MLIR  19.0.0git
Simplex.cpp
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1 //===- Simplex.cpp - MLIR Simplex Class -----------------------------------===//
2 //
3 // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
4 // See https://llvm.org/LICENSE.txt for license information.
5 // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
6 //
7 //===----------------------------------------------------------------------===//
8 
16 #include "mlir/Support/LLVM.h"
18 #include "llvm/ADT/STLExtras.h"
19 #include "llvm/ADT/SmallBitVector.h"
20 #include "llvm/ADT/SmallVector.h"
21 #include "llvm/Support/Compiler.h"
22 #include "llvm/Support/ErrorHandling.h"
23 #include "llvm/Support/raw_ostream.h"
24 #include <cassert>
25 #include <functional>
26 #include <limits>
27 #include <optional>
28 #include <tuple>
29 #include <utility>
30 
31 using namespace mlir;
32 using namespace presburger;
33 
35 
37 
38 // Return a + scale*b;
39 LLVM_ATTRIBUTE_UNUSED
42  assert(a.size() == b.size());
44  res.reserve(a.size());
45  for (unsigned i = 0, e = a.size(); i < e; ++i)
46  res.push_back(a[i] + scale * b[i]);
47  return res;
48 }
49 
50 SimplexBase::SimplexBase(unsigned nVar, bool mustUseBigM)
51  : usingBigM(mustUseBigM), nRedundant(0), nSymbol(0),
52  tableau(0, getNumFixedCols() + nVar), empty(false) {
53  colUnknown.insert(colUnknown.begin(), getNumFixedCols(), nullIndex);
54  for (unsigned i = 0; i < nVar; ++i) {
55  var.emplace_back(Orientation::Column, /*restricted=*/false,
56  /*pos=*/getNumFixedCols() + i);
57  colUnknown.push_back(i);
58  }
59 }
60 
61 SimplexBase::SimplexBase(unsigned nVar, bool mustUseBigM,
62  const llvm::SmallBitVector &isSymbol)
63  : SimplexBase(nVar, mustUseBigM) {
64  assert(isSymbol.size() == nVar && "invalid bitmask!");
65  // Invariant: nSymbol is the number of symbols that have been marked
66  // already and these occupy the columns
67  // [getNumFixedCols(), getNumFixedCols() + nSymbol).
68  for (unsigned symbolIdx : isSymbol.set_bits()) {
69  var[symbolIdx].isSymbol = true;
70  swapColumns(var[symbolIdx].pos, getNumFixedCols() + nSymbol);
71  ++nSymbol;
72  }
73 }
74 
76  assert(index != nullIndex && "nullIndex passed to unknownFromIndex");
77  return index >= 0 ? var[index] : con[~index];
78 }
79 
81  assert(col < getNumColumns() && "Invalid column");
82  return unknownFromIndex(colUnknown[col]);
83 }
84 
85 const Simplex::Unknown &SimplexBase::unknownFromRow(unsigned row) const {
86  assert(row < getNumRows() && "Invalid row");
87  return unknownFromIndex(rowUnknown[row]);
88 }
89 
91  assert(index != nullIndex && "nullIndex passed to unknownFromIndex");
92  return index >= 0 ? var[index] : con[~index];
93 }
94 
96  assert(col < getNumColumns() && "Invalid column");
97  return unknownFromIndex(colUnknown[col]);
98 }
99 
101  assert(row < getNumRows() && "Invalid row");
102  return unknownFromIndex(rowUnknown[row]);
103 }
104 
105 unsigned SimplexBase::addZeroRow(bool makeRestricted) {
106  // Resize the tableau to accommodate the extra row.
107  unsigned newRow = tableau.appendExtraRow();
108  assert(getNumRows() == getNumRows() && "Inconsistent tableau size");
109  rowUnknown.push_back(~con.size());
110  con.emplace_back(Orientation::Row, makeRestricted, newRow);
112  tableau(newRow, 0) = 1;
113  return newRow;
114 }
115 
116 /// Add a new row to the tableau corresponding to the given constant term and
117 /// list of coefficients. The coefficients are specified as a vector of
118 /// (variable index, coefficient) pairs.
119 unsigned SimplexBase::addRow(ArrayRef<MPInt> coeffs, bool makeRestricted) {
120  assert(coeffs.size() == var.size() + 1 &&
121  "Incorrect number of coefficients!");
122  assert(var.size() + getNumFixedCols() == getNumColumns() &&
123  "inconsistent column count!");
124 
125  unsigned newRow = addZeroRow(makeRestricted);
126  tableau(newRow, 1) = coeffs.back();
127  if (usingBigM) {
128  // When the lexicographic pivot rule is used, instead of the variables
129  //
130  // x, y, z ...
131  //
132  // we internally use the variables
133  //
134  // M, M + x, M + y, M + z, ...
135  //
136  // where M is the big M parameter. As such, when the user tries to add
137  // a row ax + by + cz + d, we express it in terms of our internal variables
138  // as -(a + b + c)M + a(M + x) + b(M + y) + c(M + z) + d.
139  //
140  // Symbols don't use the big M parameter since they do not get lex
141  // optimized.
142  MPInt bigMCoeff(0);
143  for (unsigned i = 0; i < coeffs.size() - 1; ++i)
144  if (!var[i].isSymbol)
145  bigMCoeff -= coeffs[i];
146  // The coefficient to the big M parameter is stored in column 2.
147  tableau(newRow, 2) = bigMCoeff;
148  }
149 
150  // Process each given variable coefficient.
151  for (unsigned i = 0; i < var.size(); ++i) {
152  unsigned pos = var[i].pos;
153  if (coeffs[i] == 0)
154  continue;
155 
156  if (var[i].orientation == Orientation::Column) {
157  // If a variable is in column position at column col, then we just add the
158  // coefficient for that variable (scaled by the common row denominator) to
159  // the corresponding entry in the new row.
160  tableau(newRow, pos) += coeffs[i] * tableau(newRow, 0);
161  continue;
162  }
163 
164  // If the variable is in row position, we need to add that row to the new
165  // row, scaled by the coefficient for the variable, accounting for the two
166  // rows potentially having different denominators. The new denominator is
167  // the lcm of the two.
168  MPInt lcm = presburger::lcm(tableau(newRow, 0), tableau(pos, 0));
169  MPInt nRowCoeff = lcm / tableau(newRow, 0);
170  MPInt idxRowCoeff = coeffs[i] * (lcm / tableau(pos, 0));
171  tableau(newRow, 0) = lcm;
172  for (unsigned col = 1, e = getNumColumns(); col < e; ++col)
173  tableau(newRow, col) =
174  nRowCoeff * tableau(newRow, col) + idxRowCoeff * tableau(pos, col);
175  }
176 
177  tableau.normalizeRow(newRow);
178  // Push to undo log along with the index of the new constraint.
179  return con.size() - 1;
180 }
181 
182 namespace {
183 bool signMatchesDirection(const MPInt &elem, Direction direction) {
184  assert(elem != 0 && "elem should not be 0");
185  return direction == Direction::Up ? elem > 0 : elem < 0;
186 }
187 
188 Direction flippedDirection(Direction direction) {
189  return direction == Direction::Up ? Direction::Down : Simplex::Direction::Up;
190 }
191 } // namespace
192 
193 /// We simply make the tableau consistent while maintaining a lexicopositive
194 /// basis transform, and then return the sample value. If the tableau becomes
195 /// empty, we return empty.
196 ///
197 /// Let the variables be x = (x_1, ... x_n).
198 /// Let the basis unknowns be y = (y_1, ... y_n).
199 /// We have that x = A*y + b for some n x n matrix A and n x 1 column vector b.
200 ///
201 /// As we will show below, A*y is either zero or lexicopositive.
202 /// Adding a lexicopositive vector to b will make it lexicographically
203 /// greater, so A*y + b is always equal to or lexicographically greater than b.
204 /// Thus, since we can attain x = b, that is the lexicographic minimum.
205 ///
206 /// We have that every column in A is lexicopositive, i.e., has at least
207 /// one non-zero element, with the first such element being positive. Since for
208 /// the tableau to be consistent we must have non-negative sample values not
209 /// only for the constraints but also for the variables, we also have x >= 0 and
210 /// y >= 0, by which we mean every element in these vectors is non-negative.
211 ///
212 /// Proof that if every column in A is lexicopositive, and y >= 0, then
213 /// A*y is zero or lexicopositive. Begin by considering A_1, the first row of A.
214 /// If this row is all zeros, then (A*y)_1 = (A_1)*y = 0; proceed to the next
215 /// row. If we run out of rows, A*y is zero and we are done; otherwise, we
216 /// encounter some row A_i that has a non-zero element. Every column is
217 /// lexicopositive and so has some positive element before any negative elements
218 /// occur, so the element in this row for any column, if non-zero, must be
219 /// positive. Consider (A*y)_i = (A_i)*y. All the elements in both vectors are
220 /// non-negative, so if this is non-zero then it must be positive. Then the
221 /// first non-zero element of A*y is positive so A*y is lexicopositive.
222 ///
223 /// Otherwise, if (A_i)*y is zero, then for every column j that had a non-zero
224 /// element in A_i, y_j is zero. Thus these columns have no contribution to A*y
225 /// and we can completely ignore these columns of A. We now continue downwards,
226 /// looking for rows of A that have a non-zero element other than in the ignored
227 /// columns. If we find one, say A_k, once again these elements must be positive
228 /// since they are the first non-zero element in each of these columns, so if
229 /// (A_k)*y is not zero then we have that A*y is lexicopositive and if not we
230 /// add these to the set of ignored columns and continue to the next row. If we
231 /// run out of rows, then A*y is zero and we are done.
233  if (restoreRationalConsistency().failed()) {
234  markEmpty();
235  return OptimumKind::Empty;
236  }
237  return getRationalSample();
238 }
239 
240 /// Given a row that has a non-integer sample value, add an inequality such
241 /// that this fractional sample value is cut away from the polytope. The added
242 /// inequality will be such that no integer points are removed. i.e., the
243 /// integer lexmin, if it exists, is the same with and without this constraint.
244 ///
245 /// Let the row be
246 /// (c + coeffM*M + a_1*s_1 + ... + a_m*s_m + b_1*y_1 + ... + b_n*y_n)/d,
247 /// where s_1, ... s_m are the symbols and
248 /// y_1, ... y_n are the other basis unknowns.
249 ///
250 /// For this to be an integer, we want
251 /// coeffM*M + a_1*s_1 + ... + a_m*s_m + b_1*y_1 + ... + b_n*y_n = -c (mod d)
252 /// Note that this constraint must always hold, independent of the basis,
253 /// becuse the row unknown's value always equals this expression, even if *we*
254 /// later compute the sample value from a different expression based on a
255 /// different basis.
256 ///
257 /// Let us assume that M has a factor of d in it. Imposing this constraint on M
258 /// does not in any way hinder us from finding a value of M that is big enough.
259 /// Moreover, this function is only called when the symbolic part of the sample,
260 /// a_1*s_1 + ... + a_m*s_m, is known to be an integer.
261 ///
262 /// Also, we can safely reduce the coefficients modulo d, so we have:
263 ///
264 /// (b_1%d)y_1 + ... + (b_n%d)y_n = (-c%d) + k*d for some integer `k`
265 ///
266 /// Note that all coefficient modulos here are non-negative. Also, all the
267 /// unknowns are non-negative here as both constraints and variables are
268 /// non-negative in LexSimplexBase. (We used the big M trick to make the
269 /// variables non-negative). Therefore, the LHS here is non-negative.
270 /// Since 0 <= (-c%d) < d, k is the quotient of dividing the LHS by d and
271 /// is therefore non-negative as well.
272 ///
273 /// So we have
274 /// ((b_1%d)y_1 + ... + (b_n%d)y_n - (-c%d))/d >= 0.
275 ///
276 /// The constraint is violated when added (it would be useless otherwise)
277 /// so we immediately try to move it to a column.
279  MPInt d = tableau(row, 0);
280  unsigned cutRow = addZeroRow(/*makeRestricted=*/true);
281  tableau(cutRow, 0) = d;
282  tableau(cutRow, 1) = -mod(-tableau(row, 1), d); // -c%d.
283  tableau(cutRow, 2) = 0;
284  for (unsigned col = 3 + nSymbol, e = getNumColumns(); col < e; ++col)
285  tableau(cutRow, col) = mod(tableau(row, col), d); // b_i%d.
286  return moveRowUnknownToColumn(cutRow);
287 }
288 
289 std::optional<unsigned> LexSimplex::maybeGetNonIntegralVarRow() const {
290  for (const Unknown &u : var) {
291  if (u.orientation == Orientation::Column)
292  continue;
293  // If the sample value is of the form (a/d)M + b/d, we need b to be
294  // divisible by d. We assume M contains all possible
295  // factors and is divisible by everything.
296  unsigned row = u.pos;
297  if (tableau(row, 1) % tableau(row, 0) != 0)
298  return row;
299  }
300  return {};
301 }
302 
304  // We first try to make the tableau consistent.
305  if (restoreRationalConsistency().failed())
306  return OptimumKind::Empty;
307 
308  // Then, if the sample value is integral, we are done.
309  while (std::optional<unsigned> maybeRow = maybeGetNonIntegralVarRow()) {
310  // Otherwise, for the variable whose row has a non-integral sample value,
311  // we add a cut, a constraint that remove this rational point
312  // while preserving all integer points, thus keeping the lexmin the same.
313  // We then again try to make the tableau with the new constraint
314  // consistent. This continues until the tableau becomes empty, in which
315  // case there is no integer point, or until there are no variables with
316  // non-integral sample values.
317  //
318  // Failure indicates that the tableau became empty, which occurs when the
319  // polytope is integer empty.
320  if (addCut(*maybeRow).failed())
321  return OptimumKind::Empty;
322  if (restoreRationalConsistency().failed())
323  return OptimumKind::Empty;
324  }
325 
326  MaybeOptimum<SmallVector<Fraction, 8>> sample = getRationalSample();
327  assert(!sample.isEmpty() && "If we reached here the sample should exist!");
328  if (sample.isUnbounded())
329  return OptimumKind::Unbounded;
330  return llvm::to_vector<8>(
331  llvm::map_range(*sample, std::mem_fn(&Fraction::getAsInteger)));
332 }
333 
335  SimplexRollbackScopeExit scopeExit(*this);
336  addInequality(coeffs);
337  return findIntegerLexMin().isEmpty();
338 }
339 
341  return isSeparateInequality(getComplementIneq(coeffs));
342 }
343 
345 SymbolicLexSimplex::getSymbolicSampleNumerator(unsigned row) const {
346  SmallVector<MPInt, 8> sample;
347  sample.reserve(nSymbol + 1);
348  for (unsigned col = 3; col < 3 + nSymbol; ++col)
349  sample.push_back(tableau(row, col));
350  sample.push_back(tableau(row, 1));
351  return sample;
352 }
353 
355 SymbolicLexSimplex::getSymbolicSampleIneq(unsigned row) const {
356  SmallVector<MPInt, 8> sample = getSymbolicSampleNumerator(row);
357  // The inequality is equivalent to the GCD-normalized one.
358  normalizeRange(sample);
359  return sample;
360 }
361 
363  appendVariable();
364  swapColumns(3 + nSymbol, getNumColumns() - 1);
365  var.back().isSymbol = true;
366  nSymbol++;
367 }
368 
369 static bool isRangeDivisibleBy(ArrayRef<MPInt> range, const MPInt &divisor) {
370  assert(divisor > 0 && "divisor must be positive!");
371  return llvm::all_of(range,
372  [divisor](const MPInt &x) { return x % divisor == 0; });
373 }
374 
375 bool SymbolicLexSimplex::isSymbolicSampleIntegral(unsigned row) const {
376  MPInt denom = tableau(row, 0);
377  return tableau(row, 1) % denom == 0 &&
378  isRangeDivisibleBy(tableau.getRow(row).slice(3, nSymbol), denom);
379 }
380 
381 /// This proceeds similarly to LexSimplexBase::addCut(). We are given a row that
382 /// has a symbolic sample value with fractional coefficients.
383 ///
384 /// Let the row be
385 /// (c + coeffM*M + sum_i a_i*s_i + sum_j b_j*y_j)/d,
386 /// where s_1, ... s_m are the symbols and
387 /// y_1, ... y_n are the other basis unknowns.
388 ///
389 /// As in LexSimplex::addCut, for this to be an integer, we want
390 ///
391 /// coeffM*M + sum_j b_j*y_j = -c + sum_i (-a_i*s_i) (mod d)
392 ///
393 /// This time, a_1*s_1 + ... + a_m*s_m may not be an integer. We find that
394 ///
395 /// sum_i (b_i%d)y_i = ((-c%d) + sum_i (-a_i%d)s_i)%d + k*d for some integer k
396 ///
397 /// where we take a modulo of the whole symbolic expression on the right to
398 /// bring it into the range [0, d - 1]. Therefore, as in addCut(),
399 /// k is the quotient on dividing the LHS by d, and since LHS >= 0, we have
400 /// k >= 0 as well. If all the a_i are divisible by d, then we can add the
401 /// constraint directly. Otherwise, we realize the modulo of the symbolic
402 /// expression by adding a division variable
403 ///
404 /// q = ((-c%d) + sum_i (-a_i%d)s_i)/d
405 ///
406 /// to the symbol domain, so the equality becomes
407 ///
408 /// sum_i (b_i%d)y_i = (-c%d) + sum_i (-a_i%d)s_i - q*d + k*d for some integer k
409 ///
410 /// So the cut is
411 /// (sum_i (b_i%d)y_i - (-c%d) - sum_i (-a_i%d)s_i + q*d)/d >= 0
412 /// This constraint is violated when added so we immediately try to move it to a
413 /// column.
414 LogicalResult SymbolicLexSimplex::addSymbolicCut(unsigned row) {
415  MPInt d = tableau(row, 0);
416  if (isRangeDivisibleBy(tableau.getRow(row).slice(3, nSymbol), d)) {
417  // The coefficients of symbols in the symbol numerator are divisible
418  // by the denominator, so we can add the constraint directly,
419  // i.e., ignore the symbols and add a regular cut as in addCut().
420  return addCut(row);
421  }
422 
423  // Construct the division variable `q = ((-c%d) + sum_i (-a_i%d)s_i)/d`.
424  SmallVector<MPInt, 8> divCoeffs;
425  divCoeffs.reserve(nSymbol + 1);
426  MPInt divDenom = d;
427  for (unsigned col = 3; col < 3 + nSymbol; ++col)
428  divCoeffs.push_back(mod(-tableau(row, col), divDenom)); // (-a_i%d)s_i
429  divCoeffs.push_back(mod(-tableau(row, 1), divDenom)); // -c%d.
430  normalizeDiv(divCoeffs, divDenom);
431 
432  domainSimplex.addDivisionVariable(divCoeffs, divDenom);
433  domainPoly.addLocalFloorDiv(divCoeffs, divDenom);
434 
435  // Update `this` to account for the additional symbol we just added.
436  appendSymbol();
437 
438  // Add the cut (sum_i (b_i%d)y_i - (-c%d) + sum_i -(-a_i%d)s_i + q*d)/d >= 0.
439  unsigned cutRow = addZeroRow(/*makeRestricted=*/true);
440  tableau(cutRow, 0) = d;
441  tableau(cutRow, 2) = 0;
442 
443  tableau(cutRow, 1) = -mod(-tableau(row, 1), d); // -(-c%d).
444  for (unsigned col = 3; col < 3 + nSymbol - 1; ++col)
445  tableau(cutRow, col) = -mod(-tableau(row, col), d); // -(-a_i%d)s_i.
446  tableau(cutRow, 3 + nSymbol - 1) = d; // q*d.
447 
448  for (unsigned col = 3 + nSymbol, e = getNumColumns(); col < e; ++col)
449  tableau(cutRow, col) = mod(tableau(row, col), d); // (b_i%d)y_i.
450  return moveRowUnknownToColumn(cutRow);
451 }
452 
453 void SymbolicLexSimplex::recordOutput(SymbolicLexOpt &result) const {
454  IntMatrix output(0, domainPoly.getNumVars() + 1);
455  output.reserveRows(result.lexopt.getNumOutputs());
456  for (const Unknown &u : var) {
457  if (u.isSymbol)
458  continue;
459 
460  if (u.orientation == Orientation::Column) {
461  // M + u has a sample value of zero so u has a sample value of -M, i.e,
462  // unbounded.
463  result.unboundedDomain.unionInPlace(domainPoly);
464  return;
465  }
466 
467  MPInt denom = tableau(u.pos, 0);
468  if (tableau(u.pos, 2) < denom) {
469  // M + u has a sample value of fM + something, where f < 1, so
470  // u = (f - 1)M + something, which has a negative coefficient for M,
471  // and so is unbounded.
472  result.unboundedDomain.unionInPlace(domainPoly);
473  return;
474  }
475  assert(tableau(u.pos, 2) == denom &&
476  "Coefficient of M should not be greater than 1!");
477 
478  SmallVector<MPInt, 8> sample = getSymbolicSampleNumerator(u.pos);
479  for (MPInt &elem : sample) {
480  assert(elem % denom == 0 && "coefficients must be integral!");
481  elem /= denom;
482  }
483  output.appendExtraRow(sample);
484  }
485 
486  // Store the output in a MultiAffineFunction and add it the result.
487  PresburgerSpace funcSpace = result.lexopt.getSpace();
488  funcSpace.insertVar(VarKind::Local, 0, domainPoly.getNumLocalVars());
489 
490  result.lexopt.addPiece(
491  {PresburgerSet(domainPoly),
492  MultiAffineFunction(funcSpace, output, domainPoly.getLocalReprs())});
493 }
494 
495 std::optional<unsigned> SymbolicLexSimplex::maybeGetAlwaysViolatedRow() {
496  // First look for rows that are clearly violated just from the big M
497  // coefficient, without needing to perform any simplex queries on the domain.
498  for (unsigned row = 0, e = getNumRows(); row < e; ++row)
499  if (tableau(row, 2) < 0)
500  return row;
501 
502  for (unsigned row = 0, e = getNumRows(); row < e; ++row) {
503  if (tableau(row, 2) > 0)
504  continue;
505  if (domainSimplex.isSeparateInequality(getSymbolicSampleIneq(row))) {
506  // Sample numerator always takes negative values in the symbol domain.
507  return row;
508  }
509  }
510  return {};
511 }
512 
513 std::optional<unsigned> SymbolicLexSimplex::maybeGetNonIntegralVarRow() {
514  for (const Unknown &u : var) {
515  if (u.orientation == Orientation::Column)
516  continue;
517  assert(!u.isSymbol && "Symbol should not be in row orientation!");
518  if (!isSymbolicSampleIntegral(u.pos))
519  return u.pos;
520  }
521  return {};
522 }
523 
524 /// The non-branching pivots are just the ones moving the rows
525 /// that are always violated in the symbol domain.
526 LogicalResult SymbolicLexSimplex::doNonBranchingPivots() {
527  while (std::optional<unsigned> row = maybeGetAlwaysViolatedRow())
528  if (moveRowUnknownToColumn(*row).failed())
529  return failure();
530  return success();
531 }
532 
535  /*numDomain=*/domainPoly.getNumDimVars(),
536  /*numRange=*/var.size() - nSymbol,
537  /*numSymbols=*/domainPoly.getNumSymbolVars()));
538 
539  /// The algorithm is more naturally expressed recursively, but we implement
540  /// it iteratively here to avoid potential issues with stack overflows in the
541  /// compiler. We explicitly maintain the stack frames in a vector.
542  ///
543  /// To "recurse", we store the current "stack frame", i.e., state variables
544  /// that we will need when we "return", into `stack`, increment `level`, and
545  /// `continue`. To "tail recurse", we just `continue`.
546  /// To "return", we decrement `level` and `continue`.
547  ///
548  /// When there is no stack frame for the current `level`, this indicates that
549  /// we have just "recursed" or "tail recursed". When there does exist one,
550  /// this indicates that we have just "returned" from recursing. There is only
551  /// one point at which non-tail calls occur so we always "return" there.
552  unsigned level = 1;
553  struct StackFrame {
554  int splitIndex;
555  unsigned snapshot;
556  unsigned domainSnapshot;
557  IntegerRelation::CountsSnapshot domainPolyCounts;
558  };
560 
561  while (level > 0) {
562  assert(level >= stack.size());
563  if (level > stack.size()) {
564  if (empty || domainSimplex.findIntegerLexMin().isEmpty()) {
565  // No integer points; return.
566  --level;
567  continue;
568  }
569 
570  if (doNonBranchingPivots().failed()) {
571  // Could not find pivots for violated constraints; return.
572  --level;
573  continue;
574  }
575 
576  SmallVector<MPInt, 8> symbolicSample;
577  unsigned splitRow = 0;
578  for (unsigned e = getNumRows(); splitRow < e; ++splitRow) {
579  if (tableau(splitRow, 2) > 0)
580  continue;
581  assert(tableau(splitRow, 2) == 0 &&
582  "Non-branching pivots should have been handled already!");
583 
584  symbolicSample = getSymbolicSampleIneq(splitRow);
585  if (domainSimplex.isRedundantInequality(symbolicSample))
586  continue;
587 
588  // It's neither redundant nor separate, so it takes both positive and
589  // negative values, and hence constitutes a row for which we need to
590  // split the domain and separately run each case.
591  assert(!domainSimplex.isSeparateInequality(symbolicSample) &&
592  "Non-branching pivots should have been handled already!");
593  break;
594  }
595 
596  if (splitRow < getNumRows()) {
597  unsigned domainSnapshot = domainSimplex.getSnapshot();
598  IntegerRelation::CountsSnapshot domainPolyCounts =
599  domainPoly.getCounts();
600 
601  // First, we consider the part of the domain where the row is not
602  // violated. We don't have to do any pivots for the row in this case,
603  // but we record the additional constraint that defines this part of
604  // the domain.
605  domainSimplex.addInequality(symbolicSample);
606  domainPoly.addInequality(symbolicSample);
607 
608  // Recurse.
609  //
610  // On return, the basis as a set is preserved but not the internal
611  // ordering within rows or columns. Thus, we take note of the index of
612  // the Unknown that caused the split, which may be in a different
613  // row when we come back from recursing. We will need this to recurse
614  // on the other part of the split domain, where the row is violated.
615  //
616  // Note that we have to capture the index above and not a reference to
617  // the Unknown itself, since the array it lives in might get
618  // reallocated.
619  int splitIndex = rowUnknown[splitRow];
620  unsigned snapshot = getSnapshot();
621  stack.push_back(
622  {splitIndex, snapshot, domainSnapshot, domainPolyCounts});
623  ++level;
624  continue;
625  }
626 
627  // The tableau is rationally consistent for the current domain.
628  // Now we look for non-integral sample values and add cuts for them.
629  if (std::optional<unsigned> row = maybeGetNonIntegralVarRow()) {
630  if (addSymbolicCut(*row).failed()) {
631  // No integral points; return.
632  --level;
633  continue;
634  }
635 
636  // Rerun this level with the added cut constraint (tail recurse).
637  continue;
638  }
639 
640  // Record output and return.
641  recordOutput(result);
642  --level;
643  continue;
644  }
645 
646  if (level == stack.size()) {
647  // We have "returned" from "recursing".
648  const StackFrame &frame = stack.back();
649  domainPoly.truncate(frame.domainPolyCounts);
650  domainSimplex.rollback(frame.domainSnapshot);
651  rollback(frame.snapshot);
652  const Unknown &u = unknownFromIndex(frame.splitIndex);
653 
654  // Drop the frame. We don't need it anymore.
655  stack.pop_back();
656 
657  // Now we consider the part of the domain where the unknown `splitIndex`
658  // was negative.
659  assert(u.orientation == Orientation::Row &&
660  "The split row should have been returned to row orientation!");
661  SmallVector<MPInt, 8> splitIneq =
662  getComplementIneq(getSymbolicSampleIneq(u.pos));
663  normalizeRange(splitIneq);
664  if (moveRowUnknownToColumn(u.pos).failed()) {
665  // The unknown can't be made non-negative; return.
666  --level;
667  continue;
668  }
669 
670  // The unknown can be made negative; recurse with the corresponding domain
671  // constraints.
672  domainSimplex.addInequality(splitIneq);
673  domainPoly.addInequality(splitIneq);
674 
675  // We are now taking care of the second half of the domain and we don't
676  // need to do anything else here after returning, so it's a tail recurse.
677  continue;
678  }
679  }
680 
681  return result;
682 }
683 
684 bool LexSimplex::rowIsViolated(unsigned row) const {
685  if (tableau(row, 2) < 0)
686  return true;
687  if (tableau(row, 2) == 0 && tableau(row, 1) < 0)
688  return true;
689  return false;
690 }
691 
692 std::optional<unsigned> LexSimplex::maybeGetViolatedRow() const {
693  for (unsigned row = 0, e = getNumRows(); row < e; ++row)
694  if (rowIsViolated(row))
695  return row;
696  return {};
697 }
698 
699 /// We simply look for violated rows and keep trying to move them to column
700 /// orientation, which always succeeds unless the constraints have no solution
701 /// in which case we just give up and return.
702 LogicalResult LexSimplex::restoreRationalConsistency() {
703  if (empty)
704  return failure();
705  while (std::optional<unsigned> maybeViolatedRow = maybeGetViolatedRow())
706  if (moveRowUnknownToColumn(*maybeViolatedRow).failed())
707  return failure();
708  return success();
709 }
710 
711 // Move the row unknown to column orientation while preserving lexicopositivity
712 // of the basis transform. The sample value of the row must be non-positive.
713 //
714 // We only consider pivots where the pivot element is positive. Suppose no such
715 // pivot exists, i.e., some violated row has no positive coefficient for any
716 // basis unknown. The row can be represented as (s + c_1*u_1 + ... + c_n*u_n)/d,
717 // where d is the denominator, s is the sample value and the c_i are the basis
718 // coefficients. If s != 0, then since any feasible assignment of the basis
719 // satisfies u_i >= 0 for all i, and we have s < 0 as well as c_i < 0 for all i,
720 // any feasible assignment would violate this row and therefore the constraints
721 // have no solution.
722 //
723 // We can preserve lexicopositivity by picking the pivot column with positive
724 // pivot element that makes the lexicographically smallest change to the sample
725 // point.
726 //
727 // Proof. Let
728 // x = (x_1, ... x_n) be the variables,
729 // z = (z_1, ... z_m) be the constraints,
730 // y = (y_1, ... y_n) be the current basis, and
731 // define w = (x_1, ... x_n, z_1, ... z_m) = B*y + s.
732 // B is basically the simplex tableau of our implementation except that instead
733 // of only describing the transform to get back the non-basis unknowns, it
734 // defines the values of all the unknowns in terms of the basis unknowns.
735 // Similarly, s is the column for the sample value.
736 //
737 // Our goal is to show that each column in B, restricted to the first n
738 // rows, is lexicopositive after the pivot if it is so before. This is
739 // equivalent to saying the columns in the whole matrix are lexicopositive;
740 // there must be some non-zero element in every column in the first n rows since
741 // the n variables cannot be spanned without using all the n basis unknowns.
742 //
743 // Consider a pivot where z_i replaces y_j in the basis. Recall the pivot
744 // transform for the tableau derived for SimplexBase::pivot:
745 //
746 // pivot col other col pivot col other col
747 // pivot row a b -> pivot row 1/a -b/a
748 // other row c d other row c/a d - bc/a
749 //
750 // Similarly, a pivot results in B changing to B' and c to c'; the difference
751 // between the tableau and these matrices B and B' is that there is no special
752 // case for the pivot row, since it continues to represent the same unknown. The
753 // same formula applies for all rows:
754 //
755 // B'.col(j) = B.col(j) / B(i,j)
756 // B'.col(k) = B.col(k) - B(i,k) * B.col(j) / B(i,j) for k != j
757 // and similarly, s' = s - s_i * B.col(j) / B(i,j).
758 //
759 // If s_i == 0, then the sample value remains unchanged. Otherwise, if s_i < 0,
760 // the change in sample value when pivoting with column a is lexicographically
761 // smaller than that when pivoting with column b iff B.col(a) / B(i, a) is
762 // lexicographically smaller than B.col(b) / B(i, b).
763 //
764 // Since B(i, j) > 0, column j remains lexicopositive.
765 //
766 // For the other columns, suppose C.col(k) is not lexicopositive.
767 // This means that for some p, for all t < p,
768 // C(t,k) = 0 => B(t,k) = B(t,j) * B(i,k) / B(i,j) and
769 // C(t,k) < 0 => B(p,k) < B(t,j) * B(i,k) / B(i,j),
770 // which is in contradiction to the fact that B.col(j) / B(i,j) must be
771 // lexicographically smaller than B.col(k) / B(i,k), since it lexicographically
772 // minimizes the change in sample value.
774  std::optional<unsigned> maybeColumn;
775  for (unsigned col = 3 + nSymbol, e = getNumColumns(); col < e; ++col) {
776  if (tableau(row, col) <= 0)
777  continue;
778  maybeColumn =
779  !maybeColumn ? col : getLexMinPivotColumn(row, *maybeColumn, col);
780  }
781 
782  if (!maybeColumn)
783  return failure();
784 
785  pivot(row, *maybeColumn);
786  return success();
787 }
788 
789 unsigned LexSimplexBase::getLexMinPivotColumn(unsigned row, unsigned colA,
790  unsigned colB) const {
791  // First, let's consider the non-symbolic case.
792  // A pivot causes the following change. (in the diagram the matrix elements
793  // are shown as rationals and there is no common denominator used)
794  //
795  // pivot col big M col const col
796  // pivot row a p b
797  // other row c q d
798  // |
799  // v
800  //
801  // pivot col big M col const col
802  // pivot row 1/a -p/a -b/a
803  // other row c/a q - pc/a d - bc/a
804  //
805  // Let the sample value of the pivot row be s = pM + b before the pivot. Since
806  // the pivot row represents a violated constraint we know that s < 0.
807  //
808  // If the variable is a non-pivot column, its sample value is zero before and
809  // after the pivot.
810  //
811  // If the variable is the pivot column, then its sample value goes from 0 to
812  // (-p/a)M + (-b/a), i.e. 0 to -(pM + b)/a. Thus the change in the sample
813  // value is -s/a.
814  //
815  // If the variable is the pivot row, its sample value goes from s to 0, for a
816  // change of -s.
817  //
818  // If the variable is a non-pivot row, its sample value changes from
819  // qM + d to qM + d + (-pc/a)M + (-bc/a). Thus the change in sample value
820  // is -(pM + b)(c/a) = -sc/a.
821  //
822  // Thus the change in sample value is either 0, -s/a, -s, or -sc/a. Here -s is
823  // fixed for all calls to this function since the row and tableau are fixed.
824  // The callee just wants to compare the return values with the return value of
825  // other invocations of the same function. So the -s is common for all
826  // comparisons involved and can be ignored, since -s is strictly positive.
827  //
828  // Thus we take away this common factor and just return 0, 1/a, 1, or c/a as
829  // appropriate. This allows us to run the entire algorithm treating M
830  // symbolically, as the pivot to be performed does not depend on the value
831  // of M, so long as the sample value s is negative. Note that this is not
832  // because of any special feature of M; by the same argument, we ignore the
833  // symbols too. The caller ensure that the sample value s is negative for
834  // all possible values of the symbols.
835  auto getSampleChangeCoeffForVar = [this, row](unsigned col,
836  const Unknown &u) -> Fraction {
837  MPInt a = tableau(row, col);
838  if (u.orientation == Orientation::Column) {
839  // Pivot column case.
840  if (u.pos == col)
841  return {1, a};
842 
843  // Non-pivot column case.
844  return {0, 1};
845  }
846 
847  // Pivot row case.
848  if (u.pos == row)
849  return {1, 1};
850 
851  // Non-pivot row case.
852  MPInt c = tableau(u.pos, col);
853  return {c, a};
854  };
855 
856  for (const Unknown &u : var) {
857  Fraction changeA = getSampleChangeCoeffForVar(colA, u);
858  Fraction changeB = getSampleChangeCoeffForVar(colB, u);
859  if (changeA < changeB)
860  return colA;
861  if (changeA > changeB)
862  return colB;
863  }
864 
865  // If we reached here, both result in exactly the same changes, so it
866  // doesn't matter which we return.
867  return colA;
868 }
869 
870 /// Find a pivot to change the sample value of the row in the specified
871 /// direction. The returned pivot row will involve `row` if and only if the
872 /// unknown is unbounded in the specified direction.
873 ///
874 /// To increase (resp. decrease) the value of a row, we need to find a live
875 /// column with a non-zero coefficient. If the coefficient is positive, we need
876 /// to increase (decrease) the value of the column, and if the coefficient is
877 /// negative, we need to decrease (increase) the value of the column. Also,
878 /// we cannot decrease the sample value of restricted columns.
879 ///
880 /// If multiple columns are valid, we break ties by considering a lexicographic
881 /// ordering where we prefer unknowns with lower index.
882 std::optional<SimplexBase::Pivot>
883 Simplex::findPivot(int row, Direction direction) const {
884  std::optional<unsigned> col;
885  for (unsigned j = 2, e = getNumColumns(); j < e; ++j) {
886  MPInt elem = tableau(row, j);
887  if (elem == 0)
888  continue;
889 
890  if (unknownFromColumn(j).restricted &&
891  !signMatchesDirection(elem, direction))
892  continue;
893  if (!col || colUnknown[j] < colUnknown[*col])
894  col = j;
895  }
896 
897  if (!col)
898  return {};
899 
900  Direction newDirection =
901  tableau(row, *col) < 0 ? flippedDirection(direction) : direction;
902  std::optional<unsigned> maybePivotRow = findPivotRow(row, newDirection, *col);
903  return Pivot{maybePivotRow.value_or(row), *col};
904 }
905 
906 /// Swap the associated unknowns for the row and the column.
907 ///
908 /// First we swap the index associated with the row and column. Then we update
909 /// the unknowns to reflect their new position and orientation.
910 void SimplexBase::swapRowWithCol(unsigned row, unsigned col) {
911  std::swap(rowUnknown[row], colUnknown[col]);
912  Unknown &uCol = unknownFromColumn(col);
913  Unknown &uRow = unknownFromRow(row);
916  uCol.pos = col;
917  uRow.pos = row;
918 }
919 
920 void SimplexBase::pivot(Pivot pair) { pivot(pair.row, pair.column); }
921 
922 /// Pivot pivotRow and pivotCol.
923 ///
924 /// Let R be the pivot row unknown and let C be the pivot col unknown.
925 /// Since initially R = a*C + sum b_i * X_i
926 /// (where the sum is over the other column's unknowns, x_i)
927 /// C = (R - (sum b_i * X_i))/a
928 ///
929 /// Let u be some other row unknown.
930 /// u = c*C + sum d_i * X_i
931 /// So u = c*(R - sum b_i * X_i)/a + sum d_i * X_i
932 ///
933 /// This results in the following transform:
934 /// pivot col other col pivot col other col
935 /// pivot row a b -> pivot row 1/a -b/a
936 /// other row c d other row c/a d - bc/a
937 ///
938 /// Taking into account the common denominators p and q:
939 ///
940 /// pivot col other col pivot col other col
941 /// pivot row a/p b/p -> pivot row p/a -b/a
942 /// other row c/q d/q other row cp/aq (da - bc)/aq
943 ///
944 /// The pivot row transform is accomplished be swapping a with the pivot row's
945 /// common denominator and negating the pivot row except for the pivot column
946 /// element.
947 void SimplexBase::pivot(unsigned pivotRow, unsigned pivotCol) {
948  assert(pivotCol >= getNumFixedCols() && "Refusing to pivot invalid column");
949  assert(!unknownFromColumn(pivotCol).isSymbol);
950 
951  swapRowWithCol(pivotRow, pivotCol);
952  std::swap(tableau(pivotRow, 0), tableau(pivotRow, pivotCol));
953  // We need to negate the whole pivot row except for the pivot column.
954  if (tableau(pivotRow, 0) < 0) {
955  // If the denominator is negative, we negate the row by simply negating the
956  // denominator.
957  tableau(pivotRow, 0) = -tableau(pivotRow, 0);
958  tableau(pivotRow, pivotCol) = -tableau(pivotRow, pivotCol);
959  } else {
960  for (unsigned col = 1, e = getNumColumns(); col < e; ++col) {
961  if (col == pivotCol)
962  continue;
963  tableau(pivotRow, col) = -tableau(pivotRow, col);
964  }
965  }
966  tableau.normalizeRow(pivotRow);
967 
968  for (unsigned row = 0, numRows = getNumRows(); row < numRows; ++row) {
969  if (row == pivotRow)
970  continue;
971  if (tableau(row, pivotCol) == 0) // Nothing to do.
972  continue;
973  tableau(row, 0) *= tableau(pivotRow, 0);
974  for (unsigned col = 1, numCols = getNumColumns(); col < numCols; ++col) {
975  if (col == pivotCol)
976  continue;
977  // Add rather than subtract because the pivot row has been negated.
978  tableau(row, col) = tableau(row, col) * tableau(pivotRow, 0) +
979  tableau(row, pivotCol) * tableau(pivotRow, col);
980  }
981  tableau(row, pivotCol) *= tableau(pivotRow, pivotCol);
982  tableau.normalizeRow(row);
983  }
984 }
985 
986 /// Perform pivots until the unknown has a non-negative sample value or until
987 /// no more upward pivots can be performed. Return success if we were able to
988 /// bring the row to a non-negative sample value, and failure otherwise.
989 LogicalResult Simplex::restoreRow(Unknown &u) {
990  assert(u.orientation == Orientation::Row &&
991  "unknown should be in row position");
992 
993  while (tableau(u.pos, 1) < 0) {
994  std::optional<Pivot> maybePivot = findPivot(u.pos, Direction::Up);
995  if (!maybePivot)
996  break;
997 
998  pivot(*maybePivot);
999  if (u.orientation == Orientation::Column)
1000  return success(); // the unknown is unbounded above.
1001  }
1002  return success(tableau(u.pos, 1) >= 0);
1003 }
1004 
1005 /// Find a row that can be used to pivot the column in the specified direction.
1006 /// This returns an empty optional if and only if the column is unbounded in the
1007 /// specified direction (ignoring skipRow, if skipRow is set).
1008 ///
1009 /// If skipRow is set, this row is not considered, and (if it is restricted) its
1010 /// restriction may be violated by the returned pivot. Usually, skipRow is set
1011 /// because we don't want to move it to column position unless it is unbounded,
1012 /// and we are either trying to increase the value of skipRow or explicitly
1013 /// trying to make skipRow negative, so we are not concerned about this.
1014 ///
1015 /// If the direction is up (resp. down) and a restricted row has a negative
1016 /// (positive) coefficient for the column, then this row imposes a bound on how
1017 /// much the sample value of the column can change. Such a row with constant
1018 /// term c and coefficient f for the column imposes a bound of c/|f| on the
1019 /// change in sample value (in the specified direction). (note that c is
1020 /// non-negative here since the row is restricted and the tableau is consistent)
1021 ///
1022 /// We iterate through the rows and pick the row which imposes the most
1023 /// stringent bound, since pivoting with a row changes the row's sample value to
1024 /// 0 and hence saturates the bound it imposes. We break ties between rows that
1025 /// impose the same bound by considering a lexicographic ordering where we
1026 /// prefer unknowns with lower index value.
1027 std::optional<unsigned> Simplex::findPivotRow(std::optional<unsigned> skipRow,
1028  Direction direction,
1029  unsigned col) const {
1030  std::optional<unsigned> retRow;
1031  // Initialize these to zero in order to silence a warning about retElem and
1032  // retConst being used uninitialized in the initialization of `diff` below. In
1033  // reality, these are always initialized when that line is reached since these
1034  // are set whenever retRow is set.
1035  MPInt retElem, retConst;
1036  for (unsigned row = nRedundant, e = getNumRows(); row < e; ++row) {
1037  if (skipRow && row == *skipRow)
1038  continue;
1039  MPInt elem = tableau(row, col);
1040  if (elem == 0)
1041  continue;
1042  if (!unknownFromRow(row).restricted)
1043  continue;
1044  if (signMatchesDirection(elem, direction))
1045  continue;
1046  MPInt constTerm = tableau(row, 1);
1047 
1048  if (!retRow) {
1049  retRow = row;
1050  retElem = elem;
1051  retConst = constTerm;
1052  continue;
1053  }
1054 
1055  MPInt diff = retConst * elem - constTerm * retElem;
1056  if ((diff == 0 && rowUnknown[row] < rowUnknown[*retRow]) ||
1057  (diff != 0 && !signMatchesDirection(diff, direction))) {
1058  retRow = row;
1059  retElem = elem;
1060  retConst = constTerm;
1061  }
1062  }
1063  return retRow;
1064 }
1065 
1066 bool SimplexBase::isEmpty() const { return empty; }
1067 
1068 void SimplexBase::swapRows(unsigned i, unsigned j) {
1069  if (i == j)
1070  return;
1071  tableau.swapRows(i, j);
1072  std::swap(rowUnknown[i], rowUnknown[j]);
1073  unknownFromRow(i).pos = i;
1074  unknownFromRow(j).pos = j;
1075 }
1076 
1077 void SimplexBase::swapColumns(unsigned i, unsigned j) {
1078  assert(i < getNumColumns() && j < getNumColumns() &&
1079  "Invalid columns provided!");
1080  if (i == j)
1081  return;
1082  tableau.swapColumns(i, j);
1083  std::swap(colUnknown[i], colUnknown[j]);
1084  unknownFromColumn(i).pos = i;
1085  unknownFromColumn(j).pos = j;
1086 }
1087 
1088 /// Mark this tableau empty and push an entry to the undo stack.
1090  // If the set is already empty, then we shouldn't add another UnmarkEmpty log
1091  // entry, since in that case the Simplex will be erroneously marked as
1092  // non-empty when rolling back past this point.
1093  if (empty)
1094  return;
1096  empty = true;
1097 }
1098 
1099 /// Add an inequality to the tableau. If coeffs is c_0, c_1, ... c_n, where n
1100 /// is the current number of variables, then the corresponding inequality is
1101 /// c_n + c_0*x_0 + c_1*x_1 + ... + c_{n-1}*x_{n-1} >= 0.
1102 ///
1103 /// We add the inequality and mark it as restricted. We then try to make its
1104 /// sample value non-negative. If this is not possible, the tableau has become
1105 /// empty and we mark it as such.
1107  unsigned conIndex = addRow(coeffs, /*makeRestricted=*/true);
1108  LogicalResult result = restoreRow(con[conIndex]);
1109  if (failed(result))
1110  markEmpty();
1111 }
1112 
1113 /// Add an equality to the tableau. If coeffs is c_0, c_1, ... c_n, where n
1114 /// is the current number of variables, then the corresponding equality is
1115 /// c_n + c_0*x_0 + c_1*x_1 + ... + c_{n-1}*x_{n-1} == 0.
1116 ///
1117 /// We simply add two opposing inequalities, which force the expression to
1118 /// be zero.
1120  addInequality(coeffs);
1121  SmallVector<MPInt, 8> negatedCoeffs;
1122  for (const MPInt &coeff : coeffs)
1123  negatedCoeffs.emplace_back(-coeff);
1124  addInequality(negatedCoeffs);
1125 }
1126 
1127 unsigned SimplexBase::getNumVariables() const { return var.size(); }
1128 unsigned SimplexBase::getNumConstraints() const { return con.size(); }
1129 
1130 /// Return a snapshot of the current state. This is just the current size of the
1131 /// undo log.
1132 unsigned SimplexBase::getSnapshot() const { return undoLog.size(); }
1133 
1135  SmallVector<int, 8> basis;
1136  for (int index : colUnknown) {
1137  if (index != nullIndex)
1138  basis.push_back(index);
1139  }
1140  savedBases.push_back(std::move(basis));
1141 
1142  undoLog.emplace_back(UndoLogEntry::RestoreBasis);
1143  return undoLog.size() - 1;
1144 }
1145 
1147  assert(con.back().orientation == Orientation::Row);
1148 
1149  // Move this unknown to the last row and remove the last row from the
1150  // tableau.
1151  swapRows(con.back().pos, getNumRows() - 1);
1152  // It is not strictly necessary to shrink the tableau, but for now we
1153  // maintain the invariant that the tableau has exactly getNumRows()
1154  // rows.
1156  rowUnknown.pop_back();
1157  con.pop_back();
1158 }
1159 
1160 // This doesn't find a pivot row only if the column has zero
1161 // coefficients for every row.
1162 //
1163 // If the unknown is a constraint, this can't happen, since it was added
1164 // initially as a row. Such a row could never have been pivoted to a column. So
1165 // a pivot row will always be found if we have a constraint.
1166 //
1167 // If we have a variable, then the column has zero coefficients for every row
1168 // iff no constraints have been added with a non-zero coefficient for this row.
1169 std::optional<unsigned> SimplexBase::findAnyPivotRow(unsigned col) {
1170  for (unsigned row = nRedundant, e = getNumRows(); row < e; ++row)
1171  if (tableau(row, col) != 0)
1172  return row;
1173  return {};
1174 }
1175 
1176 // It's not valid to remove the constraint by deleting the column since this
1177 // would result in an invalid basis.
1178 void Simplex::undoLastConstraint() {
1179  if (con.back().orientation == Orientation::Column) {
1180  // We try to find any pivot row for this column that preserves tableau
1181  // consistency (except possibly the column itself, which is going to be
1182  // deallocated anyway).
1183  //
1184  // If no pivot row is found in either direction, then the unknown is
1185  // unbounded in both directions and we are free to perform any pivot at
1186  // all. To do this, we just need to find any row with a non-zero
1187  // coefficient for the column. findAnyPivotRow will always be able to
1188  // find such a row for a constraint.
1189  unsigned column = con.back().pos;
1190  if (std::optional<unsigned> maybeRow =
1191  findPivotRow({}, Direction::Up, column)) {
1192  pivot(*maybeRow, column);
1193  } else if (std::optional<unsigned> maybeRow =
1194  findPivotRow({}, Direction::Down, column)) {
1195  pivot(*maybeRow, column);
1196  } else {
1197  std::optional<unsigned> row = findAnyPivotRow(column);
1198  assert(row && "Pivot should always exist for a constraint!");
1199  pivot(*row, column);
1200  }
1201  }
1203 }
1204 
1205 // It's not valid to remove the constraint by deleting the column since this
1206 // would result in an invalid basis.
1208  if (con.back().orientation == Orientation::Column) {
1209  // When removing the last constraint during a rollback, we just need to find
1210  // any pivot at all, i.e., any row with non-zero coefficient for the
1211  // column, because when rolling back a lexicographic simplex, we always
1212  // end by restoring the exact basis that was present at the time of the
1213  // snapshot, so what pivots we perform while undoing doesn't matter as
1214  // long as we get the unknown to row orientation and remove it.
1215  unsigned column = con.back().pos;
1216  std::optional<unsigned> row = findAnyPivotRow(column);
1217  assert(row && "Pivot should always exist for a constraint!");
1218  pivot(*row, column);
1219  }
1221 }
1222 
1224  if (entry == UndoLogEntry::RemoveLastConstraint) {
1225  // Simplex and LexSimplex handle this differently, so we call out to a
1226  // virtual function to handle this.
1228  } else if (entry == UndoLogEntry::RemoveLastVariable) {
1229  // Whenever we are rolling back the addition of a variable, it is guaranteed
1230  // that the variable will be in column position.
1231  //
1232  // We can see this as follows: any constraint that depends on this variable
1233  // was added after this variable was added, so the addition of such
1234  // constraints should already have been rolled back by the time we get to
1235  // rolling back the addition of the variable. Therefore, no constraint
1236  // currently has a component along the variable, so the variable itself must
1237  // be part of the basis.
1238  assert(var.back().orientation == Orientation::Column &&
1239  "Variable to be removed must be in column orientation!");
1240 
1241  if (var.back().isSymbol)
1242  nSymbol--;
1243 
1244  // Move this variable to the last column and remove the column from the
1245  // tableau.
1246  swapColumns(var.back().pos, getNumColumns() - 1);
1248  var.pop_back();
1249  colUnknown.pop_back();
1250  } else if (entry == UndoLogEntry::UnmarkEmpty) {
1251  empty = false;
1252  } else if (entry == UndoLogEntry::UnmarkLastRedundant) {
1253  nRedundant--;
1254  } else if (entry == UndoLogEntry::RestoreBasis) {
1255  assert(!savedBases.empty() && "No bases saved!");
1256 
1257  SmallVector<int, 8> basis = std::move(savedBases.back());
1258  savedBases.pop_back();
1259 
1260  for (int index : basis) {
1261  Unknown &u = unknownFromIndex(index);
1263  continue;
1264  for (unsigned col = getNumFixedCols(), e = getNumColumns(); col < e;
1265  col++) {
1266  assert(colUnknown[col] != nullIndex &&
1267  "Column should not be a fixed column!");
1268  if (llvm::is_contained(basis, colUnknown[col]))
1269  continue;
1270  if (tableau(u.pos, col) == 0)
1271  continue;
1272  pivot(u.pos, col);
1273  break;
1274  }
1275 
1276  assert(u.orientation == Orientation::Column && "No pivot found!");
1277  }
1278  }
1279 }
1280 
1281 /// Rollback to the specified snapshot.
1282 ///
1283 /// We undo all the log entries until the log size when the snapshot was taken
1284 /// is reached.
1285 void SimplexBase::rollback(unsigned snapshot) {
1286  while (undoLog.size() > snapshot) {
1287  undo(undoLog.back());
1288  undoLog.pop_back();
1289  }
1290 }
1291 
1292 /// We add the usual floor division constraints:
1293 /// `0 <= coeffs - denom*q <= denom - 1`, where `q` is the new division
1294 /// variable.
1295 ///
1296 /// This constrains the remainder `coeffs - denom*q` to be in the
1297 /// range `[0, denom - 1]`, which fixes the integer value of the quotient `q`.
1299  const MPInt &denom) {
1300  assert(denom > 0 && "Denominator must be positive!");
1301  appendVariable();
1302 
1303  SmallVector<MPInt, 8> ineq(coeffs.begin(), coeffs.end());
1304  MPInt constTerm = ineq.back();
1305  ineq.back() = -denom;
1306  ineq.push_back(constTerm);
1307  addInequality(ineq);
1308 
1309  for (MPInt &coeff : ineq)
1310  coeff = -coeff;
1311  ineq.back() += denom - 1;
1312  addInequality(ineq);
1313 }
1314 
1315 void SimplexBase::appendVariable(unsigned count) {
1316  if (count == 0)
1317  return;
1318  var.reserve(var.size() + count);
1319  colUnknown.reserve(colUnknown.size() + count);
1320  for (unsigned i = 0; i < count; ++i) {
1321  var.emplace_back(Orientation::Column, /*restricted=*/false,
1322  /*pos=*/getNumColumns() + i);
1323  colUnknown.push_back(var.size() - 1);
1324  }
1326  undoLog.insert(undoLog.end(), count, UndoLogEntry::RemoveLastVariable);
1327 }
1328 
1329 /// Add all the constraints from the given IntegerRelation.
1331  assert(rel.getNumVars() == getNumVariables() &&
1332  "IntegerRelation must have same dimensionality as simplex");
1333  for (unsigned i = 0, e = rel.getNumInequalities(); i < e; ++i)
1334  addInequality(rel.getInequality(i));
1335  for (unsigned i = 0, e = rel.getNumEqualities(); i < e; ++i)
1336  addEquality(rel.getEquality(i));
1337 }
1338 
1340  unsigned row) {
1341  // Keep trying to find a pivot for the row in the specified direction.
1342  while (std::optional<Pivot> maybePivot = findPivot(row, direction)) {
1343  // If findPivot returns a pivot involving the row itself, then the optimum
1344  // is unbounded, so we return std::nullopt.
1345  if (maybePivot->row == row)
1346  return OptimumKind::Unbounded;
1347  pivot(*maybePivot);
1348  }
1349 
1350  // The row has reached its optimal sample value, which we return.
1351  // The sample value is the entry in the constant column divided by the common
1352  // denominator for this row.
1353  return Fraction(tableau(row, 1), tableau(row, 0));
1354 }
1355 
1356 /// Compute the optimum of the specified expression in the specified direction,
1357 /// or std::nullopt if it is unbounded.
1359  ArrayRef<MPInt> coeffs) {
1360  if (empty)
1361  return OptimumKind::Empty;
1362 
1363  SimplexRollbackScopeExit scopeExit(*this);
1364  unsigned conIndex = addRow(coeffs);
1365  unsigned row = con[conIndex].pos;
1366  return computeRowOptimum(direction, row);
1367 }
1368 
1370  Unknown &u) {
1371  if (empty)
1372  return OptimumKind::Empty;
1373  if (u.orientation == Orientation::Column) {
1374  unsigned column = u.pos;
1375  std::optional<unsigned> pivotRow = findPivotRow({}, direction, column);
1376  // If no pivot is returned, the constraint is unbounded in the specified
1377  // direction.
1378  if (!pivotRow)
1379  return OptimumKind::Unbounded;
1380  pivot(*pivotRow, column);
1381  }
1382 
1383  unsigned row = u.pos;
1384  MaybeOptimum<Fraction> optimum = computeRowOptimum(direction, row);
1385  if (u.restricted && direction == Direction::Down &&
1386  (optimum.isUnbounded() || *optimum < Fraction(0, 1))) {
1387  if (failed(restoreRow(u)))
1388  llvm_unreachable("Could not restore row!");
1389  }
1390  return optimum;
1391 }
1392 
1393 bool Simplex::isBoundedAlongConstraint(unsigned constraintIndex) {
1394  assert(!empty && "It is not meaningful to ask whether a direction is bounded "
1395  "in an empty set.");
1396  // The constraint's perpendicular is already bounded below, since it is a
1397  // constraint. If it is also bounded above, we can return true.
1398  return computeOptimum(Direction::Up, con[constraintIndex]).isBounded();
1399 }
1400 
1401 /// Redundant constraints are those that are in row orientation and lie in
1402 /// rows 0 to nRedundant - 1.
1403 bool Simplex::isMarkedRedundant(unsigned constraintIndex) const {
1404  const Unknown &u = con[constraintIndex];
1405  return u.orientation == Orientation::Row && u.pos < nRedundant;
1406 }
1407 
1408 /// Mark the specified row redundant.
1409 ///
1410 /// This is done by moving the unknown to the end of the block of redundant
1411 /// rows (namely, to row nRedundant) and incrementing nRedundant to
1412 /// accomodate the new redundant row.
1413 void Simplex::markRowRedundant(Unknown &u) {
1414  assert(u.orientation == Orientation::Row &&
1415  "Unknown should be in row position!");
1416  assert(u.pos >= nRedundant && "Unknown is already marked redundant!");
1417  swapRows(u.pos, nRedundant);
1418  ++nRedundant;
1420 }
1421 
1422 /// Find a subset of constraints that is redundant and mark them redundant.
1423 void Simplex::detectRedundant(unsigned offset, unsigned count) {
1424  assert(offset + count <= con.size() && "invalid range!");
1425  // It is not meaningful to talk about redundancy for empty sets.
1426  if (empty)
1427  return;
1428 
1429  // Iterate through the constraints and check for each one if it can attain
1430  // negative sample values. If it can, it's not redundant. Otherwise, it is.
1431  // We mark redundant constraints redundant.
1432  //
1433  // Constraints that get marked redundant in one iteration are not respected
1434  // when checking constraints in later iterations. This prevents, for example,
1435  // two identical constraints both being marked redundant since each is
1436  // redundant given the other one. In this example, only the first of the
1437  // constraints that is processed will get marked redundant, as it should be.
1438  for (unsigned i = 0; i < count; ++i) {
1439  Unknown &u = con[offset + i];
1440  if (u.orientation == Orientation::Column) {
1441  unsigned column = u.pos;
1442  std::optional<unsigned> pivotRow =
1443  findPivotRow({}, Direction::Down, column);
1444  // If no downward pivot is returned, the constraint is unbounded below
1445  // and hence not redundant.
1446  if (!pivotRow)
1447  continue;
1448  pivot(*pivotRow, column);
1449  }
1450 
1451  unsigned row = u.pos;
1453  if (minimum.isUnbounded() || *minimum < Fraction(0, 1)) {
1454  // Constraint is unbounded below or can attain negative sample values and
1455  // hence is not redundant.
1456  if (failed(restoreRow(u)))
1457  llvm_unreachable("Could not restore non-redundant row!");
1458  continue;
1459  }
1460 
1461  markRowRedundant(u);
1462  }
1463 }
1464 
1466  if (empty)
1467  return false;
1468 
1469  SmallVector<MPInt, 8> dir(var.size() + 1);
1470  for (unsigned i = 0; i < var.size(); ++i) {
1471  dir[i] = 1;
1472 
1474  return true;
1475 
1477  return true;
1478 
1479  dir[i] = 0;
1480  }
1481  return false;
1482 }
1483 
1484 /// Make a tableau to represent a pair of points in the original tableau.
1485 ///
1486 /// The product constraints and variables are stored as: first A's, then B's.
1487 ///
1488 /// The product tableau has row layout:
1489 /// A's redundant rows, B's redundant rows, A's other rows, B's other rows.
1490 ///
1491 /// It has column layout:
1492 /// denominator, constant, A's columns, B's columns.
1494  unsigned numVar = a.getNumVariables() + b.getNumVariables();
1495  unsigned numCon = a.getNumConstraints() + b.getNumConstraints();
1496  Simplex result(numVar);
1497 
1498  result.tableau.reserveRows(numCon);
1499  result.empty = a.empty || b.empty;
1500 
1501  auto concat = [](ArrayRef<Unknown> v, ArrayRef<Unknown> w) {
1502  SmallVector<Unknown, 8> result;
1503  result.reserve(v.size() + w.size());
1504  result.insert(result.end(), v.begin(), v.end());
1505  result.insert(result.end(), w.begin(), w.end());
1506  return result;
1507  };
1508  result.con = concat(a.con, b.con);
1509  result.var = concat(a.var, b.var);
1510 
1511  auto indexFromBIndex = [&](int index) {
1512  return index >= 0 ? a.getNumVariables() + index
1513  : ~(a.getNumConstraints() + ~index);
1514  };
1515 
1516  result.colUnknown.assign(2, nullIndex);
1517  for (unsigned i = 2, e = a.getNumColumns(); i < e; ++i) {
1518  result.colUnknown.push_back(a.colUnknown[i]);
1519  result.unknownFromIndex(result.colUnknown.back()).pos =
1520  result.colUnknown.size() - 1;
1521  }
1522  for (unsigned i = 2, e = b.getNumColumns(); i < e; ++i) {
1523  result.colUnknown.push_back(indexFromBIndex(b.colUnknown[i]));
1524  result.unknownFromIndex(result.colUnknown.back()).pos =
1525  result.colUnknown.size() - 1;
1526  }
1527 
1528  auto appendRowFromA = [&](unsigned row) {
1529  unsigned resultRow = result.tableau.appendExtraRow();
1530  for (unsigned col = 0, e = a.getNumColumns(); col < e; ++col)
1531  result.tableau(resultRow, col) = a.tableau(row, col);
1532  result.rowUnknown.push_back(a.rowUnknown[row]);
1533  result.unknownFromIndex(result.rowUnknown.back()).pos =
1534  result.rowUnknown.size() - 1;
1535  };
1536 
1537  // Also fixes the corresponding entry in rowUnknown and var/con (as the case
1538  // may be).
1539  auto appendRowFromB = [&](unsigned row) {
1540  unsigned resultRow = result.tableau.appendExtraRow();
1541  result.tableau(resultRow, 0) = b.tableau(row, 0);
1542  result.tableau(resultRow, 1) = b.tableau(row, 1);
1543 
1544  unsigned offset = a.getNumColumns() - 2;
1545  for (unsigned col = 2, e = b.getNumColumns(); col < e; ++col)
1546  result.tableau(resultRow, offset + col) = b.tableau(row, col);
1547  result.rowUnknown.push_back(indexFromBIndex(b.rowUnknown[row]));
1548  result.unknownFromIndex(result.rowUnknown.back()).pos =
1549  result.rowUnknown.size() - 1;
1550  };
1551 
1552  result.nRedundant = a.nRedundant + b.nRedundant;
1553  for (unsigned row = 0; row < a.nRedundant; ++row)
1554  appendRowFromA(row);
1555  for (unsigned row = 0; row < b.nRedundant; ++row)
1556  appendRowFromB(row);
1557  for (unsigned row = a.nRedundant, e = a.getNumRows(); row < e; ++row)
1558  appendRowFromA(row);
1559  for (unsigned row = b.nRedundant, e = b.getNumRows(); row < e; ++row)
1560  appendRowFromB(row);
1561 
1562  return result;
1563 }
1564 
1565 std::optional<SmallVector<Fraction, 8>> Simplex::getRationalSample() const {
1566  if (empty)
1567  return {};
1568 
1569  SmallVector<Fraction, 8> sample;
1570  sample.reserve(var.size());
1571  // Push the sample value for each variable into the vector.
1572  for (const Unknown &u : var) {
1573  if (u.orientation == Orientation::Column) {
1574  // If the variable is in column position, its sample value is zero.
1575  sample.emplace_back(0, 1);
1576  } else {
1577  // If the variable is in row position, its sample value is the
1578  // entry in the constant column divided by the denominator.
1579  MPInt denom = tableau(u.pos, 0);
1580  sample.emplace_back(tableau(u.pos, 1), denom);
1581  }
1582  }
1583  return sample;
1584 }
1585 
1587  addRow(coeffs, /*makeRestricted=*/true);
1588 }
1589 
1590 MaybeOptimum<SmallVector<Fraction, 8>> LexSimplex::getRationalSample() const {
1591  if (empty)
1592  return OptimumKind::Empty;
1593 
1594  SmallVector<Fraction, 8> sample;
1595  sample.reserve(var.size());
1596  // Push the sample value for each variable into the vector.
1597  for (const Unknown &u : var) {
1598  // When the big M parameter is being used, each variable x is represented
1599  // as M + x, so its sample value is finite if and only if it is of the
1600  // form 1*M + c. If the coefficient of M is not one then the sample value
1601  // is infinite, and we return an empty optional.
1602 
1603  if (u.orientation == Orientation::Column) {
1604  // If the variable is in column position, the sample value of M + x is
1605  // zero, so x = -M which is unbounded.
1606  return OptimumKind::Unbounded;
1607  }
1608 
1609  // If the variable is in row position, its sample value is the
1610  // entry in the constant column divided by the denominator.
1611  MPInt denom = tableau(u.pos, 0);
1612  if (usingBigM)
1613  if (tableau(u.pos, 2) != denom)
1614  return OptimumKind::Unbounded;
1615  sample.emplace_back(tableau(u.pos, 1), denom);
1616  }
1617  return sample;
1618 }
1619 
1620 std::optional<SmallVector<MPInt, 8>> Simplex::getSamplePointIfIntegral() const {
1621  // If the tableau is empty, no sample point exists.
1622  if (empty)
1623  return {};
1624 
1625  // The value will always exist since the Simplex is non-empty.
1626  SmallVector<Fraction, 8> rationalSample = *getRationalSample();
1627  SmallVector<MPInt, 8> integerSample;
1628  integerSample.reserve(var.size());
1629  for (const Fraction &coord : rationalSample) {
1630  // If the sample is non-integral, return std::nullopt.
1631  if (coord.num % coord.den != 0)
1632  return {};
1633  integerSample.push_back(coord.num / coord.den);
1634  }
1635  return integerSample;
1636 }
1637 
1638 /// Given a simplex for a polytope, construct a new simplex whose variables are
1639 /// identified with a pair of points (x, y) in the original polytope. Supports
1640 /// some operations needed for generalized basis reduction. In what follows,
1641 /// dotProduct(x, y) = x_1 * y_1 + x_2 * y_2 + ... x_n * y_n where n is the
1642 /// dimension of the original polytope.
1643 ///
1644 /// This supports adding equality constraints dotProduct(dir, x - y) == 0. It
1645 /// also supports rolling back this addition, by maintaining a snapshot stack
1646 /// that contains a snapshot of the Simplex's state for each equality, just
1647 /// before that equality was added.
1650 
1651 public:
1652  GBRSimplex(const Simplex &originalSimplex)
1653  : simplex(Simplex::makeProduct(originalSimplex, originalSimplex)),
1654  simplexConstraintOffset(simplex.getNumConstraints()) {}
1655 
1656  /// Add an equality dotProduct(dir, x - y) == 0.
1657  /// First pushes a snapshot for the current simplex state to the stack so
1658  /// that this can be rolled back later.
1660  assert(llvm::any_of(dir, [](const MPInt &x) { return x != 0; }) &&
1661  "Direction passed is the zero vector!");
1662  snapshotStack.push_back(simplex.getSnapshot());
1663  simplex.addEquality(getCoeffsForDirection(dir));
1664  }
1665  /// Compute max(dotProduct(dir, x - y)).
1667  MaybeOptimum<Fraction> maybeWidth =
1668  simplex.computeOptimum(Direction::Up, getCoeffsForDirection(dir));
1669  assert(maybeWidth.isBounded() && "Width should be bounded!");
1670  return *maybeWidth;
1671  }
1672 
1673  /// Compute max(dotProduct(dir, x - y)) and save the dual variables for only
1674  /// the direction equalities to `dual`.
1676  SmallVectorImpl<MPInt> &dual,
1677  MPInt &dualDenom) {
1678  // We can't just call into computeWidth or computeOptimum since we need to
1679  // access the state of the tableau after computing the optimum, and these
1680  // functions rollback the insertion of the objective function into the
1681  // tableau before returning. We instead add a row for the objective function
1682  // ourselves, call into computeOptimum, compute the duals from the tableau
1683  // state, and finally rollback the addition of the row before returning.
1684  SimplexRollbackScopeExit scopeExit(simplex);
1685  unsigned conIndex = simplex.addRow(getCoeffsForDirection(dir));
1686  unsigned row = simplex.con[conIndex].pos;
1687  MaybeOptimum<Fraction> maybeWidth =
1688  simplex.computeRowOptimum(Simplex::Direction::Up, row);
1689  assert(maybeWidth.isBounded() && "Width should be bounded!");
1690  dualDenom = simplex.tableau(row, 0);
1691  dual.clear();
1692 
1693  // The increment is i += 2 because equalities are added as two inequalities,
1694  // one positive and one negative. Each iteration processes one equality.
1695  for (unsigned i = simplexConstraintOffset; i < conIndex; i += 2) {
1696  // The dual variable for an inequality in column orientation is the
1697  // negative of its coefficient at the objective row. If the inequality is
1698  // in row orientation, the corresponding dual variable is zero.
1699  //
1700  // We want the dual for the original equality, which corresponds to two
1701  // inequalities: a positive inequality, which has the same coefficients as
1702  // the equality, and a negative equality, which has negated coefficients.
1703  //
1704  // Note that at most one of these inequalities can be in column
1705  // orientation because the column unknowns should form a basis and hence
1706  // must be linearly independent. If the positive inequality is in column
1707  // position, its dual is the dual corresponding to the equality. If the
1708  // negative inequality is in column position, the negation of its dual is
1709  // the dual corresponding to the equality. If neither is in column
1710  // position, then that means that this equality is redundant, and its dual
1711  // is zero.
1712  //
1713  // Note that it is NOT valid to perform pivots during the computation of
1714  // the duals. This entire dual computation must be performed on the same
1715  // tableau configuration.
1716  assert(!(simplex.con[i].orientation == Orientation::Column &&
1717  simplex.con[i + 1].orientation == Orientation::Column) &&
1718  "Both inequalities for the equality cannot be in column "
1719  "orientation!");
1720  if (simplex.con[i].orientation == Orientation::Column)
1721  dual.push_back(-simplex.tableau(row, simplex.con[i].pos));
1722  else if (simplex.con[i + 1].orientation == Orientation::Column)
1723  dual.push_back(simplex.tableau(row, simplex.con[i + 1].pos));
1724  else
1725  dual.emplace_back(0);
1726  }
1727  return *maybeWidth;
1728  }
1729 
1730  /// Remove the last equality that was added through addEqualityForDirection.
1731  ///
1732  /// We do this by rolling back to the snapshot at the top of the stack, which
1733  /// should be a snapshot taken just before the last equality was added.
1735  assert(!snapshotStack.empty() && "Snapshot stack is empty!");
1736  simplex.rollback(snapshotStack.back());
1737  snapshotStack.pop_back();
1738  }
1739 
1740 private:
1741  /// Returns coefficients of the expression 'dot_product(dir, x - y)',
1742  /// i.e., dir_1 * x_1 + dir_2 * x_2 + ... + dir_n * x_n
1743  /// - dir_1 * y_1 - dir_2 * y_2 - ... - dir_n * y_n,
1744  /// where n is the dimension of the original polytope.
1745  SmallVector<MPInt, 8> getCoeffsForDirection(ArrayRef<MPInt> dir) {
1746  assert(2 * dir.size() == simplex.getNumVariables() &&
1747  "Direction vector has wrong dimensionality");
1748  SmallVector<MPInt, 8> coeffs(dir.begin(), dir.end());
1749  coeffs.reserve(2 * dir.size());
1750  for (const MPInt &coeff : dir)
1751  coeffs.push_back(-coeff);
1752  coeffs.emplace_back(0); // constant term
1753  return coeffs;
1754  }
1755 
1756  Simplex simplex;
1757  /// The first index of the equality constraints, the index immediately after
1758  /// the last constraint in the initial product simplex.
1759  unsigned simplexConstraintOffset;
1760  /// A stack of snapshots, used for rolling back.
1761  SmallVector<unsigned, 8> snapshotStack;
1762 };
1763 
1764 /// Reduce the basis to try and find a direction in which the polytope is
1765 /// "thin". This only works for bounded polytopes.
1766 ///
1767 /// This is an implementation of the algorithm described in the paper
1768 /// "An Implementation of Generalized Basis Reduction for Integer Programming"
1769 /// by W. Cook, T. Rutherford, H. E. Scarf, D. Shallcross.
1770 ///
1771 /// Let b_{level}, b_{level + 1}, ... b_n be the current basis.
1772 /// Let width_i(v) = max <v, x - y> where x and y are points in the original
1773 /// polytope such that <b_j, x - y> = 0 is satisfied for all level <= j < i.
1774 ///
1775 /// In every iteration, we first replace b_{i+1} with b_{i+1} + u*b_i, where u
1776 /// is the integer such that width_i(b_{i+1} + u*b_i) is minimized. Let dual_i
1777 /// be the dual variable associated with the constraint <b_i, x - y> = 0 when
1778 /// computing width_{i+1}(b_{i+1}). It can be shown that dual_i is the
1779 /// minimizing value of u, if it were allowed to be fractional. Due to
1780 /// convexity, the minimizing integer value is either floor(dual_i) or
1781 /// ceil(dual_i), so we just need to check which of these gives a lower
1782 /// width_{i+1} value. If dual_i turned out to be an integer, then u = dual_i.
1783 ///
1784 /// Now if width_i(b_{i+1}) < 0.75 * width_i(b_i), we swap b_i and (the new)
1785 /// b_{i + 1} and decrement i (unless i = level, in which case we stay at the
1786 /// same i). Otherwise, we increment i.
1787 ///
1788 /// We keep f values and duals cached and invalidate them when necessary.
1789 /// Whenever possible, we use them instead of recomputing them. We implement the
1790 /// algorithm as follows.
1791 ///
1792 /// In an iteration at i we need to compute:
1793 /// a) width_i(b_{i + 1})
1794 /// b) width_i(b_i)
1795 /// c) the integer u that minimizes width_i(b_{i + 1} + u*b_i)
1796 ///
1797 /// If width_i(b_i) is not already cached, we compute it.
1798 ///
1799 /// If the duals are not already cached, we compute width_{i+1}(b_{i+1}) and
1800 /// store the duals from this computation.
1801 ///
1802 /// We call updateBasisWithUAndGetFCandidate, which finds the minimizing value
1803 /// of u as explained before, caches the duals from this computation, sets
1804 /// b_{i+1} to b_{i+1} + u*b_i, and returns the new value of width_i(b_{i+1}).
1805 ///
1806 /// Now if width_i(b_{i+1}) < 0.75 * width_i(b_i), we swap b_i and b_{i+1} and
1807 /// decrement i, resulting in the basis
1808 /// ... b_{i - 1}, b_{i + 1} + u*b_i, b_i, b_{i+2}, ...
1809 /// with corresponding f values
1810 /// ... width_{i-1}(b_{i-1}), width_i(b_{i+1} + u*b_i), width_{i+1}(b_i), ...
1811 /// The values up to i - 1 remain unchanged. We have just gotten the middle
1812 /// value from updateBasisWithUAndGetFCandidate, so we can update that in the
1813 /// cache. The value at width_{i+1}(b_i) is unknown, so we evict this value from
1814 /// the cache. The iteration after decrementing needs exactly the duals from the
1815 /// computation of width_i(b_{i + 1} + u*b_i), so we keep these in the cache.
1816 ///
1817 /// When incrementing i, no cached f values get invalidated. However, the cached
1818 /// duals do get invalidated as the duals for the higher levels are different.
1819 void Simplex::reduceBasis(IntMatrix &basis, unsigned level) {
1820  const Fraction epsilon(3, 4);
1821 
1822  if (level == basis.getNumRows() - 1)
1823  return;
1824 
1825  GBRSimplex gbrSimplex(*this);
1827  SmallVector<MPInt, 8> dual;
1828  MPInt dualDenom;
1829 
1830  // Finds the value of u that minimizes width_i(b_{i+1} + u*b_i), caches the
1831  // duals from this computation, sets b_{i+1} to b_{i+1} + u*b_i, and returns
1832  // the new value of width_i(b_{i+1}).
1833  //
1834  // If dual_i is not an integer, the minimizing value must be either
1835  // floor(dual_i) or ceil(dual_i). We compute the expression for both and
1836  // choose the minimizing value.
1837  //
1838  // If dual_i is an integer, we don't need to perform these computations. We
1839  // know that in this case,
1840  // a) u = dual_i.
1841  // b) one can show that dual_j for j < i are the same duals we would have
1842  // gotten from computing width_i(b_{i + 1} + u*b_i), so the correct duals
1843  // are the ones already in the cache.
1844  // c) width_i(b_{i+1} + u*b_i) = min_{alpha} width_i(b_{i+1} + alpha * b_i),
1845  // which
1846  // one can show is equal to width_{i+1}(b_{i+1}). The latter value must
1847  // be in the cache, so we get it from there and return it.
1848  auto updateBasisWithUAndGetFCandidate = [&](unsigned i) -> Fraction {
1849  assert(i < level + dual.size() && "dual_i is not known!");
1850 
1851  MPInt u = floorDiv(dual[i - level], dualDenom);
1852  basis.addToRow(i, i + 1, u);
1853  if (dual[i - level] % dualDenom != 0) {
1854  SmallVector<MPInt, 8> candidateDual[2];
1855  MPInt candidateDualDenom[2];
1856  Fraction widthI[2];
1857 
1858  // Initially u is floor(dual) and basis reflects this.
1859  widthI[0] = gbrSimplex.computeWidthAndDuals(
1860  basis.getRow(i + 1), candidateDual[0], candidateDualDenom[0]);
1861 
1862  // Now try ceil(dual), i.e. floor(dual) + 1.
1863  ++u;
1864  basis.addToRow(i, i + 1, 1);
1865  widthI[1] = gbrSimplex.computeWidthAndDuals(
1866  basis.getRow(i + 1), candidateDual[1], candidateDualDenom[1]);
1867 
1868  unsigned j = widthI[0] < widthI[1] ? 0 : 1;
1869  if (j == 0)
1870  // Subtract 1 to go from u = ceil(dual) back to floor(dual).
1871  basis.addToRow(i, i + 1, -1);
1872 
1873  // width_i(b{i+1} + u*b_i) should be minimized at our value of u.
1874  // We assert that this holds by checking that the values of width_i at
1875  // u - 1 and u + 1 are greater than or equal to the value at u. If the
1876  // width is lesser at either of the adjacent values, then our computed
1877  // value of u is clearly not the minimizer. Otherwise by convexity the
1878  // computed value of u is really the minimizer.
1879 
1880  // Check the value at u - 1.
1881  assert(gbrSimplex.computeWidth(scaleAndAddForAssert(
1882  basis.getRow(i + 1), MPInt(-1), basis.getRow(i))) >=
1883  widthI[j] &&
1884  "Computed u value does not minimize the width!");
1885  // Check the value at u + 1.
1886  assert(gbrSimplex.computeWidth(scaleAndAddForAssert(
1887  basis.getRow(i + 1), MPInt(+1), basis.getRow(i))) >=
1888  widthI[j] &&
1889  "Computed u value does not minimize the width!");
1890 
1891  dual = std::move(candidateDual[j]);
1892  dualDenom = candidateDualDenom[j];
1893  return widthI[j];
1894  }
1895 
1896  assert(i + 1 - level < width.size() && "width_{i+1} wasn't saved");
1897  // f_i(b_{i+1} + dual*b_i) == width_{i+1}(b_{i+1}) when `dual` minimizes the
1898  // LHS. (note: the basis has already been updated, so b_{i+1} + dual*b_i in
1899  // the above expression is equal to basis.getRow(i+1) below.)
1900  assert(gbrSimplex.computeWidth(basis.getRow(i + 1)) ==
1901  width[i + 1 - level]);
1902  return width[i + 1 - level];
1903  };
1904 
1905  // In the ith iteration of the loop, gbrSimplex has constraints for directions
1906  // from `level` to i - 1.
1907  unsigned i = level;
1908  while (i < basis.getNumRows() - 1) {
1909  if (i >= level + width.size()) {
1910  // We don't even know the value of f_i(b_i), so let's find that first.
1911  // We have to do this first since later we assume that width already
1912  // contains values up to and including i.
1913 
1914  assert((i == 0 || i - 1 < level + width.size()) &&
1915  "We are at level i but we don't know the value of width_{i-1}");
1916 
1917  // We don't actually use these duals at all, but it doesn't matter
1918  // because this case should only occur when i is level, and there are no
1919  // duals in that case anyway.
1920  assert(i == level && "This case should only occur when i == level");
1921  width.push_back(
1922  gbrSimplex.computeWidthAndDuals(basis.getRow(i), dual, dualDenom));
1923  }
1924 
1925  if (i >= level + dual.size()) {
1926  assert(i + 1 >= level + width.size() &&
1927  "We don't know dual_i but we know width_{i+1}");
1928  // We don't know dual for our level, so let's find it.
1929  gbrSimplex.addEqualityForDirection(basis.getRow(i));
1930  width.push_back(gbrSimplex.computeWidthAndDuals(basis.getRow(i + 1), dual,
1931  dualDenom));
1932  gbrSimplex.removeLastEquality();
1933  }
1934 
1935  // This variable stores width_i(b_{i+1} + u*b_i).
1936  Fraction widthICandidate = updateBasisWithUAndGetFCandidate(i);
1937  if (widthICandidate < epsilon * width[i - level]) {
1938  basis.swapRows(i, i + 1);
1939  width[i - level] = widthICandidate;
1940  // The values of width_{i+1}(b_{i+1}) and higher may change after the
1941  // swap, so we remove the cached values here.
1942  width.resize(i - level + 1);
1943  if (i == level) {
1944  dual.clear();
1945  continue;
1946  }
1947 
1948  gbrSimplex.removeLastEquality();
1949  i--;
1950  continue;
1951  }
1952 
1953  // Invalidate duals since the higher level needs to recompute its own duals.
1954  dual.clear();
1955  gbrSimplex.addEqualityForDirection(basis.getRow(i));
1956  i++;
1957  }
1958 }
1959 
1960 /// Search for an integer sample point using a branch and bound algorithm.
1961 ///
1962 /// Each row in the basis matrix is a vector, and the set of basis vectors
1963 /// should span the space. Initially this is the identity matrix,
1964 /// i.e., the basis vectors are just the variables.
1965 ///
1966 /// In every level, a value is assigned to the level-th basis vector, as
1967 /// follows. Compute the minimum and maximum rational values of this direction.
1968 /// If only one integer point lies in this range, constrain the variable to
1969 /// have this value and recurse to the next variable.
1970 ///
1971 /// If the range has multiple values, perform generalized basis reduction via
1972 /// reduceBasis and then compute the bounds again. Now we try constraining
1973 /// this direction in the first value in this range and "recurse" to the next
1974 /// level. If we fail to find a sample, we try assigning the direction the next
1975 /// value in this range, and so on.
1976 ///
1977 /// If no integer sample is found from any of the assignments, or if the range
1978 /// contains no integer value, then of course the polytope is empty for the
1979 /// current assignment of the values in previous levels, so we return to
1980 /// the previous level.
1981 ///
1982 /// If we reach the last level where all the variables have been assigned values
1983 /// already, then we simply return the current sample point if it is integral,
1984 /// and go back to the previous level otherwise.
1985 ///
1986 /// To avoid potentially arbitrarily large recursion depths leading to stack
1987 /// overflows, this algorithm is implemented iteratively.
1988 std::optional<SmallVector<MPInt, 8>> Simplex::findIntegerSample() {
1989  if (empty)
1990  return {};
1991 
1992  unsigned nDims = var.size();
1993  IntMatrix basis = IntMatrix::identity(nDims);
1994 
1995  unsigned level = 0;
1996  // The snapshot just before constraining a direction to a value at each level.
1997  SmallVector<unsigned, 8> snapshotStack;
1998  // The maximum value in the range of the direction for each level.
1999  SmallVector<MPInt, 8> upperBoundStack;
2000  // The next value to try constraining the basis vector to at each level.
2001  SmallVector<MPInt, 8> nextValueStack;
2002 
2003  snapshotStack.reserve(basis.getNumRows());
2004  upperBoundStack.reserve(basis.getNumRows());
2005  nextValueStack.reserve(basis.getNumRows());
2006  while (level != -1u) {
2007  if (level == basis.getNumRows()) {
2008  // We've assigned values to all variables. Return if we have a sample,
2009  // or go back up to the previous level otherwise.
2010  if (auto maybeSample = getSamplePointIfIntegral())
2011  return maybeSample;
2012  level--;
2013  continue;
2014  }
2015 
2016  if (level >= upperBoundStack.size()) {
2017  // We haven't populated the stack values for this level yet, so we have
2018  // just come down a level ("recursed"). Find the lower and upper bounds.
2019  // If there is more than one integer point in the range, perform
2020  // generalized basis reduction.
2021  SmallVector<MPInt, 8> basisCoeffs =
2022  llvm::to_vector<8>(basis.getRow(level));
2023  basisCoeffs.emplace_back(0);
2024 
2025  auto [minRoundedUp, maxRoundedDown] = computeIntegerBounds(basisCoeffs);
2026 
2027  // We don't have any integer values in the range.
2028  // Pop the stack and return up a level.
2029  if (minRoundedUp.isEmpty() || maxRoundedDown.isEmpty()) {
2030  assert((minRoundedUp.isEmpty() && maxRoundedDown.isEmpty()) &&
2031  "If one bound is empty, both should be.");
2032  snapshotStack.pop_back();
2033  nextValueStack.pop_back();
2034  upperBoundStack.pop_back();
2035  level--;
2036  continue;
2037  }
2038 
2039  // We already checked the empty case above.
2040  assert((minRoundedUp.isBounded() && maxRoundedDown.isBounded()) &&
2041  "Polyhedron should be bounded!");
2042 
2043  // Heuristic: if the sample point is integral at this point, just return
2044  // it.
2045  if (auto maybeSample = getSamplePointIfIntegral())
2046  return *maybeSample;
2047 
2048  if (*minRoundedUp < *maxRoundedDown) {
2049  reduceBasis(basis, level);
2050  basisCoeffs = llvm::to_vector<8>(basis.getRow(level));
2051  basisCoeffs.emplace_back(0);
2052  std::tie(minRoundedUp, maxRoundedDown) =
2053  computeIntegerBounds(basisCoeffs);
2054  }
2055 
2056  snapshotStack.push_back(getSnapshot());
2057  // The smallest value in the range is the next value to try.
2058  // The values in the optionals are guaranteed to exist since we know the
2059  // polytope is bounded.
2060  nextValueStack.push_back(*minRoundedUp);
2061  upperBoundStack.push_back(*maxRoundedDown);
2062  }
2063 
2064  assert((snapshotStack.size() - 1 == level &&
2065  nextValueStack.size() - 1 == level &&
2066  upperBoundStack.size() - 1 == level) &&
2067  "Mismatched variable stack sizes!");
2068 
2069  // Whether we "recursed" or "returned" from a lower level, we rollback
2070  // to the snapshot of the starting state at this level. (in the "recursed"
2071  // case this has no effect)
2072  rollback(snapshotStack.back());
2073  MPInt nextValue = nextValueStack.back();
2074  ++nextValueStack.back();
2075  if (nextValue > upperBoundStack.back()) {
2076  // We have exhausted the range and found no solution. Pop the stack and
2077  // return up a level.
2078  snapshotStack.pop_back();
2079  nextValueStack.pop_back();
2080  upperBoundStack.pop_back();
2081  level--;
2082  continue;
2083  }
2084 
2085  // Try the next value in the range and "recurse" into the next level.
2086  SmallVector<MPInt, 8> basisCoeffs(basis.getRow(level).begin(),
2087  basis.getRow(level).end());
2088  basisCoeffs.push_back(-nextValue);
2089  addEquality(basisCoeffs);
2090  level++;
2091  }
2092 
2093  return {};
2094 }
2095 
2096 /// Compute the minimum and maximum integer values the expression can take. We
2097 /// compute each separately.
2098 std::pair<MaybeOptimum<MPInt>, MaybeOptimum<MPInt>>
2100  MaybeOptimum<MPInt> minRoundedUp(
2102  MaybeOptimum<MPInt> maxRoundedDown(
2104  return {minRoundedUp, maxRoundedDown};
2105 }
2106 
2108  assert(!isEmpty() && "cannot check for flatness of empty simplex!");
2109  auto upOpt = computeOptimum(Simplex::Direction::Up, coeffs);
2110  auto downOpt = computeOptimum(Simplex::Direction::Down, coeffs);
2111 
2112  if (!upOpt.isBounded())
2113  return false;
2114  if (!downOpt.isBounded())
2115  return false;
2116 
2117  return *upOpt == *downOpt;
2118 }
2119 
2120 void SimplexBase::print(raw_ostream &os) const {
2121  os << "rows = " << getNumRows() << ", columns = " << getNumColumns() << "\n";
2122  if (empty)
2123  os << "Simplex marked empty!\n";
2124  os << "var: ";
2125  for (unsigned i = 0; i < var.size(); ++i) {
2126  if (i > 0)
2127  os << ", ";
2128  var[i].print(os);
2129  }
2130  os << "\ncon: ";
2131  for (unsigned i = 0; i < con.size(); ++i) {
2132  if (i > 0)
2133  os << ", ";
2134  con[i].print(os);
2135  }
2136  os << '\n';
2137  for (unsigned row = 0, e = getNumRows(); row < e; ++row) {
2138  if (row > 0)
2139  os << ", ";
2140  os << "r" << row << ": " << rowUnknown[row];
2141  }
2142  os << '\n';
2143  os << "c0: denom, c1: const";
2144  for (unsigned col = 2, e = getNumColumns(); col < e; ++col)
2145  os << ", c" << col << ": " << colUnknown[col];
2146  os << '\n';
2147  for (unsigned row = 0, numRows = getNumRows(); row < numRows; ++row) {
2148  for (unsigned col = 0, numCols = getNumColumns(); col < numCols; ++col)
2149  os << tableau(row, col) << '\t';
2150  os << '\n';
2151  }
2152  os << '\n';
2153 }
2154 
2155 void SimplexBase::dump() const { print(llvm::errs()); }
2156 
2158  if (isEmpty())
2159  return true;
2160 
2161  for (unsigned i = 0, e = rel.getNumInequalities(); i < e; ++i)
2163  return false;
2164 
2165  for (unsigned i = 0, e = rel.getNumEqualities(); i < e; ++i)
2166  if (!isRedundantEquality(rel.getEquality(i)))
2167  return false;
2168 
2169  return true;
2170 }
2171 
2172 /// Returns the type of the inequality with coefficients `coeffs`.
2173 /// Possible types are:
2174 /// Redundant The inequality is satisfied by all points in the polytope
2175 /// Cut The inequality is satisfied by some points, but not by others
2176 /// Separate The inequality is not satisfied by any point
2177 ///
2178 /// Internally, this computes the minimum and the maximum the inequality with
2179 /// coefficients `coeffs` can take. If the minimum is >= 0, the inequality holds
2180 /// for all points in the polytope, so it is redundant. If the minimum is <= 0
2181 /// and the maximum is >= 0, the points in between the minimum and the
2182 /// inequality do not satisfy it, the points in between the inequality and the
2183 /// maximum satisfy it. Hence, it is a cut inequality. If both are < 0, no
2184 /// points of the polytope satisfy the inequality, which means it is a separate
2185 /// inequality.
2188  if (minimum.isBounded() && *minimum >= Fraction(0, 1)) {
2189  return IneqType::Redundant;
2190  }
2192  if ((!minimum.isBounded() || *minimum <= Fraction(0, 1)) &&
2193  (!maximum.isBounded() || *maximum >= Fraction(0, 1))) {
2194  return IneqType::Cut;
2195  }
2196  return IneqType::Separate;
2197 }
2198 
2199 /// Checks whether the type of the inequality with coefficients `coeffs`
2200 /// is Redundant.
2202  assert(!empty &&
2203  "It is not meaningful to ask about redundancy in an empty set!");
2204  return findIneqType(coeffs) == IneqType::Redundant;
2205 }
2206 
2207 /// Check whether the equality given by `coeffs == 0` is redundant given
2208 /// the existing constraints. This is redundant when `coeffs` is already
2209 /// always zero under the existing constraints. `coeffs` is always zero
2210 /// when the minimum and maximum value that `coeffs` can take are both zero.
2212  assert(!empty &&
2213  "It is not meaningful to ask about redundancy in an empty set!");
2216  assert((!minimum.isEmpty() && !maximum.isEmpty()) &&
2217  "Optima should be non-empty for a non-empty set");
2218  return minimum.isBounded() && maximum.isBounded() &&
2219  *maximum == Fraction(0, 1) && *minimum == Fraction(0, 1);
2220 }
static Value max(ImplicitLocOpBuilder &builder, Value value, Value bound)
static LLVM_ATTRIBUTE_UNUSED SmallVector< MPInt, 8 > scaleAndAddForAssert(ArrayRef< MPInt > a, const MPInt &scale, ArrayRef< MPInt > b)
Definition: Simplex.cpp:41
static bool isRangeDivisibleBy(ArrayRef< MPInt > range, const MPInt &divisor)
Definition: Simplex.cpp:369
const int nullIndex
Definition: Simplex.cpp:36
MPInt normalizeRow(unsigned row, unsigned nCols)
Divide the first nCols of the specified row by their GCD.
Definition: Matrix.cpp:545
static IntMatrix identity(unsigned dimension)
Return the identity matrix of the specified dimension.
Definition: Matrix.cpp:452
An IntegerRelation represents the set of points from a PresburgerSpace that satisfy a list of affine ...
ArrayRef< MPInt > getEquality(unsigned idx) const
void addInequality(ArrayRef< MPInt > inEq)
Adds an inequality (>= 0) from the coefficients specified in inEq.
void truncate(const CountsSnapshot &counts)
void addLocalFloorDiv(ArrayRef< MPInt > dividend, const MPInt &divisor)
Adds a new local variable as the floordiv of an affine function of other variables,...
ArrayRef< MPInt > getInequality(unsigned idx) const
DivisionRepr getLocalReprs(std::vector< MaybeLocalRepr > *repr=nullptr) const
Returns a DivisonRepr representing the division representation of local variables in the constraint s...
void undoLastConstraint() final
Undo the addition of the last constraint.
Definition: Simplex.cpp:1207
LogicalResult moveRowUnknownToColumn(unsigned row)
Try to move the specified row to column orientation while preserving the lexicopositivity of the basi...
Definition: Simplex.cpp:773
void addInequality(ArrayRef< MPInt > coeffs) final
Add an inequality to the tableau.
Definition: Simplex.cpp:1586
LogicalResult addCut(unsigned row)
Given a row that has a non-integer sample value, add an inequality to cut away this fractional sample...
Definition: Simplex.cpp:278
unsigned getLexMinPivotColumn(unsigned row, unsigned colA, unsigned colB) const
Given two potential pivot columns for a row, return the one that results in the lexicographically sma...
Definition: Simplex.cpp:789
unsigned getSnapshot()
Get a snapshot of the current state. This is used for rolling back.
Definition: Simplex.h:429
void appendSymbol()
Add new symbolic variables to the end of the list of variables.
Definition: Simplex.cpp:362
MaybeOptimum< SmallVector< MPInt, 8 > > findIntegerLexMin()
Return the lexicographically minimum integer solution to the constraints.
Definition: Simplex.cpp:303
bool isRedundantInequality(ArrayRef< MPInt > coeffs)
Definition: Simplex.cpp:340
MaybeOptimum< SmallVector< Fraction, 8 > > findRationalLexMin()
Return the lexicographically minimum rational solution to the constraints.
Definition: Simplex.cpp:232
bool isSeparateInequality(ArrayRef< MPInt > coeffs)
Return whether the specified inequality is redundant/separate for the polytope.
Definition: Simplex.cpp:334
This class provides support for multi-precision arithmetic.
Definition: MPInt.h:87
unsigned getNumRows() const
Definition: Matrix.h:85
void swapColumns(unsigned column, unsigned otherColumn)
Swap the given columns.
Definition: Matrix.cpp:122
unsigned appendExtraRow()
Add an extra row at the bottom of the matrix and return its position.
Definition: Matrix.cpp:67
MutableArrayRef< T > getRow(unsigned row)
Get a [Mutable]ArrayRef corresponding to the specified row.
Definition: Matrix.cpp:132
void resizeVertically(unsigned newNRows)
Definition: Matrix.cpp:106
void swapRows(unsigned row, unsigned otherRow)
Swap the given rows.
Definition: Matrix.cpp:112
void resizeHorizontally(unsigned newNColumns)
Definition: Matrix.cpp:92
void reserveRows(unsigned rows)
Reserve enough space to resize to the specified number of rows without reallocations.
Definition: Matrix.cpp:62
void addToRow(unsigned sourceRow, unsigned targetRow, const T &scale)
Add scale multiples of the source row to the target row.
Definition: Matrix.cpp:301
bool isBounded() const
Definition: Utils.h:52
bool isUnbounded() const
Definition: Utils.h:53
This class represents a multi-affine function with the domain as Z^d, where d is the number of domain...
Definition: PWMAFunction.h:41
const PresburgerSpace & getSpace() const
Definition: PWMAFunction.h:168
void addPiece(const Piece &piece)
unsigned getNumOutputs() const
Definition: PWMAFunction.h:178
void unionInPlace(const IntegerRelation &disjunct)
Mutate this set, turning it into the union of this set and the given disjunct.
PresburgerSpace is the space of all possible values of a tuple of integer valued variables/variables.
static PresburgerSpace getRelationSpace(unsigned numDomain=0, unsigned numRange=0, unsigned numSymbols=0, unsigned numLocals=0)
unsigned insertVar(VarKind kind, unsigned pos, unsigned num=1)
Insert num variables of the specified kind at position pos.
The Simplex class implements a version of the Simplex and Generalized Basis Reduction algorithms,...
Definition: Simplex.h:157
unsigned addZeroRow(bool makeRestricted=false)
Add a new row to the tableau and the associated data structures.
Definition: Simplex.cpp:105
bool isEmpty() const
Returns true if the tableau is empty (has conflicting constraints), false otherwise.
Definition: Simplex.cpp:1066
void appendVariable(unsigned count=1)
Add new variables to the end of the list of variables.
Definition: Simplex.cpp:1315
virtual void undoLastConstraint()=0
Undo the addition of the last constraint.
SmallVector< int, 8 > rowUnknown
These hold the indexes of the unknown at a given row or column position.
Definition: Simplex.h:362
SmallVector< SmallVector< int, 8 >, 8 > savedBases
Holds a vector of bases.
Definition: Simplex.h:353
void intersectIntegerRelation(const IntegerRelation &rel)
Add all the constraints from the given IntegerRelation.
Definition: Simplex.cpp:1330
SmallVector< UndoLogEntry, 8 > undoLog
Holds a log of operations, used for rolling back to a previous state.
Definition: Simplex.h:348
bool usingBigM
Stores whether or not a big M column is present in the tableau.
Definition: Simplex.h:330
unsigned getSnapshot() const
Get a snapshot of the current state.
Definition: Simplex.cpp:1132
void print(raw_ostream &os) const
Print the tableau's internal state.
Definition: Simplex.cpp:2120
UndoLogEntry
Enum to denote operations that need to be undone during rollback.
Definition: Simplex.h:305
unsigned getNumRows() const
Definition: Simplex.h:326
unsigned addRow(ArrayRef< MPInt > coeffs, bool makeRestricted=false)
Add a new row to the tableau and the associated data structures.
Definition: Simplex.cpp:119
const Unknown & unknownFromRow(unsigned row) const
Returns the unknown associated with row.
Definition: Simplex.cpp:85
SmallVector< int, 8 > colUnknown
Definition: Simplex.h:362
SmallVector< Unknown, 8 > var
Definition: Simplex.h:365
virtual void addInequality(ArrayRef< MPInt > coeffs)=0
Add an inequality to the tableau.
unsigned getSnapshotBasis()
Get a snapshot of the current state including the basis.
Definition: Simplex.cpp:1134
unsigned getNumFixedCols() const
Return the number of fixed columns, as described in the constructor above, this is the number of colu...
Definition: Simplex.h:325
SmallVector< Unknown, 8 > con
These hold information about each unknown.
Definition: Simplex.h:365
void markEmpty()
Mark the tableau as being empty.
Definition: Simplex.cpp:1089
bool empty
This is true if the tableau has been detected to be empty, false otherwise.
Definition: Simplex.h:345
void swapColumns(unsigned i, unsigned j)
Definition: Simplex.cpp:1077
void removeLastConstraintRowOrientation()
Remove the last constraint, which must be in row orientation.
Definition: Simplex.cpp:1146
std::optional< unsigned > findAnyPivotRow(unsigned col)
Return any row that this column can be pivoted with, ignoring tableau consistency.
Definition: Simplex.cpp:1169
void addEquality(ArrayRef< MPInt > coeffs)
Add an equality to the tableau.
Definition: Simplex.cpp:1119
const Unknown & unknownFromColumn(unsigned col) const
Returns the unknown associated with col.
Definition: Simplex.cpp:80
void rollback(unsigned snapshot)
Rollback to a snapshot. This invalidates all later snapshots.
Definition: Simplex.cpp:1285
IntMatrix tableau
The matrix representing the tableau.
Definition: Simplex.h:341
void pivot(unsigned row, unsigned col)
Pivot the row with the column.
Definition: Simplex.cpp:947
void swapRows(unsigned i, unsigned j)
Swap the two rows/columns in the tableau and associated data structures.
Definition: Simplex.cpp:1068
void undo(UndoLogEntry entry)
Undo the operation represented by the log entry.
Definition: Simplex.cpp:1223
void addDivisionVariable(ArrayRef< MPInt > coeffs, const MPInt &denom)
Append a new variable to the simplex and constrain it such that its only integer value is the floor d...
Definition: Simplex.cpp:1298
const Unknown & unknownFromIndex(int index) const
Returns the unknown associated with index.
Definition: Simplex.cpp:75
unsigned nSymbol
The number of parameters.
Definition: Simplex.h:338
unsigned nRedundant
The number of redundant rows in the tableau.
Definition: Simplex.h:334
unsigned getNumVariables() const
Returns the number of variables in the tableau.
Definition: Simplex.cpp:1127
void swapRowWithCol(unsigned row, unsigned col)
Swap the row with the column in the tableau's data structures but not the tableau itself.
Definition: Simplex.cpp:910
unsigned getNumColumns() const
Definition: Simplex.h:327
unsigned getNumConstraints() const
Returns the number of constraints in the tableau.
Definition: Simplex.cpp:1128
Takes a snapshot of the simplex state on construction and rolls back to the snapshot on destruction.
Definition: Simplex.h:878
The Simplex class uses the Normal pivot rule and supports integer emptiness checks as well as detecti...
Definition: Simplex.h:695
MaybeOptimum< Fraction > computeOptimum(Direction direction, ArrayRef< MPInt > coeffs)
Compute the maximum or minimum value of the given expression, depending on direction.
Definition: Simplex.cpp:1358
bool isMarkedRedundant(unsigned constraintIndex) const
Returns whether the specified constraint has been marked as redundant.
Definition: Simplex.cpp:1403
void addInequality(ArrayRef< MPInt > coeffs) final
Add an inequality to the tableau.
Definition: Simplex.cpp:1106
std::optional< SmallVector< MPInt, 8 > > findIntegerSample()
Returns an integer sample point if one exists, or std::nullopt otherwise.
Definition: Simplex.cpp:1988
bool isFlatAlong(ArrayRef< MPInt > coeffs)
Check if the simplex takes only one rational value along the direction of coeffs.
Definition: Simplex.cpp:2107
bool isRedundantEquality(ArrayRef< MPInt > coeffs)
Check if the specified equality already holds in the polytope.
Definition: Simplex.cpp:2211
static Simplex makeProduct(const Simplex &a, const Simplex &b)
Make a tableau to represent a pair of points in the given tableaus, one in tableau A and one in B.
Definition: Simplex.cpp:1493
MaybeOptimum< Fraction > computeRowOptimum(Direction direction, unsigned row)
Compute the maximum or minimum value of the given row, depending on direction.
Definition: Simplex.cpp:1339
bool isRationalSubsetOf(const IntegerRelation &rel)
Returns true if this Simplex's polytope is a rational subset of rel.
Definition: Simplex.cpp:2157
std::pair< MaybeOptimum< MPInt >, MaybeOptimum< MPInt > > computeIntegerBounds(ArrayRef< MPInt > coeffs)
Returns a (min, max) pair denoting the minimum and maximum integer values of the given expression.
Definition: Simplex.cpp:2099
bool isRedundantInequality(ArrayRef< MPInt > coeffs)
Check if the specified inequality already holds in the polytope.
Definition: Simplex.cpp:2201
bool isBoundedAlongConstraint(unsigned constraintIndex)
Returns whether the perpendicular of the specified constraint is a is a direction along which the pol...
Definition: Simplex.cpp:1393
bool isUnbounded()
Returns true if the polytope is unbounded, i.e., extends to infinity in some direction.
Definition: Simplex.cpp:1465
IneqType findIneqType(ArrayRef< MPInt > coeffs)
Returns the type of the inequality with coefficients coeffs.
Definition: Simplex.cpp:2186
std::optional< SmallVector< Fraction, 8 > > getRationalSample() const
Returns the current sample point, which may contain non-integer (rational) coordinates.
Definition: Simplex.cpp:1565
std::optional< SmallVector< MPInt, 8 > > getSamplePointIfIntegral() const
Returns the current sample point if it is integral.
Definition: Simplex.cpp:1620
SymbolicLexOpt computeSymbolicIntegerLexMin()
The lexmin will be stored as a function lexopt from symbols to non-symbols in the result.
Definition: Simplex.cpp:533
Given a simplex for a polytope, construct a new simplex whose variables are identified with a pair of...
Definition: Simplex.cpp:1648
Fraction computeWidth(ArrayRef< MPInt > dir)
Compute max(dotProduct(dir, x - y)).
Definition: Simplex.cpp:1666
Fraction computeWidthAndDuals(ArrayRef< MPInt > dir, SmallVectorImpl< MPInt > &dual, MPInt &dualDenom)
Compute max(dotProduct(dir, x - y)) and save the dual variables for only the direction equalities to ...
Definition: Simplex.cpp:1675
void removeLastEquality()
Remove the last equality that was added through addEqualityForDirection.
Definition: Simplex.cpp:1734
void addEqualityForDirection(ArrayRef< MPInt > dir)
Add an equality dotProduct(dir, x - y) == 0.
Definition: Simplex.cpp:1659
GBRSimplex(const Simplex &originalSimplex)
Definition: Simplex.cpp:1652
SmallVector< AffineExpr, 4 > concat(ArrayRef< AffineExpr > a, ArrayRef< AffineExpr > b)
Return the vector that is the concatenation of a and b.
Definition: LinalgOps.cpp:2035
LLVM_ATTRIBUTE_ALWAYS_INLINE MPInt mod(const MPInt &lhs, const MPInt &rhs)
is always non-negative.
Definition: MPInt.h:393
void normalizeDiv(MutableArrayRef< MPInt > num, MPInt &denom)
Normalize the given (numerator, denominator) pair by dividing out the common factors between them.
Definition: Utils.cpp:365
MPInt ceil(const Fraction &f)
Definition: Fraction.h:76
MPInt normalizeRange(MutableArrayRef< MPInt > range)
Divide the range by its gcd and return the gcd.
Definition: Utils.cpp:356
MPInt floor(const Fraction &f)
Definition: Fraction.h:74
SmallVector< MPInt, 8 > getComplementIneq(ArrayRef< MPInt > ineq)
Return the complement of the given inequality.
Definition: Utils.cpp:381
LLVM_ATTRIBUTE_ALWAYS_INLINE MPInt lcm(const MPInt &a, const MPInt &b)
Returns the least common multiple of 'a' and 'b'.
Definition: MPInt.h:407
LLVM_ATTRIBUTE_ALWAYS_INLINE MPInt floorDiv(const MPInt &lhs, const MPInt &rhs)
Definition: MPInt.h:382
Include the generated interface declarations.
LogicalResult failure(bool isFailure=true)
Utility function to generate a LogicalResult.
Definition: LogicalResult.h:62
LogicalResult success(bool isSuccess=true)
Utility function to generate a LogicalResult.
Definition: LogicalResult.h:56
bool failed(LogicalResult result)
Utility function that returns true if the provided LogicalResult corresponds to a failure value.
Definition: LogicalResult.h:72
This class represents an efficient way to signal success or failure.
Definition: LogicalResult.h:26
bool failed() const
Returns true if the provided LogicalResult corresponds to a failure value.
Definition: LogicalResult.h:44
A class to represent fractions.
Definition: Fraction.h:28
MPInt getAsInteger() const
Definition: Fraction.h:48
The struct CountsSnapshot stores the count of each VarKind, and also of each constraint type.
An Unknown is either a variable or a constraint.
Definition: Simplex.h:238
Represents the result of a symbolic lexicographic optimization computation.
Definition: Simplex.h:533
PWMAFunction lexopt
This maps assignments of symbols to the corresponding lexopt.
Definition: Simplex.h:541
PresburgerSet unboundedDomain
Contains all assignments to the symbols that made the lexopt unbounded.
Definition: Simplex.h:545
Eliminates variable at the specified position using Fourier-Motzkin variable elimination.