (!lbMap || lbMap.getNumResults() > 1) {
LLVM_DEBUG(llvm::dbgs()
<< "WARNING: Potentially over-approximating slice lb\n");
auto lbConst = getConstantLowerBound(pos + offset);
if (lbConst.hasValue()) {
lbMap = AffineMap::get(
numMapDims, numMapSymbols,
getAffineConstantExpr(lbConst.getValue(), context));
}
}
if (!ubMap || ubMap.getNumResults() > 1) {
LLVM_DEBUG(llvm::dbgs()
<< "WARNING: Potentially over-approximating slice ub\n");
auto ubConst = getConstantUpperBound(pos + offset);
if (ubConst.hasValue()) {
(ubMap) = AffineMap::get(
numMapDims, numMapSymbols,
getAffineConstantExpr(ubConst.getValue() + 1, context));
}
}
}
LLVM_DEBUG(llvm::dbgs()
<< "lb map for pos = " << Twine(pos + offset) << ", expr: ");
LLVM_DEBUG(lbMap.dump(););
LLVM_DEBUG(llvm::dbgs()
<< "ub map for pos = " << Twine(pos + offset) << ", expr: ");
LLVM_DEBUG(ubMap.dump(););
}
}
LogicalResult
FlatAffineConstraints::addLowerOrUpperBound(unsigned pos, AffineMap boundMap,
ValueRange boundOperands, bool eq,
bool lower) {
assert(pos < getNumDimAndSymbolIds() && "invalid position");
// Equality follows the logic of lower bound except that we add an equality
// instead of an inequality.
assert((!eq || boundMap.getNumResults() == 1) && "single result expected");
if (eq)
lower = true;
// Fully compose map and operands; canonicalize and simplify so that we
// transitively get to terminal symbols or loop IVs.
auto map = boundMap;
SmallVector<Value, 4> operands(boundOperands.begin(), boundOperands.end());
fullyComposeAffineMapAndOperands(&map, &operands);
map = simplifyAffineMap(map);
canonicalizeMapAndOperands(&map, &operands);
for (auto operand : operands)
addInductionVarOrTerminalSymbol(operand);
FlatAffineConstraints localVarCst;
std::vector<SmallVector<int64_t, 8>> flatExprs;
if (failed(getFlattenedAffineExprs(map, &flatExprs, &localVarCst))) {
LLVM_DEBUG(llvm::dbgs() << "semi-affine expressions not yet supported\n");
return failure();
}
// Merge and align with localVarCst.
if (localVarCst.getNumLocalIds() > 0) {
// Set values for localVarCst.
localVarCst.setIdValues(0, localVarCst.getNumDimAndSymbolIds(), operands);
for (auto operand : operands) {
unsigned pos;
if (findId(operand, &pos)) {
if (pos >= getNumDimIds() && pos < getNumDimAndSymbolIds()) {
// If the local var cst has this as a dim, turn it into its symbol.
turnDimIntoSymbol(&localVarCst, operand);
} else if (pos < getNumDimIds()) {
// Or vice versa.
turnSymbolIntoDim(&localVarCst, operand);
}
}
}
mergeAndAlignIds(/*offset=*/0, this, &localVarCst);
append(localVarCst);
}
// Record positions of the operands in the constraint system. Need to do
// this here since the constraint system changes after a bound is added.
SmallVector<unsigned, 8> positions;
unsigned numOperands = operands.size();
for (auto operand : operands) {
unsigned pos;
if (!findId(operand, &pos))
assert(0 && "expected to be found");
positions.push_back(pos);
}
for (const auto &flatExpr : flatExprs) {
SmallVector<int64_t, 4> ineq(getNumCols(), 0);
ineq[pos] = lower ? 1 : -1;
// Dims and symbols.
for (unsigned j = 0, e = map.getNumInputs(); j < e; j++) {
ineq[positions[j]] = lower ? -flatExpr[j] : flatExpr[j];
}
// Copy over the local id coefficients.
unsigned numLocalIds = flatExpr.size() - 1 - numOperands;
for (unsigned jj = 0, j = getNumIds() - numLocalIds; jj < numLocalIds;
jj++, j++) {
ineq[j] =
lower ? -flatExpr[numOperands + jj] : flatExpr[numOperands + jj];
}
// Constant term.
ineq[getNumCols() - 1] =
lower ? -flatExpr[flatExpr.size() - 1]
// Upper bound in flattenedExpr is an exclusive one.
: flatExpr[flatExpr.size() - 1] - 1;
eq ? addEquality(ineq) : addInequality(ineq);
}
return success();
}
// Adds slice lower bounds represented by lower bounds in 'lbMaps' and upper
// bounds in 'ubMaps' to each value in `values' that appears in the constraint
// system. Note that both lower/upper bounds share the same operand list
// 'operands'.
// This function assumes 'values.size' == 'lbMaps.size' == 'ubMaps.size', and
// skips any null AffineMaps in 'lbMaps' or 'ubMaps'.
// Note that both lower/upper bounds use operands from 'operands'.
// Returns failure for unimplemented cases such as semi-affine expressions or
// expressions with mod/floordiv.
LogicalResult FlatAffineConstraints::addSliceBounds(ArrayRef<Value> values,
ArrayRef<AffineMap> lbMaps,
ArrayRef<AffineMap> ubMaps,
ArrayRef<Value> operands) {
assert(values.size() == lbMaps.size());
assert(lbMaps.size() == ubMaps.size());
for (unsigned i = 0, e = lbMaps.size(); i < e; ++i) {
unsigned pos;
if (!findId(values[i], &pos))
continue;
AffineMap lbMap = lbMaps[i];
AffineMap ubMap = ubMaps[i];
assert(!lbMap || lbMap.getNumInputs() == operands.size());
assert(!ubMap || ubMap.getNumInputs() == operands.size());
// Check if this slice is just an equality along this dimension.
if (lbMap && ubMap && lbMap.getNumResults() == 1 &&
ubMap.getNumResults() == 1 &&
lbMap.getResult(0) + 1 == ubMap.getResult(0)) {
if (failed(addLowerOrUpperBound(pos, lbMap, operands, /*eq=*/true,
/*lower=*/true)))
return failure();
continue;
}
if (lbMap && failed(addLowerOrUpperBound(pos, lbMap, operands, /*eq=*/false,
/*lower=*/true)))
return failure();
if (ubMap && failed(addLowerOrUpperBound(pos, ubMap, operands, /*eq=*/false,
/*lower=*/false)))
return failure();
}
return success();
}
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;
}
/// Adds a new local identifier as the floordiv of an affine function of other
/// identifiers, the coefficients of which are provided in 'dividend' and with
/// respect to a positive constant 'divisor'. Two constraints are added to the
/// system to capture equivalence with the floordiv.
/// q = expr floordiv c <=> c*q <= expr <= c*q + c - 1.
void FlatAffineConstraints::addLocalFloorDiv(ArrayRef<int64_t> dividend,
int64_t divisor) {
assert(dividend.size() == getNumCols() && "incorrect dividend size");
assert(divisor > 0 && "positive divisor expected");
addLocalId(getNumLocalIds());
// Add two constraints for this new identifier 'q'.
SmallVector<int64_t, 8> bound(dividend.size() + 1);
// dividend - q * divisor >= 0
std::copy(dividend.begin(), dividend.begin() + dividend.size() - 1,
bound.begin());
bound.back() = dividend.back();
bound[getNumIds() - 1] = -divisor;
addInequality(bound);
// -dividend +qdivisor * q + divisor - 1 >= 0
std::transform(bound.begin(), bound.end(), bound.begin(),
std::negate<int64_t>());
bound[bound.size() - 1] += divisor - 1;
addInequality(bound);
}
bool FlatAffineConstraints::findId(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;
}
bool FlatAffineConstraints::containsId(Value id) const {
return llvm::any_of(ids, [&](const Optional<Value> &mayBeId) {
return mayBeId.hasValue() && mayBeId.getValue() == id;
});
}
void FlatAffineConstraints::setDimSymbolSeparation(unsigned newSymbolCount) {
assert(newSymbolCount <= numDims + numSymbols &&
"invalid separation position");
numDims = numDims + numSymbols - newSymbolCount;
numSymbols = newSymbolCount;
}
/// Sets the specified identifier 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 identifier to a constant value; asserts if the id is not
/// found.
void FlatAffineConstraints::setIdToConstant(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);
assert(equalities.size() >= numReservedCols);
equalities.resize(equalities.size() - numReservedCols);
}
void FlatAffineConstraints::removeInequality(unsigned pos) {
unsigned numInequalities = getNumInequalities();
assert(pos < numInequalities && "invalid position");
unsigned outputIndex = pos * numReservedCols;
unsigned inputIndex = (pos + 1) * numReservedCols;
unsigned numElemsToCopy = (numInequalities - pos - 1) * numReservedCols;
std::copy(inequalities.begin() + inputIndex,
inequalities.begin() + inputIndex + numElemsToCopy,
inequalities.begin() + outputIndex);
assert(inequalities.size() >= numReservedCols);
inequalities.resize(inequalities.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. Returns -1 if such an equality
/// could not be found.
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);
}
LogicalResult FlatAffineConstraints::constantFoldId(unsigned pos) {
assert(pos < getNumIds() && "invalid position");
int rowIdx;
if ((rowIdx = findEqualityToConstant(*this, pos)) == -1)
return failure();
// 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 success();
}
void FlatAffineConstraints::constantFoldIdRange(unsigned pos, unsigned num) {
for (unsigned s = pos, t = pos, e = pos + num; s < e; s++) {
if (failed(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)
// s0 + s1 + 16 <= d0 <= s0 + s1 + 31, returns 16.
// s0 - 7 <= 8*j <= s0 returns 1 with lb = s0, lbDivisor = 8 (since lb =
// ceil(s0 - 7 / 8) = floor(s0 / 8)).
Optional<int64_t> FlatAffineConstraints::getConstantBoundOnDimSize(
unsigned pos, SmallVectorImpl<int64_t> *lb, int64_t *boundFloorDivisor,
SmallVectorImpl<int64_t> *ub, unsigned *minLbPos,
unsigned *minUbPos) const {
assert(pos < getNumDimIds() && "Invalid identifier position");
// Find an equality for 'pos'^th identifier that equates it to some function
// of the symbolic identifiers (+ constant).
int eqPos = findEqualityToConstant(*this, pos, /*symbolic=*/true);
if (eqPos != -1) {
auto eq = getEquality(eqPos);
// If the equality involves a local var, punt for now.
// TODO: this can be handled in the future by using the explicit
// representation of the local vars.
if (!std::all_of(eq.begin() + getNumDimAndSymbolIds(), eq.end() - 1,
[](int64_t coeff) { return coeff == 0; }))
return None;
// This identifier can only take a single value.
if (lb) {
// Set lb to that symbolic value.
lb->resize(getNumSymbolIds() + 1);
if (ub)
ub->resize(getNumSymbolIds() + 1);
for (unsigned c = 0, f = getNumSymbolIds() + 1; c < f; c++) {
int64_t v = atEq(eqPos, pos);
// atEq(eqRow, pos) is either -1 or 1.
assert(v * v == 1);
(*lb)[c] = v < 0 ? atEq(eqPos, getNumDimIds() + c) / -v
: -atEq(eqPos, getNumDimIds() + c) / v;
// Since this is an equality, ub = lb.
if (ub)
(*ub)[c] = (*lb)[c];
}
assert(boundFloorDivisor &&
"both lb and divisor or none should be provided");
*boundFloorDivisor = 1;
}
if (minLbPos)
*minLbPos = eqPos;
if (minUbPos)
*minUbPos = eqPos;
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, there isn't a bound on it.
return None;
// Positions of constraints that are lower/upper bounds on the variable.
SmallVector<unsigned, 4> lbIndices, ubIndices;
// Gather all symbolic lower bounds and upper bounds of the variable, i.e.,
// the bounds can only involve symbolic (and local) identifiers. 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.
getLowerAndUpperBoundIndices(pos, &lbIndices, &ubIndices,
/*eqIndices=*/nullptr, /*offset=*/0,
/*num=*/getNumDimIds());
Optional<int64_t> minDiff = None;
unsigned minLbPosition = 0, minUbPosition = 0;
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 diff = ceilDiv(atIneq(ubPos, getNumCols() - 1) +
atIneq(lbPos, getNumCols() - 1) + 1,
atIneq(lbPos, pos));
if (minDiff == None || diff < minDiff) {
minDiff = diff;
minLbPosition = lbPos;
minUbPosition = ubPos;
}
}
}
if (lb && minDiff.hasValue()) {
// Set lb to the symbolic lower bound.
lb->resize(getNumSymbolIds() + 1);
if (ub)
ub->resize(getNumSymbolIds() + 1);
// The lower bound is the ceildiv of the lb constraint over the coefficient
// of the variable at 'pos'. We express the ceildiv equivalently as a floor
// for uniformity. For eg., if the lower bound constraint was: 32*d0 - N +
// 31 >= 0, the lower bound for d0 is ceil(N - 31, 32), i.e., floor(N, 32).
*boundFloorDivisor = atIneq(minLbPosition, pos);
assert(*boundFloorDivisor == -atIneq(minUbPosition, pos));
for (unsigned c = 0, e = getNumSymbolIds() + 1; c < e; c++) {
(*lb)[c] = -atIneq(minLbPosition, getNumDimIds() + c);
}
if (ub) {
for (unsigned c = 0, e = getNumSymbolIds() + 1; c < e; c++)
(*ub)[c] = atIneq(minUbPosition, getNumDimIds() + c);
}
// The lower bound leads to a ceildiv while the upper bound is a floordiv
// whenever the coefficient at pos != 1. ceildiv (val / d) = floordiv (val +
// d - 1 / d); hence, the addition of 'atIneq(minLbPosition, pos) - 1' to
// the constant term for the lower bound.
(*lb)[getNumSymbolIds()] += atIneq(minLbPosition, pos) - 1;
}
if (minLbPos)
*minLbPos = minLbPosition;
if (minUbPos)
*minUbPos = minUbPosition;
return minDiff;
}
template <bool isLower>
Optional<int64_t>
FlatAffineConstraints::computeConstantLowerOrUpperBound(unsigned pos) {
assert(pos < getNumIds() && "invalid position");
// Project to 'pos'.
projectOut(0, pos);
projectOut(1, getNumIds() - 1);
// Check if there's an equality equating the '0'^th identifier to a constant.
int eqRowIdx = findEqualityToConstant(*this, 0, /*symbolic=*/false);
if (eqRowIdx != -1)
// atEq(rowIdx, 0) is either -1 or 1.
return -atEq(eqRowIdx, getNumCols() - 1) / atEq(eqRowIdx, 0);
// 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, 0) != 0)
break;
}
if (r == e)
// If it doesn't, there isn't a bound on it.
return None;
Optional<int64_t> minOrMaxConst = None;
// Take the max across all const lower bounds (or min across all constant
// upper bounds).
for (unsigned r = 0, e = getNumInequalities(); r < e; r++) {
if (isLower) {
if (atIneq(r, 0) <= 0)
// Not a lower bound.
continue;
} else if (atIneq(r, 0) >= 0) {
// Not an upper bound.
continue;
}
unsigned c, f;
for (c = 0, f = getNumCols() - 1; c < f; c++)
if (c != 0 && atIneq(r, c) != 0)
break;
if (c < getNumCols() - 1)
// Not a constant bound.
continue;
int64_t boundConst =
isLower ? mlir::ceilDiv(-atIneq(r, getNumCols() - 1), atIneq(r, 0))
: mlir::floorDiv(atIneq(r, getNumCols() - 1), -atIneq(r, 0));
if (isLower) {
if (minOrMaxConst == None || boundConst > minOrMaxConst)
minOrMaxConst = boundConst;
} else {
if (minOrMaxConst == None || boundConst < minOrMaxConst)
minOrMaxConst = boundConst;
}
}
return minOrMaxConst;
}
Optional<int64_t>
FlatAffineConstraints::getConstantLowerBound(unsigned pos) const {
FlatAffineConstraints tmpCst(*this);
return tmpCst.computeConstantLowerOrUpperBound</*isLower=*/true>(pos);
}
Optional<int64_t>
FlatAffineConstraints::getConstantUpperBound(unsigned pos) const {
FlatAffineConstraints tmpCst(*this);
return tmpCst.computeConstantLowerOrUpperBound</*isLower=*/false>(pos);
}
// A simple (naive and conservative) check for hyper-rectangularity.
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, trivially true constraints, and constraints
/// that can be detected as redundant as a result of differing only in their
/// constant term part. A constraint of the form <non-negative constant> >= 0 is
/// considered trivially true.
// Uses a DenseSet to hash and detect duplicates followed by a linear scan to
// remove duplicates in place.
void FlatAffineConstraints::removeTrivialRedundancy() {
GCDTightenInequalities();
normalizeConstraintsByGCD();
// A map used to detect redundancy stemming from constraints that only differ
// in their constant term. The value stored is <row position, const term>
// for a given row.
SmallDenseMap<ArrayRef<int64_t>, std::pair<unsigned, int64_t>>
rowsWithoutConstTerm;
// To unique rows.
SmallDenseSet<ArrayRef<int64_t>, 8> 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.
SmallVector<bool, 256> 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;
continue;
}
// Among constraints that only differ in the constant term part, mark
// everything other than the one with the smallest constant term redundant.
// (eg: among i - 16j - 5 >= 0, i - 16j - 1 >=0, i - 16j - 7 >= 0, the
// former two are redundant).
int64_t constTerm = atIneq(r, getNumCols() - 1);
auto rowWithoutConstTerm = ArrayRef<int64_t>(rowStart, getNumCols() - 1);
const auto &ret =
rowsWithoutConstTerm.insert({rowWithoutConstTerm, {r, constTerm}});
if (!ret.second) {
// Check if the other constraint has a higher constant term.
auto &val = ret.first->second;
if (val.second > constTerm) {
// The stored row is redundant. Mark it so, and update with this one.
redunIneq[val.first] = true;
val = {r, constTerm};
} else {
// The one stored makes this one redundant.
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: 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};
}
#undef DEBUG_TYPE
#define DEBUG_TYPE "fm"
/// 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: 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).
LogicalResult ret = gaussianEliminateId(pos);
(void)ret;
assert(succeeded(ret) && "Gaussian elimination guaranteed to succeed");
LLVM_DEBUG(llvm::dbgs() << "FM output (through Gaussian elimination):\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: 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.append(ids.begin(), ids.begin() + pos);
newIds.append(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: 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);
}
}
LLVM_DEBUG(llvm::dbgs() << "FM isResultIntegerExact: " << (lcmProducts == 1)
<< "\n");
if (lcmProducts == 1 && isResultIntegerExact)
*isResultIntegerExact = true;
// 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);
}
// GCD tightening and normalization allows detection of more trivially
// redundant constraints.
newFac.GCDTightenInequalities();
newFac.normalizeConstraintsByGCD();
newFac.removeTrivialRedundancy();
clearAndCopyFrom(newFac);
LLVM_DEBUG(llvm::dbgs() << "FM output:\n");
LLVM_DEBUG(dump());
}
#undef DEBUG_TYPE
#define DEBUG_TYPE "affine-structures"
void FlatAffineConstraints::projectOut(unsigned pos, unsigned num) {
if (num == 0)
return;
// 'pos' can be at most getNumCols() - 2 if num > 0.
assert((getNumCols() < 2 || 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;
while (currentPos < getNumIds()) {
unsigned curNumEliminated =
gaussianEliminateIds(currentPos, currentPos + numToEliminate);
++currentPos;
numToEliminate -= curNumEliminated + 1;
numGaussianEliminated += curNumEliminated;
}
// Eliminate the remaining using Fourier-Motzkin.
for (unsigned i = 0; i < num - numGaussianEliminated; i++) {
unsigned numToEliminate = num - numGaussianEliminated - i;
FourierMotzkinEliminate(
getBestIdToEliminate(*this, pos, pos + numToEliminate));
}
// 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);
}
void FlatAffineConstraints::clearConstraints() {
equalities.clear();
inequalities.clear();
}
namespace {
enum BoundCmpResult { Greater, Less, Equal, Unknown };
/// Compares two affine bounds whose coefficients are provided in 'first' and
/// 'second'. The last coefficient is the constant term.
static BoundCmpResult compareBounds(ArrayRef<int64_t> a, ArrayRef<int64_t> b) {
assert(a.size() == b.size());
// For the bounds to be comparable, their corresponding identifier
// coefficients should be equal; the constant terms are then compared to
// determine less/greater/equal.
if (!std::equal(a.begin(), a.end() - 1, b.begin()))
return Unknown;
if (a.back() == b.back())
return Equal;
return a.back() < b.back() ? Less : Greater;
}
} // namespace
// Returns constraints that are common to both A & B.
static void getCommonConstraints(const FlatAffineConstraints &A,
const FlatAffineConstraints &B,
FlatAffineConstraints &C) {
C.reset(A.getNumDimIds(), A.getNumSymbolIds(), A.getNumLocalIds());
// A naive O(n^2) check should be enough here given the input sizes.
for (unsigned r = 0, e = A.getNumInequalities(); r < e; ++r) {
for (unsigned s = 0, f = B.getNumInequalities(); s < f; ++s) {
if (A.getInequality(r) == B.getInequality(s)) {
C.addInequality(A.getInequality(r));
break;
}
}
}
for (unsigned r = 0, e = A.getNumEqualities(); r < e; ++r) {
for (unsigned s = 0, f = B.getNumEqualities(); s < f; ++s) {
if (A.getEquality(r) == B.getEquality(s)) {
C.addEquality(A.getEquality(r));
break;
}
}
}
}
// Computes the bounding box with respect to 'other' by finding the min of the
// lower bounds and the max of the upper bounds along each of the dimensions.
LogicalResult
FlatAffineConstraints::unionBoundingBox(const FlatAffineConstraints &otherCst) {
assert(otherCst.getNumDimIds() == numDims && "dims mismatch");
assert(otherCst.getIds()
.slice(0, getNumDimIds())
.equals(getIds().slice(0, getNumDimIds())) &&
"dim values mismatch");
assert(otherCst.getNumLocalIds() == 0 && "local ids not supported here");
assert(getNumLocalIds() == 0 && "local ids not supported yet here");
// Align `other` to this.
Optional<FlatAffineConstraints> otherCopy;
if (!areIdsAligned(*this, otherCst)) {
otherCopy.emplace(FlatAffineConstraints(otherCst));
mergeAndAlignIds(/*offset=*/numDims, this, &otherCopy.getValue());
}
const auto &otherAligned = otherCopy ? *otherCopy : otherCst;
// Get the constraints common to both systems; these will be added as is to
// the union.
FlatAffineConstraints commonCst;
getCommonConstraints(*this, otherAligned, commonCst);
std::vector<SmallVector<int64_t, 8>> boundingLbs;
std::vector<SmallVector<int64_t, 8>> boundingUbs;
boundingLbs.reserve(2 * getNumDimIds());
boundingUbs.reserve(2 * getNumDimIds());
// To hold lower and upper bounds for each dimension.
SmallVector<int64_t, 4> lb, otherLb, ub, otherUb;
// To compute min of lower bounds and max of upper bounds for each dimension.
SmallVector<int64_t, 4> minLb(getNumSymbolIds() + 1);
SmallVector<int64_t, 4> maxUb(getNumSymbolIds() + 1);
// To compute final new lower and upper bounds for the union.
SmallVector<int64_t, 8> newLb(getNumCols()), newUb(getNumCols());
int64_t lbFloorDivisor, otherLbFloorDivisor;
for (unsigned d = 0, e = getNumDimIds(); d < e; ++d) {
auto extent = getConstantBoundOnDimSize(d, &lb, &lbFloorDivisor, &ub);
if (!extent.hasValue())
// TODO: symbolic extents when necessary.
// TODO: handle union if a dimension is unbounded.
return failure();
auto otherExtent = otherAligned.getConstantBoundOnDimSize(
d, &otherLb, &otherLbFloorDivisor, &otherUb);
if (!otherExtent.hasValue() || lbFloorDivisor != otherLbFloorDivisor)
// TODO: symbolic extents when necessary.
return failure();
assert(lbFloorDivisor > 0 && "divisor always expected to be positive");
auto res = compareBounds(lb, otherLb);
// Identify min.
if (res == BoundCmpResult::Less || res == BoundCmpResult::Equal) {
minLb = lb;
// Since the divisor is for a floordiv, we need to convert to ceildiv,
// i.e., i >= expr floordiv div <=> i >= (expr - div + 1) ceildiv div <=>
// div * i >= expr - div + 1.
minLb.back() -= lbFloorDivisor - 1;
} else if (res == BoundCmpResult::Greater) {
minLb = otherLb;
minLb.back() -= otherLbFloorDivisor - 1;
} else {
// Uncomparable - check for constant lower/upper bounds.
auto constLb = getConstantLowerBound(d);
auto constOtherLb = otherAligned.getConstantLowerBound(d);
if (!constLb.hasValue() || !constOtherLb.hasValue())
return failure();
std::fill(minLb.begin(), minLb.end(), 0);
minLb.back() = std::min(constLb.getValue(), constOtherLb.getValue());
}
// Do the same for ub's but max of upper bounds. Identify max.
auto uRes = compareBounds(ub, otherUb);
if (uRes == BoundCmpResult::Greater || uRes == BoundCmpResult::Equal) {
maxUb = ub;
} else if (uRes == BoundCmpResult::Less) {
maxUb = otherUb;
} else {
// Uncomparable - check for constant lower/upper bounds.
auto constUb = getConstantUpperBound(d);
auto constOtherUb = otherAligned.getConstantUpperBound(d);
if (!constUb.hasValue() || !constOtherUb.hasValue())
return failure();
std::fill(maxUb.begin(), maxUb.end(), 0);
maxUb.back() = std::max(constUb.getValue(), constOtherUb.getValue());
}
std::fill(newLb.begin(), newLb.end(), 0);
std::fill(newUb.begin(), newUb.end(), 0);
// The divisor for lb, ub, otherLb, otherUb at this point is lbDivisor,
// and so it's the divisor for newLb and newUb as well.
newLb[d] = lbFloorDivisor;
newUb[d] = -lbFloorDivisor;
// Copy over the symbolic part + constant term.
std::copy(minLb.begin(), minLb.end(), newLb.begin() + getNumDimIds());
std::transform(newLb.begin() + getNumDimIds(), newLb.end(),
newLb.begin() + getNumDimIds(), std::negate<int64_t>());
std::copy(maxUb.begin(), maxUb.end(), newUb.begin() + getNumDimIds());
boundingLbs.push_back(newLb);
boundingUbs.push_back(newUb);
}
// Clear all constraints and add the lower/upper bounds for the bounding box.
clearConstraints();
for (unsigned d = 0, e = getNumDimIds(); d < e; ++d) {
addInequality(boundingLbs[d]);
addInequality(boundingUbs[d]);
}
// Add the constraints that were common to both systems.
append(commonCst);
removeTrivialRedundancy();
// TODO: copy over pure symbolic constraints from this and 'other' over to the
// union (since the above are just the union along dimensions); we shouldn't
// be discarding any other constraints on the symbols.
return success();
}
/// Compute an explicit representation for local vars. For all systems coming
/// from MLIR integer sets, maps, or expressions where local vars were
/// introduced to model floordivs and mods, this always succeeds.
static LogicalResult computeLocalVars(const FlatAffineConstraints &cst,
SmallVectorImpl<AffineExpr> &memo,
MLIRContext *context) {
unsigned numDims = cst.getNumDimIds();
unsigned numSyms = cst.getNumSymbolIds();
// Initialize dimensional and symbolic identifiers.
for (unsigned i = 0; i < numDims; i++)
memo[i] = getAffineDimExpr(i, context);
for (unsigned i = numDims, e = numDims + numSyms; i < e; i++)
memo[i] = getAffineSymbolExpr(i - numDims, context);
bool changed;
do {
// Each time `changed` is true at the end of this iteration, one or more
// local vars would have been detected as floordivs and set in memo; so the
// number of null entries in memo[...] strictly reduces; so this converges.
changed = false;
for (unsigned i = 0, e = cst.getNumLocalIds(); i < e; ++i)
if (!memo[numDims + numSyms + i] &&
detectAsFloorDiv(cst, /*pos=*/numDims + numSyms + i, context, memo))
changed = true;
} while (changed);
ArrayRef<AffineExpr> localExprs =
ArrayRef<AffineExpr>(memo).take_back(cst.getNumLocalIds());
return success(
llvm::all_of(localExprs, [](AffineExpr expr) { return expr; }));
}
void FlatAffineConstraints::getIneqAsAffineValueMap(
unsigned pos, unsigned ineqPos, AffineValueMap &vmap,
MLIRContext *context) const {
unsigned numDims = getNumDimIds();
unsigned numSyms = getNumSymbolIds();
assert(pos < numDims && "invalid position");
assert(ineqPos < getNumInequalities() && "invalid inequality position");
// Get expressions for local vars.
SmallVector<AffineExpr, 8> memo(getNumIds(), AffineExpr());
if (failed(computeLocalVars(*this, memo, context)))
assert(false &&
"one or more local exprs do not have an explicit representation");
auto localExprs = ArrayRef<AffineExpr>(memo).take_back(getNumLocalIds());
// Compute the AffineExpr lower/upper bound for this inequality.
ArrayRef<int64_t> inequality = getInequality(ineqPos);
SmallVector<int64_t, 8> bound;
bound.reserve(getNumCols() - 1);
// Everything other than the coefficient at `pos`.
bound.append(inequality.begin(), inequality.begin() + pos);
bound.append(inequality.begin() + pos + 1, inequality.end());
if (inequality[pos] > 0)
// Lower bound.
std::transform(bound.begin(), bound.end(), bound.begin(),
std::negate<int64_t>());
else
// Upper bound (which is exclusive).
bound.back() += 1;
// Convert to AffineExpr (tree) form.
auto boundExpr = getAffineExprFromFlatForm(bound, numDims - 1, numSyms,
localExprs, context);
// Get the values to bind to this affine expr (all dims and symbols).
SmallVector<Value, 4> operands;
getIdValues(0, pos, &operands);
SmallVector<Value, 4> trailingOperands;
getIdValues(pos + 1, getNumDimAndSymbolIds(), &trailingOperands);
operands.append(trailingOperands.begin(), trailingOperands.end());
vmap.reset(AffineMap::get(numDims - 1, numSyms, boundExpr), operands);
}
/// Returns true if the pos^th column is all zero for both inequalities and
/// equalities..
static bool isColZero(const FlatAffineConstraints &cst, unsigned pos) {
unsigned rowPos;
return !findConstraintWithNonZeroAt(cst, pos, /*isEq=*/false, &rowPos) &&
!findConstraintWithNonZeroAt(cst, pos, /*isEq=*/true, &rowPos);
}
IntegerSet FlatAffineConstraints::getAsIntegerSet(MLIRContext *context) const {
if (getNumConstraints() == 0)
// Return universal set (always true): 0 == 0.
return IntegerSet::get(getNumDimIds(), getNumSymbolIds(),
getAffineConstantExpr(/*constant=*/0, context),
/*eqFlags=*/true);
// Construct local references.
SmallVector<AffineExpr, 8> memo(getNumIds(), AffineExpr());
if (failed(computeLocalVars(*this, memo, context))) {
// Check if the local variables without an explicit representation have
// zero coefficients everywhere.
for (unsigned i = getNumDimAndSymbolIds(), e = getNumIds(); i < e; ++i) {
if (!memo[i] && !isColZero(*this, /*pos=*/i)) {
LLVM_DEBUG(llvm::dbgs() << "one or more local exprs do not have an "
"explicit representation");
return IntegerSet();
}
}
}
ArrayRef<AffineExpr> localExprs =
ArrayRef<AffineExpr>(memo).take_back(getNumLocalIds());
// Construct the IntegerSet from the equalities/inequalities.
unsigned numDims = getNumDimIds();
unsigned numSyms = getNumSymbolIds();
SmallVector<bool, 16> eqFlags(getNumConstraints());
std::fill(eqFlags.begin(), eqFlags.begin() + getNumEqualities(), true);
std::fill(eqFlags.begin() + getNumEqualities(), eqFlags.end(), false);
SmallVector<AffineExpr, 8> exprs;
exprs.reserve(getNumConstraints());
for (unsigned i = 0, e = getNumEqualities(); i < e; ++i)
exprs.push_back(getAffineExprFromFlatForm(getEquality(i), numDims, numSyms,
localExprs, context));
for (unsigned i = 0, e = getNumInequalities(); i < e; ++i)
exprs.push_back(getAffineExprFromFlatForm(getInequality(i), numDims,
numSyms, localExprs, context));
return IntegerSet::get(numDims, numSyms, exprs, eqFlags);
}
/// Find positions of inequalities and equalities that do not have a coefficient
/// for [pos, pos + num) identifiers.
static void getIndependentConstraints(const FlatAffineConstraints &cst,
unsigned pos, unsigned num,
SmallVectorImpl<unsigned> &nbIneqIndices,
SmallVectorImpl<unsigned> &nbEqIndices) {
assert(pos < cst.getNumIds() && "invalid start position");
assert(pos + num <= cst.getNumIds() && "invalid limit");
for (unsigned r = 0, e = cst.getNumInequalities(); r < e; r++) {
// The bounds are to be independent of [offset, offset + num) columns.
unsigned c;
for (c = pos; c < pos + num; ++c) {
if (cst.atIneq(r, c) != 0)
break;
}
if (c == pos + num)
nbIneqIndices.push_back(r);
}
for (unsigned r = 0, e = cst.getNumEqualities(); r < e; r++) {
// The bounds are to be independent of [offset, offset + num) columns.
unsigned c;
for (c = pos; c < pos + num; ++c) {
if (cst.atEq(r, c) != 0)
break;
}
if (c == pos + num)
nbEqIndices.push_back(r);
}
}
void FlatAffineConstraints::removeIndependentConstraints(unsigned pos,
unsigned num) {
assert(pos + num <= getNumIds() && "invalid range");
// Remove constraints that are independent of these identifiers.
SmallVector<unsigned, 4> nbIneqIndices, nbEqIndices;
getIndependentConstraints(*this, /*pos=*/0, num, nbIneqIndices, nbEqIndices);
// Iterate in reverse so that indices don't have to be updated.
// TODO: This method can be made more efficient (because removal of each
// inequality leads to much shifting/copying in the underlying buffer).
for (auto nbIndex : llvm::reverse(nbIneqIndices))
removeInequality(nbIndex);
for (auto nbIndex : llvm::reverse(nbEqIndices))
removeEquality(nbIndex);
}