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//===- ConstraintSytem.cpp - A system of linear constraints. ----*- C++ -*-===//
//
// 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
//
//===----------------------------------------------------------------------===//
#include "llvm/Analysis/ConstraintSystem.h"
#include "llvm/ADT/SmallVector.h"
#include "llvm/ADT/StringExtras.h"
#include "llvm/IR/Value.h"
#include "llvm/Support/Debug.h"
#include "llvm/Support/MathExtras.h"
#include <string>
using namespace llvm;
#define DEBUG_TYPE "constraint-system"
bool ConstraintSystem::eliminateUsingFM() {
// Implementation of Fourier–Motzkin elimination, with some tricks from the
// paper Pugh, William. "The Omega test: a fast and practical integer
// programming algorithm for dependence
// analysis."
// Supercomputing'91: Proceedings of the 1991 ACM/
// IEEE conference on Supercomputing. IEEE, 1991.
assert(!Constraints.empty() &&
"should only be called for non-empty constraint systems");
unsigned LastIdx = NumVariables - 1;
// First, either remove the variable in place if it is 0 or add the row to
// RemainingRows and remove it from the system.
SmallVector<SmallVector<Entry, 8>, 4> RemainingRows;
for (unsigned R1 = 0; R1 < Constraints.size();) {
SmallVector<Entry, 8> &Row1 = Constraints[R1];
if (getLastCoefficient(Row1, LastIdx) == 0) {
if (Row1.size() > 0 && Row1.back().Id == LastIdx)
Row1.pop_back();
R1++;
} else {
std::swap(Constraints[R1], Constraints.back());
RemainingRows.push_back(std::move(Constraints.back()));
Constraints.pop_back();
}
}
// Process rows where the variable is != 0.
unsigned NumRemainingConstraints = RemainingRows.size();
for (unsigned R1 = 0; R1 < NumRemainingConstraints; R1++) {
// FIXME do not use copy
for (unsigned R2 = R1 + 1; R2 < NumRemainingConstraints; R2++) {
// Examples of constraints stored as {Constant, Coeff_x, Coeff_y}
// R1: 0 >= 1 * x + (-2) * y => { 0, 1, -2 }
// R2: 3 >= 2 * x + 3 * y => { 3, 2, 3 }
// LastIdx = 2 (tracking coefficient of y)
// UpperLast: 3
// LowerLast: -2
int64_t UpperLast = getLastCoefficient(RemainingRows[R2], LastIdx);
int64_t LowerLast = getLastCoefficient(RemainingRows[R1], LastIdx);
assert(
UpperLast != 0 && LowerLast != 0 &&
"RemainingRows should only contain rows where the variable is != 0");
if ((LowerLast < 0 && UpperLast < 0) || (LowerLast > 0 && UpperLast > 0))
continue;
unsigned LowerR = R1;
unsigned UpperR = R2;
if (UpperLast < 0) {
std::swap(LowerR, UpperR);
std::swap(LowerLast, UpperLast);
}
SmallVector<Entry, 8> NR;
unsigned IdxUpper = 0;
unsigned IdxLower = 0;
auto &LowerRow = RemainingRows[LowerR];
auto &UpperRow = RemainingRows[UpperR];
// Update constant and coefficients of both constraints.
// Stops until every coefficient is updated or overflows.
while (true) {
if (IdxUpper >= UpperRow.size() || IdxLower >= LowerRow.size())
break;
int64_t M1, M2, N;
// Starts with index 0 and updates every coefficients.
int64_t UpperV = 0;
int64_t LowerV = 0;
uint16_t CurrentId = std::numeric_limits<uint16_t>::max();
if (IdxUpper < UpperRow.size()) {
CurrentId = std::min(UpperRow[IdxUpper].Id, CurrentId);
}
if (IdxLower < LowerRow.size()) {
CurrentId = std::min(LowerRow[IdxLower].Id, CurrentId);
}
if (IdxUpper < UpperRow.size() && UpperRow[IdxUpper].Id == CurrentId) {
UpperV = UpperRow[IdxUpper].Coefficient;
IdxUpper++;
}
if (MulOverflow(UpperV, -1 * LowerLast, M1))
return false;
if (IdxLower < LowerRow.size() && LowerRow[IdxLower].Id == CurrentId) {
LowerV = LowerRow[IdxLower].Coefficient;
IdxLower++;
}
if (MulOverflow(LowerV, UpperLast, M2))
return false;
// This algorithm is a variant of sparse Gaussian elimination.
//
// The new coefficient for CurrentId is
// N = UpperV * (-1) * LowerLast + LowerV * UpperLast
//
// UpperRow: { 3, 2, 3 }, LowerLast: -2
// LowerRow: { 0, 1, -2 }, UpperLast: 3
//
// After multiplication:
// UpperRow: { 6, 4, 6 }
// LowerRow: { 0, 3, -6 }
//
// Eliminates y after addition:
// N: { 6, 7, 0 } => 6 >= 7 * x
if (AddOverflow(M1, M2, N))
return false;
// Skip variable that is completely eliminated.
if (N == 0)
continue;
NR.emplace_back(N, CurrentId);
}
if (NR.empty())
continue;
Constraints.push_back(std::move(NR));
// Give up if the new system gets too big.
if (Constraints.size() > 500)
return false;
}
}
NumVariables -= 1;
return true;
}
bool ConstraintSystem::mayHaveSolutionImpl() {
while (!Constraints.empty() && NumVariables > 1) {
if (!eliminateUsingFM())
return true;
}
if (Constraints.empty() || NumVariables > 1)
return true;
return all_of(Constraints, [](auto &R) {
if (R.empty())
return true;
if (R[0].Id == 0)
return R[0].Coefficient >= 0;
return true;
});
}
SmallVector<std::string> ConstraintSystem::getVarNamesList() const {
SmallVector<std::string> Names(Value2Index.size(), "");
#ifndef NDEBUG
for (auto &[V, Index] : Value2Index) {
std::string OperandName;
if (V->getName().empty())
OperandName = V->getNameOrAsOperand();
else
OperandName = std::string("%") + V->getName().str();
Names[Index - 1] = OperandName;
}
#endif
return Names;
}
void ConstraintSystem::dump() const {
#ifndef NDEBUG
if (Constraints.empty())
return;
SmallVector<std::string> Names = getVarNamesList();
for (const auto &Row : Constraints) {
SmallVector<std::string, 16> Parts;
for (const Entry &E : Row) {
if (E.Id >= NumVariables)
break;
if (E.Id == 0)
continue;
std::string Coefficient;
if (E.Coefficient != 1)
Coefficient = std::to_string(E.Coefficient) + " * ";
Parts.push_back(Coefficient + Names[E.Id - 1]);
}
// assert(!Parts.empty() && "need to have at least some parts");
int64_t ConstPart = 0;
if (Row[0].Id == 0)
ConstPart = Row[0].Coefficient;
LLVM_DEBUG(dbgs() << join(Parts, std::string(" + "))
<< " <= " << std::to_string(ConstPart) << "\n");
}
#endif
}
bool ConstraintSystem::mayHaveSolution() {
LLVM_DEBUG(dbgs() << "---\n");
LLVM_DEBUG(dump());
bool HasSolution = mayHaveSolutionImpl();
LLVM_DEBUG(dbgs() << (HasSolution ? "sat" : "unsat") << "\n");
return HasSolution;
}
bool ConstraintSystem::isConditionImplied(SmallVector<int64_t, 8> R) const {
// If all variable coefficients are 0, we have 'C >= 0'. If the constant is >=
// 0, R is always true, regardless of the system.
if (all_of(ArrayRef(R).drop_front(1), [](int64_t C) { return C == 0; }))
return R[0] >= 0;
// If there is no solution with the negation of R added to the system, the
// condition must hold based on the existing constraints.
R = ConstraintSystem::negate(R);
if (R.empty())
return false;
auto NewSystem = *this;
NewSystem.addVariableRow(R);
return !NewSystem.mayHaveSolution();
}
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