MLIR  20.0.0git
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 
15 #include "llvm/ADT/DynamicAPInt.h"
16 #include "llvm/ADT/STLExtras.h"
17 #include "llvm/ADT/SmallBitVector.h"
18 #include "llvm/ADT/SmallVector.h"
19 #include "llvm/Support/Compiler.h"
20 #include "llvm/Support/ErrorHandling.h"
21 #include "llvm/Support/LogicalResult.h"
22 #include "llvm/Support/raw_ostream.h"
23 #include <cassert>
24 #include <functional>
25 #include <limits>
26 #include <optional>
27 #include <tuple>
28 #include <utility>
29 
30 using namespace mlir;
31 using namespace presburger;
32 
34 
36 
37 // Return a + scale*b;
38 LLVM_ATTRIBUTE_UNUSED
40 scaleAndAddForAssert(ArrayRef<DynamicAPInt> a, const DynamicAPInt &scale,
42  assert(a.size() == b.size());
44  res.reserve(a.size());
45  for (unsigned i = 0, e = a.size(); i < e; ++i)
46  res.emplace_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  var.reserve(nVar);
54  colUnknown.reserve(nVar + 1);
55  colUnknown.insert(colUnknown.begin(), getNumFixedCols(), nullIndex);
56  for (unsigned i = 0; i < nVar; ++i) {
57  var.emplace_back(Orientation::Column, /*restricted=*/false,
58  /*pos=*/getNumFixedCols() + i);
59  colUnknown.emplace_back(i);
60  }
61 }
62 
63 SimplexBase::SimplexBase(unsigned nVar, bool mustUseBigM,
64  const llvm::SmallBitVector &isSymbol)
65  : SimplexBase(nVar, mustUseBigM) {
66  assert(isSymbol.size() == nVar && "invalid bitmask!");
67  // Invariant: nSymbol is the number of symbols that have been marked
68  // already and these occupy the columns
69  // [getNumFixedCols(), getNumFixedCols() + nSymbol).
70  for (unsigned symbolIdx : isSymbol.set_bits()) {
71  var[symbolIdx].isSymbol = true;
72  swapColumns(var[symbolIdx].pos, getNumFixedCols() + nSymbol);
73  ++nSymbol;
74  }
75 }
76 
78  assert(index != nullIndex && "nullIndex passed to unknownFromIndex");
79  return index >= 0 ? var[index] : con[~index];
80 }
81 
83  assert(col < getNumColumns() && "Invalid column");
84  return unknownFromIndex(colUnknown[col]);
85 }
86 
87 const Simplex::Unknown &SimplexBase::unknownFromRow(unsigned row) const {
88  assert(row < getNumRows() && "Invalid row");
89  return unknownFromIndex(rowUnknown[row]);
90 }
91 
93  assert(index != nullIndex && "nullIndex passed to unknownFromIndex");
94  return index >= 0 ? var[index] : con[~index];
95 }
96 
98  assert(col < getNumColumns() && "Invalid column");
99  return unknownFromIndex(colUnknown[col]);
100 }
101 
103  assert(row < getNumRows() && "Invalid row");
104  return unknownFromIndex(rowUnknown[row]);
105 }
106 
107 unsigned SimplexBase::addZeroRow(bool makeRestricted) {
108  // Resize the tableau to accommodate the extra row.
109  unsigned newRow = tableau.appendExtraRow();
110  assert(getNumRows() == getNumRows() && "Inconsistent tableau size");
111  rowUnknown.emplace_back(~con.size());
112  con.emplace_back(Orientation::Row, makeRestricted, newRow);
114  tableau(newRow, 0) = 1;
115  return newRow;
116 }
117 
118 /// Add a new row to the tableau corresponding to the given constant term and
119 /// list of coefficients. The coefficients are specified as a vector of
120 /// (variable index, coefficient) pairs.
122  bool makeRestricted) {
123  assert(coeffs.size() == var.size() + 1 &&
124  "Incorrect number of coefficients!");
125  assert(var.size() + getNumFixedCols() == getNumColumns() &&
126  "inconsistent column count!");
127 
128  unsigned newRow = addZeroRow(makeRestricted);
129  tableau(newRow, 1) = coeffs.back();
130  if (usingBigM) {
131  // When the lexicographic pivot rule is used, instead of the variables
132  //
133  // x, y, z ...
134  //
135  // we internally use the variables
136  //
137  // M, M + x, M + y, M + z, ...
138  //
139  // where M is the big M parameter. As such, when the user tries to add
140  // a row ax + by + cz + d, we express it in terms of our internal variables
141  // as -(a + b + c)M + a(M + x) + b(M + y) + c(M + z) + d.
142  //
143  // Symbols don't use the big M parameter since they do not get lex
144  // optimized.
145  DynamicAPInt bigMCoeff(0);
146  for (unsigned i = 0; i < coeffs.size() - 1; ++i)
147  if (!var[i].isSymbol)
148  bigMCoeff -= coeffs[i];
149  // The coefficient to the big M parameter is stored in column 2.
150  tableau(newRow, 2) = bigMCoeff;
151  }
152 
153  // Process each given variable coefficient.
154  for (unsigned i = 0; i < var.size(); ++i) {
155  unsigned pos = var[i].pos;
156  if (coeffs[i] == 0)
157  continue;
158 
159  if (var[i].orientation == Orientation::Column) {
160  // If a variable is in column position at column col, then we just add the
161  // coefficient for that variable (scaled by the common row denominator) to
162  // the corresponding entry in the new row.
163  tableau(newRow, pos) += coeffs[i] * tableau(newRow, 0);
164  continue;
165  }
166 
167  // If the variable is in row position, we need to add that row to the new
168  // row, scaled by the coefficient for the variable, accounting for the two
169  // rows potentially having different denominators. The new denominator is
170  // the lcm of the two.
171  DynamicAPInt lcm = llvm::lcm(tableau(newRow, 0), tableau(pos, 0));
172  DynamicAPInt nRowCoeff = lcm / tableau(newRow, 0);
173  DynamicAPInt idxRowCoeff = coeffs[i] * (lcm / tableau(pos, 0));
174  tableau(newRow, 0) = lcm;
175  for (unsigned col = 1, e = getNumColumns(); col < e; ++col)
176  tableau(newRow, col) =
177  nRowCoeff * tableau(newRow, col) + idxRowCoeff * tableau(pos, col);
178  }
179 
180  tableau.normalizeRow(newRow);
181  // Push to undo log along with the index of the new constraint.
182  return con.size() - 1;
183 }
184 
185 namespace {
186 bool signMatchesDirection(const DynamicAPInt &elem, Direction direction) {
187  assert(elem != 0 && "elem should not be 0");
188  return direction == Direction::Up ? elem > 0 : elem < 0;
189 }
190 
191 Direction flippedDirection(Direction direction) {
192  return direction == Direction::Up ? Direction::Down : Simplex::Direction::Up;
193 }
194 } // namespace
195 
196 /// We simply make the tableau consistent while maintaining a lexicopositive
197 /// basis transform, and then return the sample value. If the tableau becomes
198 /// empty, we return empty.
199 ///
200 /// Let the variables be x = (x_1, ... x_n).
201 /// Let the basis unknowns be y = (y_1, ... y_n).
202 /// We have that x = A*y + b for some n x n matrix A and n x 1 column vector b.
203 ///
204 /// As we will show below, A*y is either zero or lexicopositive.
205 /// Adding a lexicopositive vector to b will make it lexicographically
206 /// greater, so A*y + b is always equal to or lexicographically greater than b.
207 /// Thus, since we can attain x = b, that is the lexicographic minimum.
208 ///
209 /// We have that every column in A is lexicopositive, i.e., has at least
210 /// one non-zero element, with the first such element being positive. Since for
211 /// the tableau to be consistent we must have non-negative sample values not
212 /// only for the constraints but also for the variables, we also have x >= 0 and
213 /// y >= 0, by which we mean every element in these vectors is non-negative.
214 ///
215 /// Proof that if every column in A is lexicopositive, and y >= 0, then
216 /// A*y is zero or lexicopositive. Begin by considering A_1, the first row of A.
217 /// If this row is all zeros, then (A*y)_1 = (A_1)*y = 0; proceed to the next
218 /// row. If we run out of rows, A*y is zero and we are done; otherwise, we
219 /// encounter some row A_i that has a non-zero element. Every column is
220 /// lexicopositive and so has some positive element before any negative elements
221 /// occur, so the element in this row for any column, if non-zero, must be
222 /// positive. Consider (A*y)_i = (A_i)*y. All the elements in both vectors are
223 /// non-negative, so if this is non-zero then it must be positive. Then the
224 /// first non-zero element of A*y is positive so A*y is lexicopositive.
225 ///
226 /// Otherwise, if (A_i)*y is zero, then for every column j that had a non-zero
227 /// element in A_i, y_j is zero. Thus these columns have no contribution to A*y
228 /// and we can completely ignore these columns of A. We now continue downwards,
229 /// looking for rows of A that have a non-zero element other than in the ignored
230 /// columns. If we find one, say A_k, once again these elements must be positive
231 /// since they are the first non-zero element in each of these columns, so if
232 /// (A_k)*y is not zero then we have that A*y is lexicopositive and if not we
233 /// add these to the set of ignored columns and continue to the next row. If we
234 /// run out of rows, then A*y is zero and we are done.
236  if (restoreRationalConsistency().failed()) {
237  markEmpty();
238  return OptimumKind::Empty;
239  }
240  return getRationalSample();
241 }
242 
243 /// Given a row that has a non-integer sample value, add an inequality such
244 /// that this fractional sample value is cut away from the polytope. The added
245 /// inequality will be such that no integer points are removed. i.e., the
246 /// integer lexmin, if it exists, is the same with and without this constraint.
247 ///
248 /// Let the row be
249 /// (c + coeffM*M + a_1*s_1 + ... + a_m*s_m + b_1*y_1 + ... + b_n*y_n)/d,
250 /// where s_1, ... s_m are the symbols and
251 /// y_1, ... y_n are the other basis unknowns.
252 ///
253 /// For this to be an integer, we want
254 /// coeffM*M + a_1*s_1 + ... + a_m*s_m + b_1*y_1 + ... + b_n*y_n = -c (mod d)
255 /// Note that this constraint must always hold, independent of the basis,
256 /// becuse the row unknown's value always equals this expression, even if *we*
257 /// later compute the sample value from a different expression based on a
258 /// different basis.
259 ///
260 /// Let us assume that M has a factor of d in it. Imposing this constraint on M
261 /// does not in any way hinder us from finding a value of M that is big enough.
262 /// Moreover, this function is only called when the symbolic part of the sample,
263 /// a_1*s_1 + ... + a_m*s_m, is known to be an integer.
264 ///
265 /// Also, we can safely reduce the coefficients modulo d, so we have:
266 ///
267 /// (b_1%d)y_1 + ... + (b_n%d)y_n = (-c%d) + k*d for some integer `k`
268 ///
269 /// Note that all coefficient modulos here are non-negative. Also, all the
270 /// unknowns are non-negative here as both constraints and variables are
271 /// non-negative in LexSimplexBase. (We used the big M trick to make the
272 /// variables non-negative). Therefore, the LHS here is non-negative.
273 /// Since 0 <= (-c%d) < d, k is the quotient of dividing the LHS by d and
274 /// is therefore non-negative as well.
275 ///
276 /// So we have
277 /// ((b_1%d)y_1 + ... + (b_n%d)y_n - (-c%d))/d >= 0.
278 ///
279 /// The constraint is violated when added (it would be useless otherwise)
280 /// so we immediately try to move it to a column.
281 LogicalResult LexSimplexBase::addCut(unsigned row) {
282  DynamicAPInt d = tableau(row, 0);
283  unsigned cutRow = addZeroRow(/*makeRestricted=*/true);
284  tableau(cutRow, 0) = d;
285  tableau(cutRow, 1) = -mod(-tableau(row, 1), d); // -c%d.
286  tableau(cutRow, 2) = 0;
287  for (unsigned col = 3 + nSymbol, e = getNumColumns(); col < e; ++col)
288  tableau(cutRow, col) = mod(tableau(row, col), d); // b_i%d.
289  return moveRowUnknownToColumn(cutRow);
290 }
291 
292 std::optional<unsigned> LexSimplex::maybeGetNonIntegralVarRow() const {
293  for (const Unknown &u : var) {
294  if (u.orientation == Orientation::Column)
295  continue;
296  // If the sample value is of the form (a/d)M + b/d, we need b to be
297  // divisible by d. We assume M contains all possible
298  // factors and is divisible by everything.
299  unsigned row = u.pos;
300  if (tableau(row, 1) % tableau(row, 0) != 0)
301  return row;
302  }
303  return {};
304 }
305 
307  // We first try to make the tableau consistent.
308  if (restoreRationalConsistency().failed())
309  return OptimumKind::Empty;
310 
311  // Then, if the sample value is integral, we are done.
312  while (std::optional<unsigned> maybeRow = maybeGetNonIntegralVarRow()) {
313  // Otherwise, for the variable whose row has a non-integral sample value,
314  // we add a cut, a constraint that remove this rational point
315  // while preserving all integer points, thus keeping the lexmin the same.
316  // We then again try to make the tableau with the new constraint
317  // consistent. This continues until the tableau becomes empty, in which
318  // case there is no integer point, or until there are no variables with
319  // non-integral sample values.
320  //
321  // Failure indicates that the tableau became empty, which occurs when the
322  // polytope is integer empty.
323  if (addCut(*maybeRow).failed())
324  return OptimumKind::Empty;
325  if (restoreRationalConsistency().failed())
326  return OptimumKind::Empty;
327  }
328 
329  MaybeOptimum<SmallVector<Fraction, 8>> sample = getRationalSample();
330  assert(!sample.isEmpty() && "If we reached here the sample should exist!");
331  if (sample.isUnbounded())
332  return OptimumKind::Unbounded;
333  return llvm::to_vector<8>(
334  llvm::map_range(*sample, std::mem_fn(&Fraction::getAsInteger)));
335 }
336 
338  SimplexRollbackScopeExit scopeExit(*this);
339  addInequality(coeffs);
340  return findIntegerLexMin().isEmpty();
341 }
342 
344  return isSeparateInequality(getComplementIneq(coeffs));
345 }
346 
348 SymbolicLexSimplex::getSymbolicSampleNumerator(unsigned row) const {
350  sample.reserve(nSymbol + 1);
351  for (unsigned col = 3; col < 3 + nSymbol; ++col)
352  sample.emplace_back(tableau(row, col));
353  sample.emplace_back(tableau(row, 1));
354  return sample;
355 }
356 
358 SymbolicLexSimplex::getSymbolicSampleIneq(unsigned row) const {
359  SmallVector<DynamicAPInt, 8> sample = getSymbolicSampleNumerator(row);
360  // The inequality is equivalent to the GCD-normalized one.
361  normalizeRange(sample);
362  return sample;
363 }
364 
366  appendVariable();
367  swapColumns(3 + nSymbol, getNumColumns() - 1);
368  var.back().isSymbol = true;
369  nSymbol++;
370 }
371 
373  const DynamicAPInt &divisor) {
374  assert(divisor > 0 && "divisor must be positive!");
375  return llvm::all_of(
376  range, [divisor](const DynamicAPInt &x) { return x % divisor == 0; });
377 }
378 
379 bool SymbolicLexSimplex::isSymbolicSampleIntegral(unsigned row) const {
380  DynamicAPInt denom = tableau(row, 0);
381  return tableau(row, 1) % denom == 0 &&
382  isRangeDivisibleBy(tableau.getRow(row).slice(3, nSymbol), denom);
383 }
384 
385 /// This proceeds similarly to LexSimplexBase::addCut(). We are given a row that
386 /// has a symbolic sample value with fractional coefficients.
387 ///
388 /// Let the row be
389 /// (c + coeffM*M + sum_i a_i*s_i + sum_j b_j*y_j)/d,
390 /// where s_1, ... s_m are the symbols and
391 /// y_1, ... y_n are the other basis unknowns.
392 ///
393 /// As in LexSimplex::addCut, for this to be an integer, we want
394 ///
395 /// coeffM*M + sum_j b_j*y_j = -c + sum_i (-a_i*s_i) (mod d)
396 ///
397 /// This time, a_1*s_1 + ... + a_m*s_m may not be an integer. We find that
398 ///
399 /// sum_i (b_i%d)y_i = ((-c%d) + sum_i (-a_i%d)s_i)%d + k*d for some integer k
400 ///
401 /// where we take a modulo of the whole symbolic expression on the right to
402 /// bring it into the range [0, d - 1]. Therefore, as in addCut(),
403 /// k is the quotient on dividing the LHS by d, and since LHS >= 0, we have
404 /// k >= 0 as well. If all the a_i are divisible by d, then we can add the
405 /// constraint directly. Otherwise, we realize the modulo of the symbolic
406 /// expression by adding a division variable
407 ///
408 /// q = ((-c%d) + sum_i (-a_i%d)s_i)/d
409 ///
410 /// to the symbol domain, so the equality becomes
411 ///
412 /// sum_i (b_i%d)y_i = (-c%d) + sum_i (-a_i%d)s_i - q*d + k*d for some integer k
413 ///
414 /// So the cut is
415 /// (sum_i (b_i%d)y_i - (-c%d) - sum_i (-a_i%d)s_i + q*d)/d >= 0
416 /// This constraint is violated when added so we immediately try to move it to a
417 /// column.
418 LogicalResult SymbolicLexSimplex::addSymbolicCut(unsigned row) {
419  DynamicAPInt d = tableau(row, 0);
420  if (isRangeDivisibleBy(tableau.getRow(row).slice(3, nSymbol), d)) {
421  // The coefficients of symbols in the symbol numerator are divisible
422  // by the denominator, so we can add the constraint directly,
423  // i.e., ignore the symbols and add a regular cut as in addCut().
424  return addCut(row);
425  }
426 
427  // Construct the division variable `q = ((-c%d) + sum_i (-a_i%d)s_i)/d`.
429  divCoeffs.reserve(nSymbol + 1);
430  DynamicAPInt divDenom = d;
431  for (unsigned col = 3; col < 3 + nSymbol; ++col)
432  divCoeffs.emplace_back(mod(-tableau(row, col), divDenom)); // (-a_i%d)s_i
433  divCoeffs.emplace_back(mod(-tableau(row, 1), divDenom)); // -c%d.
434  normalizeDiv(divCoeffs, divDenom);
435 
436  domainSimplex.addDivisionVariable(divCoeffs, divDenom);
437  domainPoly.addLocalFloorDiv(divCoeffs, divDenom);
438 
439  // Update `this` to account for the additional symbol we just added.
440  appendSymbol();
441 
442  // Add the cut (sum_i (b_i%d)y_i - (-c%d) + sum_i -(-a_i%d)s_i + q*d)/d >= 0.
443  unsigned cutRow = addZeroRow(/*makeRestricted=*/true);
444  tableau(cutRow, 0) = d;
445  tableau(cutRow, 2) = 0;
446 
447  tableau(cutRow, 1) = -mod(-tableau(row, 1), d); // -(-c%d).
448  for (unsigned col = 3; col < 3 + nSymbol - 1; ++col)
449  tableau(cutRow, col) = -mod(-tableau(row, col), d); // -(-a_i%d)s_i.
450  tableau(cutRow, 3 + nSymbol - 1) = d; // q*d.
451 
452  for (unsigned col = 3 + nSymbol, e = getNumColumns(); col < e; ++col)
453  tableau(cutRow, col) = mod(tableau(row, col), d); // (b_i%d)y_i.
454  return moveRowUnknownToColumn(cutRow);
455 }
456 
457 void SymbolicLexSimplex::recordOutput(SymbolicLexOpt &result) const {
458  IntMatrix output(0, domainPoly.getNumVars() + 1);
459  output.reserveRows(result.lexopt.getNumOutputs());
460  for (const Unknown &u : var) {
461  if (u.isSymbol)
462  continue;
463 
464  if (u.orientation == Orientation::Column) {
465  // M + u has a sample value of zero so u has a sample value of -M, i.e,
466  // unbounded.
467  result.unboundedDomain.unionInPlace(domainPoly);
468  return;
469  }
470 
471  DynamicAPInt denom = tableau(u.pos, 0);
472  if (tableau(u.pos, 2) < denom) {
473  // M + u has a sample value of fM + something, where f < 1, so
474  // u = (f - 1)M + something, which has a negative coefficient for M,
475  // and so is unbounded.
476  result.unboundedDomain.unionInPlace(domainPoly);
477  return;
478  }
479  assert(tableau(u.pos, 2) == denom &&
480  "Coefficient of M should not be greater than 1!");
481 
482  SmallVector<DynamicAPInt, 8> sample = getSymbolicSampleNumerator(u.pos);
483  for (DynamicAPInt &elem : sample) {
484  assert(elem % denom == 0 && "coefficients must be integral!");
485  elem /= denom;
486  }
487  output.appendExtraRow(sample);
488  }
489 
490  // Store the output in a MultiAffineFunction and add it the result.
491  PresburgerSpace funcSpace = result.lexopt.getSpace();
492  funcSpace.insertVar(VarKind::Local, 0, domainPoly.getNumLocalVars());
493 
494  result.lexopt.addPiece(
495  {PresburgerSet(domainPoly),
496  MultiAffineFunction(funcSpace, output, domainPoly.getLocalReprs())});
497 }
498 
499 std::optional<unsigned> SymbolicLexSimplex::maybeGetAlwaysViolatedRow() {
500  // First look for rows that are clearly violated just from the big M
501  // coefficient, without needing to perform any simplex queries on the domain.
502  for (unsigned row = 0, e = getNumRows(); row < e; ++row)
503  if (tableau(row, 2) < 0)
504  return row;
505 
506  for (unsigned row = 0, e = getNumRows(); row < e; ++row) {
507  if (tableau(row, 2) > 0)
508  continue;
509  if (domainSimplex.isSeparateInequality(getSymbolicSampleIneq(row))) {
510  // Sample numerator always takes negative values in the symbol domain.
511  return row;
512  }
513  }
514  return {};
515 }
516 
517 std::optional<unsigned> SymbolicLexSimplex::maybeGetNonIntegralVarRow() {
518  for (const Unknown &u : var) {
519  if (u.orientation == Orientation::Column)
520  continue;
521  assert(!u.isSymbol && "Symbol should not be in row orientation!");
522  if (!isSymbolicSampleIntegral(u.pos))
523  return u.pos;
524  }
525  return {};
526 }
527 
528 /// The non-branching pivots are just the ones moving the rows
529 /// that are always violated in the symbol domain.
530 LogicalResult SymbolicLexSimplex::doNonBranchingPivots() {
531  while (std::optional<unsigned> row = maybeGetAlwaysViolatedRow())
532  if (moveRowUnknownToColumn(*row).failed())
533  return failure();
534  return success();
535 }
536 
539  /*numDomain=*/domainPoly.getNumDimVars(),
540  /*numRange=*/var.size() - nSymbol,
541  /*numSymbols=*/domainPoly.getNumSymbolVars()));
542 
543  /// The algorithm is more naturally expressed recursively, but we implement
544  /// it iteratively here to avoid potential issues with stack overflows in the
545  /// compiler. We explicitly maintain the stack frames in a vector.
546  ///
547  /// To "recurse", we store the current "stack frame", i.e., state variables
548  /// that we will need when we "return", into `stack`, increment `level`, and
549  /// `continue`. To "tail recurse", we just `continue`.
550  /// To "return", we decrement `level` and `continue`.
551  ///
552  /// When there is no stack frame for the current `level`, this indicates that
553  /// we have just "recursed" or "tail recursed". When there does exist one,
554  /// this indicates that we have just "returned" from recursing. There is only
555  /// one point at which non-tail calls occur so we always "return" there.
556  unsigned level = 1;
557  struct StackFrame {
558  int splitIndex;
559  unsigned snapshot;
560  unsigned domainSnapshot;
561  IntegerRelation::CountsSnapshot domainPolyCounts;
562  };
564 
565  while (level > 0) {
566  assert(level >= stack.size());
567  if (level > stack.size()) {
568  if (empty || domainSimplex.findIntegerLexMin().isEmpty()) {
569  // No integer points; return.
570  --level;
571  continue;
572  }
573 
574  if (doNonBranchingPivots().failed()) {
575  // Could not find pivots for violated constraints; return.
576  --level;
577  continue;
578  }
579 
580  SmallVector<DynamicAPInt, 8> symbolicSample;
581  unsigned splitRow = 0;
582  for (unsigned e = getNumRows(); splitRow < e; ++splitRow) {
583  if (tableau(splitRow, 2) > 0)
584  continue;
585  assert(tableau(splitRow, 2) == 0 &&
586  "Non-branching pivots should have been handled already!");
587 
588  symbolicSample = getSymbolicSampleIneq(splitRow);
589  if (domainSimplex.isRedundantInequality(symbolicSample))
590  continue;
591 
592  // It's neither redundant nor separate, so it takes both positive and
593  // negative values, and hence constitutes a row for which we need to
594  // split the domain and separately run each case.
595  assert(!domainSimplex.isSeparateInequality(symbolicSample) &&
596  "Non-branching pivots should have been handled already!");
597  break;
598  }
599 
600  if (splitRow < getNumRows()) {
601  unsigned domainSnapshot = domainSimplex.getSnapshot();
602  IntegerRelation::CountsSnapshot domainPolyCounts =
603  domainPoly.getCounts();
604 
605  // First, we consider the part of the domain where the row is not
606  // violated. We don't have to do any pivots for the row in this case,
607  // but we record the additional constraint that defines this part of
608  // the domain.
609  domainSimplex.addInequality(symbolicSample);
610  domainPoly.addInequality(symbolicSample);
611 
612  // Recurse.
613  //
614  // On return, the basis as a set is preserved but not the internal
615  // ordering within rows or columns. Thus, we take note of the index of
616  // the Unknown that caused the split, which may be in a different
617  // row when we come back from recursing. We will need this to recurse
618  // on the other part of the split domain, where the row is violated.
619  //
620  // Note that we have to capture the index above and not a reference to
621  // the Unknown itself, since the array it lives in might get
622  // reallocated.
623  int splitIndex = rowUnknown[splitRow];
624  unsigned snapshot = getSnapshot();
625  stack.emplace_back(
626  StackFrame{splitIndex, snapshot, domainSnapshot, domainPolyCounts});
627  ++level;
628  continue;
629  }
630 
631  // The tableau is rationally consistent for the current domain.
632  // Now we look for non-integral sample values and add cuts for them.
633  if (std::optional<unsigned> row = maybeGetNonIntegralVarRow()) {
634  if (addSymbolicCut(*row).failed()) {
635  // No integral points; return.
636  --level;
637  continue;
638  }
639 
640  // Rerun this level with the added cut constraint (tail recurse).
641  continue;
642  }
643 
644  // Record output and return.
645  recordOutput(result);
646  --level;
647  continue;
648  }
649 
650  if (level == stack.size()) {
651  // We have "returned" from "recursing".
652  const StackFrame &frame = stack.back();
653  domainPoly.truncate(frame.domainPolyCounts);
654  domainSimplex.rollback(frame.domainSnapshot);
655  rollback(frame.snapshot);
656  const Unknown &u = unknownFromIndex(frame.splitIndex);
657 
658  // Drop the frame. We don't need it anymore.
659  stack.pop_back();
660 
661  // Now we consider the part of the domain where the unknown `splitIndex`
662  // was negative.
663  assert(u.orientation == Orientation::Row &&
664  "The split row should have been returned to row orientation!");
665  SmallVector<DynamicAPInt, 8> splitIneq =
666  getComplementIneq(getSymbolicSampleIneq(u.pos));
667  normalizeRange(splitIneq);
668  if (moveRowUnknownToColumn(u.pos).failed()) {
669  // The unknown can't be made non-negative; return.
670  --level;
671  continue;
672  }
673 
674  // The unknown can be made negative; recurse with the corresponding domain
675  // constraints.
676  domainSimplex.addInequality(splitIneq);
677  domainPoly.addInequality(splitIneq);
678 
679  // We are now taking care of the second half of the domain and we don't
680  // need to do anything else here after returning, so it's a tail recurse.
681  continue;
682  }
683  }
684 
685  return result;
686 }
687 
688 bool LexSimplex::rowIsViolated(unsigned row) const {
689  if (tableau(row, 2) < 0)
690  return true;
691  if (tableau(row, 2) == 0 && tableau(row, 1) < 0)
692  return true;
693  return false;
694 }
695 
696 std::optional<unsigned> LexSimplex::maybeGetViolatedRow() const {
697  for (unsigned row = 0, e = getNumRows(); row < e; ++row)
698  if (rowIsViolated(row))
699  return row;
700  return {};
701 }
702 
703 /// We simply look for violated rows and keep trying to move them to column
704 /// orientation, which always succeeds unless the constraints have no solution
705 /// in which case we just give up and return.
706 LogicalResult LexSimplex::restoreRationalConsistency() {
707  if (empty)
708  return failure();
709  while (std::optional<unsigned> maybeViolatedRow = maybeGetViolatedRow())
710  if (moveRowUnknownToColumn(*maybeViolatedRow).failed())
711  return failure();
712  return success();
713 }
714 
715 // Move the row unknown to column orientation while preserving lexicopositivity
716 // of the basis transform. The sample value of the row must be non-positive.
717 //
718 // We only consider pivots where the pivot element is positive. Suppose no such
719 // pivot exists, i.e., some violated row has no positive coefficient for any
720 // basis unknown. The row can be represented as (s + c_1*u_1 + ... + c_n*u_n)/d,
721 // where d is the denominator, s is the sample value and the c_i are the basis
722 // coefficients. If s != 0, then since any feasible assignment of the basis
723 // satisfies u_i >= 0 for all i, and we have s < 0 as well as c_i < 0 for all i,
724 // any feasible assignment would violate this row and therefore the constraints
725 // have no solution.
726 //
727 // We can preserve lexicopositivity by picking the pivot column with positive
728 // pivot element that makes the lexicographically smallest change to the sample
729 // point.
730 //
731 // Proof. Let
732 // x = (x_1, ... x_n) be the variables,
733 // z = (z_1, ... z_m) be the constraints,
734 // y = (y_1, ... y_n) be the current basis, and
735 // define w = (x_1, ... x_n, z_1, ... z_m) = B*y + s.
736 // B is basically the simplex tableau of our implementation except that instead
737 // of only describing the transform to get back the non-basis unknowns, it
738 // defines the values of all the unknowns in terms of the basis unknowns.
739 // Similarly, s is the column for the sample value.
740 //
741 // Our goal is to show that each column in B, restricted to the first n
742 // rows, is lexicopositive after the pivot if it is so before. This is
743 // equivalent to saying the columns in the whole matrix are lexicopositive;
744 // there must be some non-zero element in every column in the first n rows since
745 // the n variables cannot be spanned without using all the n basis unknowns.
746 //
747 // Consider a pivot where z_i replaces y_j in the basis. Recall the pivot
748 // transform for the tableau derived for SimplexBase::pivot:
749 //
750 // pivot col other col pivot col other col
751 // pivot row a b -> pivot row 1/a -b/a
752 // other row c d other row c/a d - bc/a
753 //
754 // Similarly, a pivot results in B changing to B' and c to c'; the difference
755 // between the tableau and these matrices B and B' is that there is no special
756 // case for the pivot row, since it continues to represent the same unknown. The
757 // same formula applies for all rows:
758 //
759 // B'.col(j) = B.col(j) / B(i,j)
760 // B'.col(k) = B.col(k) - B(i,k) * B.col(j) / B(i,j) for k != j
761 // and similarly, s' = s - s_i * B.col(j) / B(i,j).
762 //
763 // If s_i == 0, then the sample value remains unchanged. Otherwise, if s_i < 0,
764 // the change in sample value when pivoting with column a is lexicographically
765 // smaller than that when pivoting with column b iff B.col(a) / B(i, a) is
766 // lexicographically smaller than B.col(b) / B(i, b).
767 //
768 // Since B(i, j) > 0, column j remains lexicopositive.
769 //
770 // For the other columns, suppose C.col(k) is not lexicopositive.
771 // This means that for some p, for all t < p,
772 // C(t,k) = 0 => B(t,k) = B(t,j) * B(i,k) / B(i,j) and
773 // C(t,k) < 0 => B(p,k) < B(t,j) * B(i,k) / B(i,j),
774 // which is in contradiction to the fact that B.col(j) / B(i,j) must be
775 // lexicographically smaller than B.col(k) / B(i,k), since it lexicographically
776 // minimizes the change in sample value.
777 LogicalResult LexSimplexBase::moveRowUnknownToColumn(unsigned row) {
778  std::optional<unsigned> maybeColumn;
779  for (unsigned col = 3 + nSymbol, e = getNumColumns(); col < e; ++col) {
780  if (tableau(row, col) <= 0)
781  continue;
782  maybeColumn =
783  !maybeColumn ? col : getLexMinPivotColumn(row, *maybeColumn, col);
784  }
785 
786  if (!maybeColumn)
787  return failure();
788 
789  pivot(row, *maybeColumn);
790  return success();
791 }
792 
793 unsigned LexSimplexBase::getLexMinPivotColumn(unsigned row, unsigned colA,
794  unsigned colB) const {
795  // First, let's consider the non-symbolic case.
796  // A pivot causes the following change. (in the diagram the matrix elements
797  // are shown as rationals and there is no common denominator used)
798  //
799  // pivot col big M col const col
800  // pivot row a p b
801  // other row c q d
802  // |
803  // v
804  //
805  // pivot col big M col const col
806  // pivot row 1/a -p/a -b/a
807  // other row c/a q - pc/a d - bc/a
808  //
809  // Let the sample value of the pivot row be s = pM + b before the pivot. Since
810  // the pivot row represents a violated constraint we know that s < 0.
811  //
812  // If the variable is a non-pivot column, its sample value is zero before and
813  // after the pivot.
814  //
815  // If the variable is the pivot column, then its sample value goes from 0 to
816  // (-p/a)M + (-b/a), i.e. 0 to -(pM + b)/a. Thus the change in the sample
817  // value is -s/a.
818  //
819  // If the variable is the pivot row, its sample value goes from s to 0, for a
820  // change of -s.
821  //
822  // If the variable is a non-pivot row, its sample value changes from
823  // qM + d to qM + d + (-pc/a)M + (-bc/a). Thus the change in sample value
824  // is -(pM + b)(c/a) = -sc/a.
825  //
826  // Thus the change in sample value is either 0, -s/a, -s, or -sc/a. Here -s is
827  // fixed for all calls to this function since the row and tableau are fixed.
828  // The callee just wants to compare the return values with the return value of
829  // other invocations of the same function. So the -s is common for all
830  // comparisons involved and can be ignored, since -s is strictly positive.
831  //
832  // Thus we take away this common factor and just return 0, 1/a, 1, or c/a as
833  // appropriate. This allows us to run the entire algorithm treating M
834  // symbolically, as the pivot to be performed does not depend on the value
835  // of M, so long as the sample value s is negative. Note that this is not
836  // because of any special feature of M; by the same argument, we ignore the
837  // symbols too. The caller ensure that the sample value s is negative for
838  // all possible values of the symbols.
839  auto getSampleChangeCoeffForVar = [this, row](unsigned col,
840  const Unknown &u) -> Fraction {
841  DynamicAPInt a = tableau(row, col);
842  if (u.orientation == Orientation::Column) {
843  // Pivot column case.
844  if (u.pos == col)
845  return {1, a};
846 
847  // Non-pivot column case.
848  return {0, 1};
849  }
850 
851  // Pivot row case.
852  if (u.pos == row)
853  return {1, 1};
854 
855  // Non-pivot row case.
856  DynamicAPInt c = tableau(u.pos, col);
857  return {c, a};
858  };
859 
860  for (const Unknown &u : var) {
861  Fraction changeA = getSampleChangeCoeffForVar(colA, u);
862  Fraction changeB = getSampleChangeCoeffForVar(colB, u);
863  if (changeA < changeB)
864  return colA;
865  if (changeA > changeB)
866  return colB;
867  }
868 
869  // If we reached here, both result in exactly the same changes, so it
870  // doesn't matter which we return.
871  return colA;
872 }
873 
874 /// Find a pivot to change the sample value of the row in the specified
875 /// direction. The returned pivot row will involve `row` if and only if the
876 /// unknown is unbounded in the specified direction.
877 ///
878 /// To increase (resp. decrease) the value of a row, we need to find a live
879 /// column with a non-zero coefficient. If the coefficient is positive, we need
880 /// to increase (decrease) the value of the column, and if the coefficient is
881 /// negative, we need to decrease (increase) the value of the column. Also,
882 /// we cannot decrease the sample value of restricted columns.
883 ///
884 /// If multiple columns are valid, we break ties by considering a lexicographic
885 /// ordering where we prefer unknowns with lower index.
886 std::optional<SimplexBase::Pivot>
887 Simplex::findPivot(int row, Direction direction) const {
888  std::optional<unsigned> col;
889  for (unsigned j = 2, e = getNumColumns(); j < e; ++j) {
890  DynamicAPInt elem = tableau(row, j);
891  if (elem == 0)
892  continue;
893 
894  if (unknownFromColumn(j).restricted &&
895  !signMatchesDirection(elem, direction))
896  continue;
897  if (!col || colUnknown[j] < colUnknown[*col])
898  col = j;
899  }
900 
901  if (!col)
902  return {};
903 
904  Direction newDirection =
905  tableau(row, *col) < 0 ? flippedDirection(direction) : direction;
906  std::optional<unsigned> maybePivotRow = findPivotRow(row, newDirection, *col);
907  return Pivot{maybePivotRow.value_or(row), *col};
908 }
909 
910 /// Swap the associated unknowns for the row and the column.
911 ///
912 /// First we swap the index associated with the row and column. Then we update
913 /// the unknowns to reflect their new position and orientation.
914 void SimplexBase::swapRowWithCol(unsigned row, unsigned col) {
915  std::swap(rowUnknown[row], colUnknown[col]);
916  Unknown &uCol = unknownFromColumn(col);
917  Unknown &uRow = unknownFromRow(row);
920  uCol.pos = col;
921  uRow.pos = row;
922 }
923 
924 void SimplexBase::pivot(Pivot pair) { pivot(pair.row, pair.column); }
925 
926 /// Pivot pivotRow and pivotCol.
927 ///
928 /// Let R be the pivot row unknown and let C be the pivot col unknown.
929 /// Since initially R = a*C + sum b_i * X_i
930 /// (where the sum is over the other column's unknowns, x_i)
931 /// C = (R - (sum b_i * X_i))/a
932 ///
933 /// Let u be some other row unknown.
934 /// u = c*C + sum d_i * X_i
935 /// So u = c*(R - sum b_i * X_i)/a + sum d_i * X_i
936 ///
937 /// This results in the following transform:
938 /// pivot col other col pivot col other col
939 /// pivot row a b -> pivot row 1/a -b/a
940 /// other row c d other row c/a d - bc/a
941 ///
942 /// Taking into account the common denominators p and q:
943 ///
944 /// pivot col other col pivot col other col
945 /// pivot row a/p b/p -> pivot row p/a -b/a
946 /// other row c/q d/q other row cp/aq (da - bc)/aq
947 ///
948 /// The pivot row transform is accomplished be swapping a with the pivot row's
949 /// common denominator and negating the pivot row except for the pivot column
950 /// element.
951 void SimplexBase::pivot(unsigned pivotRow, unsigned pivotCol) {
952  assert(pivotCol >= getNumFixedCols() && "Refusing to pivot invalid column");
953  assert(!unknownFromColumn(pivotCol).isSymbol);
954 
955  swapRowWithCol(pivotRow, pivotCol);
956  std::swap(tableau(pivotRow, 0), tableau(pivotRow, pivotCol));
957  // We need to negate the whole pivot row except for the pivot column.
958  if (tableau(pivotRow, 0) < 0) {
959  // If the denominator is negative, we negate the row by simply negating the
960  // denominator.
961  tableau(pivotRow, 0) = -tableau(pivotRow, 0);
962  tableau(pivotRow, pivotCol) = -tableau(pivotRow, pivotCol);
963  } else {
964  for (unsigned col = 1, e = getNumColumns(); col < e; ++col) {
965  if (col == pivotCol)
966  continue;
967  tableau(pivotRow, col) = -tableau(pivotRow, col);
968  }
969  }
970  tableau.normalizeRow(pivotRow);
971 
972  for (unsigned row = 0, numRows = getNumRows(); row < numRows; ++row) {
973  if (row == pivotRow)
974  continue;
975  if (tableau(row, pivotCol) == 0) // Nothing to do.
976  continue;
977  tableau(row, 0) *= tableau(pivotRow, 0);
978  for (unsigned col = 1, numCols = getNumColumns(); col < numCols; ++col) {
979  if (col == pivotCol)
980  continue;
981  // Add rather than subtract because the pivot row has been negated.
982  tableau(row, col) = tableau(row, col) * tableau(pivotRow, 0) +
983  tableau(row, pivotCol) * tableau(pivotRow, col);
984  }
985  tableau(row, pivotCol) *= tableau(pivotRow, pivotCol);
986  tableau.normalizeRow(row);
987  }
988 }
989 
990 /// Perform pivots until the unknown has a non-negative sample value or until
991 /// no more upward pivots can be performed. Return success if we were able to
992 /// bring the row to a non-negative sample value, and failure otherwise.
993 LogicalResult Simplex::restoreRow(Unknown &u) {
994  assert(u.orientation == Orientation::Row &&
995  "unknown should be in row position");
996 
997  while (tableau(u.pos, 1) < 0) {
998  std::optional<Pivot> maybePivot = findPivot(u.pos, Direction::Up);
999  if (!maybePivot)
1000  break;
1001 
1002  pivot(*maybePivot);
1003  if (u.orientation == Orientation::Column)
1004  return success(); // the unknown is unbounded above.
1005  }
1006  return success(tableau(u.pos, 1) >= 0);
1007 }
1008 
1009 /// Find a row that can be used to pivot the column in the specified direction.
1010 /// This returns an empty optional if and only if the column is unbounded in the
1011 /// specified direction (ignoring skipRow, if skipRow is set).
1012 ///
1013 /// If skipRow is set, this row is not considered, and (if it is restricted) its
1014 /// restriction may be violated by the returned pivot. Usually, skipRow is set
1015 /// because we don't want to move it to column position unless it is unbounded,
1016 /// and we are either trying to increase the value of skipRow or explicitly
1017 /// trying to make skipRow negative, so we are not concerned about this.
1018 ///
1019 /// If the direction is up (resp. down) and a restricted row has a negative
1020 /// (positive) coefficient for the column, then this row imposes a bound on how
1021 /// much the sample value of the column can change. Such a row with constant
1022 /// term c and coefficient f for the column imposes a bound of c/|f| on the
1023 /// change in sample value (in the specified direction). (note that c is
1024 /// non-negative here since the row is restricted and the tableau is consistent)
1025 ///
1026 /// We iterate through the rows and pick the row which imposes the most
1027 /// stringent bound, since pivoting with a row changes the row's sample value to
1028 /// 0 and hence saturates the bound it imposes. We break ties between rows that
1029 /// impose the same bound by considering a lexicographic ordering where we
1030 /// prefer unknowns with lower index value.
1031 std::optional<unsigned> Simplex::findPivotRow(std::optional<unsigned> skipRow,
1032  Direction direction,
1033  unsigned col) const {
1034  std::optional<unsigned> retRow;
1035  // Initialize these to zero in order to silence a warning about retElem and
1036  // retConst being used uninitialized in the initialization of `diff` below. In
1037  // reality, these are always initialized when that line is reached since these
1038  // are set whenever retRow is set.
1039  DynamicAPInt retElem, retConst;
1040  for (unsigned row = nRedundant, e = getNumRows(); row < e; ++row) {
1041  if (skipRow && row == *skipRow)
1042  continue;
1043  DynamicAPInt elem = tableau(row, col);
1044  if (elem == 0)
1045  continue;
1046  if (!unknownFromRow(row).restricted)
1047  continue;
1048  if (signMatchesDirection(elem, direction))
1049  continue;
1050  DynamicAPInt constTerm = tableau(row, 1);
1051 
1052  if (!retRow) {
1053  retRow = row;
1054  retElem = elem;
1055  retConst = constTerm;
1056  continue;
1057  }
1058 
1059  DynamicAPInt diff = retConst * elem - constTerm * retElem;
1060  if ((diff == 0 && rowUnknown[row] < rowUnknown[*retRow]) ||
1061  (diff != 0 && !signMatchesDirection(diff, direction))) {
1062  retRow = row;
1063  retElem = elem;
1064  retConst = constTerm;
1065  }
1066  }
1067  return retRow;
1068 }
1069 
1070 bool SimplexBase::isEmpty() const { return empty; }
1071 
1072 void SimplexBase::swapRows(unsigned i, unsigned j) {
1073  if (i == j)
1074  return;
1075  tableau.swapRows(i, j);
1076  std::swap(rowUnknown[i], rowUnknown[j]);
1077  unknownFromRow(i).pos = i;
1078  unknownFromRow(j).pos = j;
1079 }
1080 
1081 void SimplexBase::swapColumns(unsigned i, unsigned j) {
1082  assert(i < getNumColumns() && j < getNumColumns() &&
1083  "Invalid columns provided!");
1084  if (i == j)
1085  return;
1086  tableau.swapColumns(i, j);
1087  std::swap(colUnknown[i], colUnknown[j]);
1088  unknownFromColumn(i).pos = i;
1089  unknownFromColumn(j).pos = j;
1090 }
1091 
1092 /// Mark this tableau empty and push an entry to the undo stack.
1094  // If the set is already empty, then we shouldn't add another UnmarkEmpty log
1095  // entry, since in that case the Simplex will be erroneously marked as
1096  // non-empty when rolling back past this point.
1097  if (empty)
1098  return;
1099  undoLog.emplace_back(UndoLogEntry::UnmarkEmpty);
1100  empty = true;
1101 }
1102 
1103 /// Add an inequality to the tableau. If coeffs is c_0, c_1, ... c_n, where n
1104 /// is the current number of variables, then the corresponding inequality is
1105 /// c_n + c_0*x_0 + c_1*x_1 + ... + c_{n-1}*x_{n-1} >= 0.
1106 ///
1107 /// We add the inequality and mark it as restricted. We then try to make its
1108 /// sample value non-negative. If this is not possible, the tableau has become
1109 /// empty and we mark it as such.
1111  unsigned conIndex = addRow(coeffs, /*makeRestricted=*/true);
1112  LogicalResult result = restoreRow(con[conIndex]);
1113  if (result.failed())
1114  markEmpty();
1115 }
1116 
1117 /// Add an equality to the tableau. If coeffs is c_0, c_1, ... c_n, where n
1118 /// is the current number of variables, then the corresponding equality is
1119 /// c_n + c_0*x_0 + c_1*x_1 + ... + c_{n-1}*x_{n-1} == 0.
1120 ///
1121 /// We simply add two opposing inequalities, which force the expression to
1122 /// be zero.
1124  addInequality(coeffs);
1125  SmallVector<DynamicAPInt, 8> negatedCoeffs;
1126  negatedCoeffs.reserve(coeffs.size());
1127  for (const DynamicAPInt &coeff : coeffs)
1128  negatedCoeffs.emplace_back(-coeff);
1129  addInequality(negatedCoeffs);
1130 }
1131 
1132 unsigned SimplexBase::getNumVariables() const { return var.size(); }
1133 unsigned SimplexBase::getNumConstraints() const { return con.size(); }
1134 
1135 /// Return a snapshot of the current state. This is just the current size of the
1136 /// undo log.
1137 unsigned SimplexBase::getSnapshot() const { return undoLog.size(); }
1138 
1140  SmallVector<int, 8> basis;
1141  basis.reserve(colUnknown.size());
1142  for (int index : colUnknown) {
1143  if (index != nullIndex)
1144  basis.emplace_back(index);
1145  }
1146  savedBases.emplace_back(std::move(basis));
1147 
1148  undoLog.emplace_back(UndoLogEntry::RestoreBasis);
1149  return undoLog.size() - 1;
1150 }
1151 
1153  assert(con.back().orientation == Orientation::Row);
1154 
1155  // Move this unknown to the last row and remove the last row from the
1156  // tableau.
1157  swapRows(con.back().pos, getNumRows() - 1);
1158  // It is not strictly necessary to shrink the tableau, but for now we
1159  // maintain the invariant that the tableau has exactly getNumRows()
1160  // rows.
1162  rowUnknown.pop_back();
1163  con.pop_back();
1164 }
1165 
1166 // This doesn't find a pivot row only if the column has zero
1167 // coefficients for every row.
1168 //
1169 // If the unknown is a constraint, this can't happen, since it was added
1170 // initially as a row. Such a row could never have been pivoted to a column. So
1171 // a pivot row will always be found if we have a constraint.
1172 //
1173 // If we have a variable, then the column has zero coefficients for every row
1174 // iff no constraints have been added with a non-zero coefficient for this row.
1175 std::optional<unsigned> SimplexBase::findAnyPivotRow(unsigned col) {
1176  for (unsigned row = nRedundant, e = getNumRows(); row < e; ++row)
1177  if (tableau(row, col) != 0)
1178  return row;
1179  return {};
1180 }
1181 
1182 // It's not valid to remove the constraint by deleting the column since this
1183 // would result in an invalid basis.
1184 void Simplex::undoLastConstraint() {
1185  if (con.back().orientation == Orientation::Column) {
1186  // We try to find any pivot row for this column that preserves tableau
1187  // consistency (except possibly the column itself, which is going to be
1188  // deallocated anyway).
1189  //
1190  // If no pivot row is found in either direction, then the unknown is
1191  // unbounded in both directions and we are free to perform any pivot at
1192  // all. To do this, we just need to find any row with a non-zero
1193  // coefficient for the column. findAnyPivotRow will always be able to
1194  // find such a row for a constraint.
1195  unsigned column = con.back().pos;
1196  if (std::optional<unsigned> maybeRow =
1197  findPivotRow({}, Direction::Up, column)) {
1198  pivot(*maybeRow, column);
1199  } else if (std::optional<unsigned> maybeRow =
1200  findPivotRow({}, Direction::Down, column)) {
1201  pivot(*maybeRow, column);
1202  } else {
1203  std::optional<unsigned> row = findAnyPivotRow(column);
1204  assert(row && "Pivot should always exist for a constraint!");
1205  pivot(*row, column);
1206  }
1207  }
1209 }
1210 
1211 // It's not valid to remove the constraint by deleting the column since this
1212 // would result in an invalid basis.
1214  if (con.back().orientation == Orientation::Column) {
1215  // When removing the last constraint during a rollback, we just need to find
1216  // any pivot at all, i.e., any row with non-zero coefficient for the
1217  // column, because when rolling back a lexicographic simplex, we always
1218  // end by restoring the exact basis that was present at the time of the
1219  // snapshot, so what pivots we perform while undoing doesn't matter as
1220  // long as we get the unknown to row orientation and remove it.
1221  unsigned column = con.back().pos;
1222  std::optional<unsigned> row = findAnyPivotRow(column);
1223  assert(row && "Pivot should always exist for a constraint!");
1224  pivot(*row, column);
1225  }
1227 }
1228 
1230  if (entry == UndoLogEntry::RemoveLastConstraint) {
1231  // Simplex and LexSimplex handle this differently, so we call out to a
1232  // virtual function to handle this.
1234  } else if (entry == UndoLogEntry::RemoveLastVariable) {
1235  // Whenever we are rolling back the addition of a variable, it is guaranteed
1236  // that the variable will be in column position.
1237  //
1238  // We can see this as follows: any constraint that depends on this variable
1239  // was added after this variable was added, so the addition of such
1240  // constraints should already have been rolled back by the time we get to
1241  // rolling back the addition of the variable. Therefore, no constraint
1242  // currently has a component along the variable, so the variable itself must
1243  // be part of the basis.
1244  assert(var.back().orientation == Orientation::Column &&
1245  "Variable to be removed must be in column orientation!");
1246 
1247  if (var.back().isSymbol)
1248  nSymbol--;
1249 
1250  // Move this variable to the last column and remove the column from the
1251  // tableau.
1252  swapColumns(var.back().pos, getNumColumns() - 1);
1254  var.pop_back();
1255  colUnknown.pop_back();
1256  } else if (entry == UndoLogEntry::UnmarkEmpty) {
1257  empty = false;
1258  } else if (entry == UndoLogEntry::UnmarkLastRedundant) {
1259  nRedundant--;
1260  } else if (entry == UndoLogEntry::RestoreBasis) {
1261  assert(!savedBases.empty() && "No bases saved!");
1262 
1263  SmallVector<int, 8> basis = std::move(savedBases.back());
1264  savedBases.pop_back();
1265 
1266  for (int index : basis) {
1267  Unknown &u = unknownFromIndex(index);
1269  continue;
1270  for (unsigned col = getNumFixedCols(), e = getNumColumns(); col < e;
1271  col++) {
1272  assert(colUnknown[col] != nullIndex &&
1273  "Column should not be a fixed column!");
1274  if (llvm::is_contained(basis, colUnknown[col]))
1275  continue;
1276  if (tableau(u.pos, col) == 0)
1277  continue;
1278  pivot(u.pos, col);
1279  break;
1280  }
1281 
1282  assert(u.orientation == Orientation::Column && "No pivot found!");
1283  }
1284  }
1285 }
1286 
1287 /// Rollback to the specified snapshot.
1288 ///
1289 /// We undo all the log entries until the log size when the snapshot was taken
1290 /// is reached.
1291 void SimplexBase::rollback(unsigned snapshot) {
1292  while (undoLog.size() > snapshot) {
1293  undo(undoLog.back());
1294  undoLog.pop_back();
1295  }
1296 }
1297 
1298 /// We add the usual floor division constraints:
1299 /// `0 <= coeffs - denom*q <= denom - 1`, where `q` is the new division
1300 /// variable.
1301 ///
1302 /// This constrains the remainder `coeffs - denom*q` to be in the
1303 /// range `[0, denom - 1]`, which fixes the integer value of the quotient `q`.
1305  const DynamicAPInt &denom) {
1306  assert(denom > 0 && "Denominator must be positive!");
1307  appendVariable();
1308 
1309  SmallVector<DynamicAPInt, 8> ineq(coeffs);
1310  DynamicAPInt constTerm = ineq.back();
1311  ineq.back() = -denom;
1312  ineq.emplace_back(constTerm);
1313  addInequality(ineq);
1314 
1315  for (DynamicAPInt &coeff : ineq)
1316  coeff = -coeff;
1317  ineq.back() += denom - 1;
1318  addInequality(ineq);
1319 }
1320 
1321 void SimplexBase::appendVariable(unsigned count) {
1322  if (count == 0)
1323  return;
1324  var.reserve(var.size() + count);
1325  colUnknown.reserve(colUnknown.size() + count);
1326  for (unsigned i = 0; i < count; ++i) {
1327  var.emplace_back(Orientation::Column, /*restricted=*/false,
1328  /*pos=*/getNumColumns() + i);
1329  colUnknown.emplace_back(var.size() - 1);
1330  }
1332  undoLog.insert(undoLog.end(), count, UndoLogEntry::RemoveLastVariable);
1333 }
1334 
1335 /// Add all the constraints from the given IntegerRelation.
1337  assert(rel.getNumVars() == getNumVariables() &&
1338  "IntegerRelation must have same dimensionality as simplex");
1339  for (unsigned i = 0, e = rel.getNumInequalities(); i < e; ++i)
1340  addInequality(rel.getInequality(i));
1341  for (unsigned i = 0, e = rel.getNumEqualities(); i < e; ++i)
1342  addEquality(rel.getEquality(i));
1343 }
1344 
1346  unsigned row) {
1347  // Keep trying to find a pivot for the row in the specified direction.
1348  while (std::optional<Pivot> maybePivot = findPivot(row, direction)) {
1349  // If findPivot returns a pivot involving the row itself, then the optimum
1350  // is unbounded, so we return std::nullopt.
1351  if (maybePivot->row == row)
1352  return OptimumKind::Unbounded;
1353  pivot(*maybePivot);
1354  }
1355 
1356  // The row has reached its optimal sample value, which we return.
1357  // The sample value is the entry in the constant column divided by the common
1358  // denominator for this row.
1359  return Fraction(tableau(row, 1), tableau(row, 0));
1360 }
1361 
1362 /// Compute the optimum of the specified expression in the specified direction,
1363 /// or std::nullopt if it is unbounded.
1365  ArrayRef<DynamicAPInt> coeffs) {
1366  if (empty)
1367  return OptimumKind::Empty;
1368 
1369  SimplexRollbackScopeExit scopeExit(*this);
1370  unsigned conIndex = addRow(coeffs);
1371  unsigned row = con[conIndex].pos;
1372  return computeRowOptimum(direction, row);
1373 }
1374 
1376  Unknown &u) {
1377  if (empty)
1378  return OptimumKind::Empty;
1379  if (u.orientation == Orientation::Column) {
1380  unsigned column = u.pos;
1381  std::optional<unsigned> pivotRow = findPivotRow({}, direction, column);
1382  // If no pivot is returned, the constraint is unbounded in the specified
1383  // direction.
1384  if (!pivotRow)
1385  return OptimumKind::Unbounded;
1386  pivot(*pivotRow, column);
1387  }
1388 
1389  unsigned row = u.pos;
1390  MaybeOptimum<Fraction> optimum = computeRowOptimum(direction, row);
1391  if (u.restricted && direction == Direction::Down &&
1392  (optimum.isUnbounded() || *optimum < Fraction(0, 1))) {
1393  if (restoreRow(u).failed())
1394  llvm_unreachable("Could not restore row!");
1395  }
1396  return optimum;
1397 }
1398 
1399 bool Simplex::isBoundedAlongConstraint(unsigned constraintIndex) {
1400  assert(!empty && "It is not meaningful to ask whether a direction is bounded "
1401  "in an empty set.");
1402  // The constraint's perpendicular is already bounded below, since it is a
1403  // constraint. If it is also bounded above, we can return true.
1404  return computeOptimum(Direction::Up, con[constraintIndex]).isBounded();
1405 }
1406 
1407 /// Redundant constraints are those that are in row orientation and lie in
1408 /// rows 0 to nRedundant - 1.
1409 bool Simplex::isMarkedRedundant(unsigned constraintIndex) const {
1410  const Unknown &u = con[constraintIndex];
1411  return u.orientation == Orientation::Row && u.pos < nRedundant;
1412 }
1413 
1414 /// Mark the specified row redundant.
1415 ///
1416 /// This is done by moving the unknown to the end of the block of redundant
1417 /// rows (namely, to row nRedundant) and incrementing nRedundant to
1418 /// accomodate the new redundant row.
1419 void Simplex::markRowRedundant(Unknown &u) {
1420  assert(u.orientation == Orientation::Row &&
1421  "Unknown should be in row position!");
1422  assert(u.pos >= nRedundant && "Unknown is already marked redundant!");
1423  swapRows(u.pos, nRedundant);
1424  ++nRedundant;
1426 }
1427 
1428 /// Find a subset of constraints that is redundant and mark them redundant.
1429 void Simplex::detectRedundant(unsigned offset, unsigned count) {
1430  assert(offset + count <= con.size() && "invalid range!");
1431  // It is not meaningful to talk about redundancy for empty sets.
1432  if (empty)
1433  return;
1434 
1435  // Iterate through the constraints and check for each one if it can attain
1436  // negative sample values. If it can, it's not redundant. Otherwise, it is.
1437  // We mark redundant constraints redundant.
1438  //
1439  // Constraints that get marked redundant in one iteration are not respected
1440  // when checking constraints in later iterations. This prevents, for example,
1441  // two identical constraints both being marked redundant since each is
1442  // redundant given the other one. In this example, only the first of the
1443  // constraints that is processed will get marked redundant, as it should be.
1444  for (unsigned i = 0; i < count; ++i) {
1445  Unknown &u = con[offset + i];
1446  if (u.orientation == Orientation::Column) {
1447  unsigned column = u.pos;
1448  std::optional<unsigned> pivotRow =
1449  findPivotRow({}, Direction::Down, column);
1450  // If no downward pivot is returned, the constraint is unbounded below
1451  // and hence not redundant.
1452  if (!pivotRow)
1453  continue;
1454  pivot(*pivotRow, column);
1455  }
1456 
1457  unsigned row = u.pos;
1459  if (minimum.isUnbounded() || *minimum < Fraction(0, 1)) {
1460  // Constraint is unbounded below or can attain negative sample values and
1461  // hence is not redundant.
1462  if (restoreRow(u).failed())
1463  llvm_unreachable("Could not restore non-redundant row!");
1464  continue;
1465  }
1466 
1467  markRowRedundant(u);
1468  }
1469 }
1470 
1472  if (empty)
1473  return false;
1474 
1475  SmallVector<DynamicAPInt, 8> dir(var.size() + 1);
1476  for (unsigned i = 0; i < var.size(); ++i) {
1477  dir[i] = 1;
1478 
1480  return true;
1481 
1483  return true;
1484 
1485  dir[i] = 0;
1486  }
1487  return false;
1488 }
1489 
1490 /// Make a tableau to represent a pair of points in the original tableau.
1491 ///
1492 /// The product constraints and variables are stored as: first A's, then B's.
1493 ///
1494 /// The product tableau has row layout:
1495 /// A's redundant rows, B's redundant rows, A's other rows, B's other rows.
1496 ///
1497 /// It has column layout:
1498 /// denominator, constant, A's columns, B's columns.
1500  unsigned numVar = a.getNumVariables() + b.getNumVariables();
1501  unsigned numCon = a.getNumConstraints() + b.getNumConstraints();
1502  Simplex result(numVar);
1503 
1504  result.tableau.reserveRows(numCon);
1505  result.empty = a.empty || b.empty;
1506 
1507  auto concat = [](ArrayRef<Unknown> v, ArrayRef<Unknown> w) {
1508  SmallVector<Unknown, 8> result;
1509  result.reserve(v.size() + w.size());
1510  result.insert(result.end(), v.begin(), v.end());
1511  result.insert(result.end(), w.begin(), w.end());
1512  return result;
1513  };
1514  result.con = concat(a.con, b.con);
1515  result.var = concat(a.var, b.var);
1516 
1517  auto indexFromBIndex = [&](int index) {
1518  return index >= 0 ? a.getNumVariables() + index
1519  : ~(a.getNumConstraints() + ~index);
1520  };
1521 
1522  result.colUnknown.assign(2, nullIndex);
1523  for (unsigned i = 2, e = a.getNumColumns(); i < e; ++i) {
1524  result.colUnknown.emplace_back(a.colUnknown[i]);
1525  result.unknownFromIndex(result.colUnknown.back()).pos =
1526  result.colUnknown.size() - 1;
1527  }
1528  for (unsigned i = 2, e = b.getNumColumns(); i < e; ++i) {
1529  result.colUnknown.emplace_back(indexFromBIndex(b.colUnknown[i]));
1530  result.unknownFromIndex(result.colUnknown.back()).pos =
1531  result.colUnknown.size() - 1;
1532  }
1533 
1534  auto appendRowFromA = [&](unsigned row) {
1535  unsigned resultRow = result.tableau.appendExtraRow();
1536  for (unsigned col = 0, e = a.getNumColumns(); col < e; ++col)
1537  result.tableau(resultRow, col) = a.tableau(row, col);
1538  result.rowUnknown.emplace_back(a.rowUnknown[row]);
1539  result.unknownFromIndex(result.rowUnknown.back()).pos =
1540  result.rowUnknown.size() - 1;
1541  };
1542 
1543  // Also fixes the corresponding entry in rowUnknown and var/con (as the case
1544  // may be).
1545  auto appendRowFromB = [&](unsigned row) {
1546  unsigned resultRow = result.tableau.appendExtraRow();
1547  result.tableau(resultRow, 0) = b.tableau(row, 0);
1548  result.tableau(resultRow, 1) = b.tableau(row, 1);
1549 
1550  unsigned offset = a.getNumColumns() - 2;
1551  for (unsigned col = 2, e = b.getNumColumns(); col < e; ++col)
1552  result.tableau(resultRow, offset + col) = b.tableau(row, col);
1553  result.rowUnknown.emplace_back(indexFromBIndex(b.rowUnknown[row]));
1554  result.unknownFromIndex(result.rowUnknown.back()).pos =
1555  result.rowUnknown.size() - 1;
1556  };
1557 
1558  result.nRedundant = a.nRedundant + b.nRedundant;
1559  for (unsigned row = 0; row < a.nRedundant; ++row)
1560  appendRowFromA(row);
1561  for (unsigned row = 0; row < b.nRedundant; ++row)
1562  appendRowFromB(row);
1563  for (unsigned row = a.nRedundant, e = a.getNumRows(); row < e; ++row)
1564  appendRowFromA(row);
1565  for (unsigned row = b.nRedundant, e = b.getNumRows(); row < e; ++row)
1566  appendRowFromB(row);
1567 
1568  return result;
1569 }
1570 
1571 std::optional<SmallVector<Fraction, 8>> Simplex::getRationalSample() const {
1572  if (empty)
1573  return {};
1574 
1575  SmallVector<Fraction, 8> sample;
1576  sample.reserve(var.size());
1577  // Push the sample value for each variable into the vector.
1578  for (const Unknown &u : var) {
1579  if (u.orientation == Orientation::Column) {
1580  // If the variable is in column position, its sample value is zero.
1581  sample.emplace_back(0, 1);
1582  } else {
1583  // If the variable is in row position, its sample value is the
1584  // entry in the constant column divided by the denominator.
1585  DynamicAPInt denom = tableau(u.pos, 0);
1586  sample.emplace_back(tableau(u.pos, 1), denom);
1587  }
1588  }
1589  return sample;
1590 }
1591 
1593  addRow(coeffs, /*makeRestricted=*/true);
1594 }
1595 
1596 MaybeOptimum<SmallVector<Fraction, 8>> LexSimplex::getRationalSample() const {
1597  if (empty)
1598  return OptimumKind::Empty;
1599 
1600  SmallVector<Fraction, 8> sample;
1601  sample.reserve(var.size());
1602  // Push the sample value for each variable into the vector.
1603  for (const Unknown &u : var) {
1604  // When the big M parameter is being used, each variable x is represented
1605  // as M + x, so its sample value is finite if and only if it is of the
1606  // form 1*M + c. If the coefficient of M is not one then the sample value
1607  // is infinite, and we return an empty optional.
1608 
1609  if (u.orientation == Orientation::Column) {
1610  // If the variable is in column position, the sample value of M + x is
1611  // zero, so x = -M which is unbounded.
1612  return OptimumKind::Unbounded;
1613  }
1614 
1615  // If the variable is in row position, its sample value is the
1616  // entry in the constant column divided by the denominator.
1617  DynamicAPInt denom = tableau(u.pos, 0);
1618  if (usingBigM)
1619  if (tableau(u.pos, 2) != denom)
1620  return OptimumKind::Unbounded;
1621  sample.emplace_back(tableau(u.pos, 1), denom);
1622  }
1623  return sample;
1624 }
1625 
1626 std::optional<SmallVector<DynamicAPInt, 8>>
1628  // If the tableau is empty, no sample point exists.
1629  if (empty)
1630  return {};
1631 
1632  // The value will always exist since the Simplex is non-empty.
1633  SmallVector<Fraction, 8> rationalSample = *getRationalSample();
1634  SmallVector<DynamicAPInt, 8> integerSample;
1635  integerSample.reserve(var.size());
1636  for (const Fraction &coord : rationalSample) {
1637  // If the sample is non-integral, return std::nullopt.
1638  if (coord.num % coord.den != 0)
1639  return {};
1640  integerSample.emplace_back(coord.num / coord.den);
1641  }
1642  return integerSample;
1643 }
1644 
1645 /// Given a simplex for a polytope, construct a new simplex whose variables are
1646 /// identified with a pair of points (x, y) in the original polytope. Supports
1647 /// some operations needed for generalized basis reduction. In what follows,
1648 /// dotProduct(x, y) = x_1 * y_1 + x_2 * y_2 + ... x_n * y_n where n is the
1649 /// dimension of the original polytope.
1650 ///
1651 /// This supports adding equality constraints dotProduct(dir, x - y) == 0. It
1652 /// also supports rolling back this addition, by maintaining a snapshot stack
1653 /// that contains a snapshot of the Simplex's state for each equality, just
1654 /// before that equality was added.
1657 
1658 public:
1659  GBRSimplex(const Simplex &originalSimplex)
1660  : simplex(Simplex::makeProduct(originalSimplex, originalSimplex)),
1661  simplexConstraintOffset(simplex.getNumConstraints()) {}
1662 
1663  /// Add an equality dotProduct(dir, x - y) == 0.
1664  /// First pushes a snapshot for the current simplex state to the stack so
1665  /// that this can be rolled back later.
1667  assert(llvm::any_of(dir, [](const DynamicAPInt &x) { return x != 0; }) &&
1668  "Direction passed is the zero vector!");
1669  snapshotStack.emplace_back(simplex.getSnapshot());
1670  simplex.addEquality(getCoeffsForDirection(dir));
1671  }
1672  /// Compute max(dotProduct(dir, x - y)).
1674  MaybeOptimum<Fraction> maybeWidth =
1675  simplex.computeOptimum(Direction::Up, getCoeffsForDirection(dir));
1676  assert(maybeWidth.isBounded() && "Width should be bounded!");
1677  return *maybeWidth;
1678  }
1679 
1680  /// Compute max(dotProduct(dir, x - y)) and save the dual variables for only
1681  /// the direction equalities to `dual`.
1684  DynamicAPInt &dualDenom) {
1685  // We can't just call into computeWidth or computeOptimum since we need to
1686  // access the state of the tableau after computing the optimum, and these
1687  // functions rollback the insertion of the objective function into the
1688  // tableau before returning. We instead add a row for the objective function
1689  // ourselves, call into computeOptimum, compute the duals from the tableau
1690  // state, and finally rollback the addition of the row before returning.
1691  SimplexRollbackScopeExit scopeExit(simplex);
1692  unsigned conIndex = simplex.addRow(getCoeffsForDirection(dir));
1693  unsigned row = simplex.con[conIndex].pos;
1694  MaybeOptimum<Fraction> maybeWidth =
1695  simplex.computeRowOptimum(Simplex::Direction::Up, row);
1696  assert(maybeWidth.isBounded() && "Width should be bounded!");
1697  dualDenom = simplex.tableau(row, 0);
1698  dual.clear();
1699  dual.reserve((conIndex - simplexConstraintOffset) / 2);
1700 
1701  // The increment is i += 2 because equalities are added as two inequalities,
1702  // one positive and one negative. Each iteration processes one equality.
1703  for (unsigned i = simplexConstraintOffset; i < conIndex; i += 2) {
1704  // The dual variable for an inequality in column orientation is the
1705  // negative of its coefficient at the objective row. If the inequality is
1706  // in row orientation, the corresponding dual variable is zero.
1707  //
1708  // We want the dual for the original equality, which corresponds to two
1709  // inequalities: a positive inequality, which has the same coefficients as
1710  // the equality, and a negative equality, which has negated coefficients.
1711  //
1712  // Note that at most one of these inequalities can be in column
1713  // orientation because the column unknowns should form a basis and hence
1714  // must be linearly independent. If the positive inequality is in column
1715  // position, its dual is the dual corresponding to the equality. If the
1716  // negative inequality is in column position, the negation of its dual is
1717  // the dual corresponding to the equality. If neither is in column
1718  // position, then that means that this equality is redundant, and its dual
1719  // is zero.
1720  //
1721  // Note that it is NOT valid to perform pivots during the computation of
1722  // the duals. This entire dual computation must be performed on the same
1723  // tableau configuration.
1724  assert((simplex.con[i].orientation != Orientation::Column ||
1725  simplex.con[i + 1].orientation != Orientation::Column) &&
1726  "Both inequalities for the equality cannot be in column "
1727  "orientation!");
1728  if (simplex.con[i].orientation == Orientation::Column)
1729  dual.emplace_back(-simplex.tableau(row, simplex.con[i].pos));
1730  else if (simplex.con[i + 1].orientation == Orientation::Column)
1731  dual.emplace_back(simplex.tableau(row, simplex.con[i + 1].pos));
1732  else
1733  dual.emplace_back(0);
1734  }
1735  return *maybeWidth;
1736  }
1737 
1738  /// Remove the last equality that was added through addEqualityForDirection.
1739  ///
1740  /// We do this by rolling back to the snapshot at the top of the stack, which
1741  /// should be a snapshot taken just before the last equality was added.
1743  assert(!snapshotStack.empty() && "Snapshot stack is empty!");
1744  simplex.rollback(snapshotStack.back());
1745  snapshotStack.pop_back();
1746  }
1747 
1748 private:
1749  /// Returns coefficients of the expression 'dot_product(dir, x - y)',
1750  /// i.e., dir_1 * x_1 + dir_2 * x_2 + ... + dir_n * x_n
1751  /// - dir_1 * y_1 - dir_2 * y_2 - ... - dir_n * y_n,
1752  /// where n is the dimension of the original polytope.
1754  getCoeffsForDirection(ArrayRef<DynamicAPInt> dir) {
1755  assert(2 * dir.size() == simplex.getNumVariables() &&
1756  "Direction vector has wrong dimensionality");
1757  SmallVector<DynamicAPInt, 8> coeffs(dir);
1758  coeffs.reserve(dir.size() + 1);
1759  for (const DynamicAPInt &coeff : dir)
1760  coeffs.emplace_back(-coeff);
1761  coeffs.emplace_back(0); // constant term
1762  return coeffs;
1763  }
1764 
1765  Simplex simplex;
1766  /// The first index of the equality constraints, the index immediately after
1767  /// the last constraint in the initial product simplex.
1768  unsigned simplexConstraintOffset;
1769  /// A stack of snapshots, used for rolling back.
1770  SmallVector<unsigned, 8> snapshotStack;
1771 };
1772 
1773 /// Reduce the basis to try and find a direction in which the polytope is
1774 /// "thin". This only works for bounded polytopes.
1775 ///
1776 /// This is an implementation of the algorithm described in the paper
1777 /// "An Implementation of Generalized Basis Reduction for Integer Programming"
1778 /// by W. Cook, T. Rutherford, H. E. Scarf, D. Shallcross.
1779 ///
1780 /// Let b_{level}, b_{level + 1}, ... b_n be the current basis.
1781 /// Let width_i(v) = max <v, x - y> where x and y are points in the original
1782 /// polytope such that <b_j, x - y> = 0 is satisfied for all level <= j < i.
1783 ///
1784 /// In every iteration, we first replace b_{i+1} with b_{i+1} + u*b_i, where u
1785 /// is the integer such that width_i(b_{i+1} + u*b_i) is minimized. Let dual_i
1786 /// be the dual variable associated with the constraint <b_i, x - y> = 0 when
1787 /// computing width_{i+1}(b_{i+1}). It can be shown that dual_i is the
1788 /// minimizing value of u, if it were allowed to be fractional. Due to
1789 /// convexity, the minimizing integer value is either floor(dual_i) or
1790 /// ceil(dual_i), so we just need to check which of these gives a lower
1791 /// width_{i+1} value. If dual_i turned out to be an integer, then u = dual_i.
1792 ///
1793 /// Now if width_i(b_{i+1}) < 0.75 * width_i(b_i), we swap b_i and (the new)
1794 /// b_{i + 1} and decrement i (unless i = level, in which case we stay at the
1795 /// same i). Otherwise, we increment i.
1796 ///
1797 /// We keep f values and duals cached and invalidate them when necessary.
1798 /// Whenever possible, we use them instead of recomputing them. We implement the
1799 /// algorithm as follows.
1800 ///
1801 /// In an iteration at i we need to compute:
1802 /// a) width_i(b_{i + 1})
1803 /// b) width_i(b_i)
1804 /// c) the integer u that minimizes width_i(b_{i + 1} + u*b_i)
1805 ///
1806 /// If width_i(b_i) is not already cached, we compute it.
1807 ///
1808 /// If the duals are not already cached, we compute width_{i+1}(b_{i+1}) and
1809 /// store the duals from this computation.
1810 ///
1811 /// We call updateBasisWithUAndGetFCandidate, which finds the minimizing value
1812 /// of u as explained before, caches the duals from this computation, sets
1813 /// b_{i+1} to b_{i+1} + u*b_i, and returns the new value of width_i(b_{i+1}).
1814 ///
1815 /// Now if width_i(b_{i+1}) < 0.75 * width_i(b_i), we swap b_i and b_{i+1} and
1816 /// decrement i, resulting in the basis
1817 /// ... b_{i - 1}, b_{i + 1} + u*b_i, b_i, b_{i+2}, ...
1818 /// with corresponding f values
1819 /// ... width_{i-1}(b_{i-1}), width_i(b_{i+1} + u*b_i), width_{i+1}(b_i), ...
1820 /// The values up to i - 1 remain unchanged. We have just gotten the middle
1821 /// value from updateBasisWithUAndGetFCandidate, so we can update that in the
1822 /// cache. The value at width_{i+1}(b_i) is unknown, so we evict this value from
1823 /// the cache. The iteration after decrementing needs exactly the duals from the
1824 /// computation of width_i(b_{i + 1} + u*b_i), so we keep these in the cache.
1825 ///
1826 /// When incrementing i, no cached f values get invalidated. However, the cached
1827 /// duals do get invalidated as the duals for the higher levels are different.
1828 void Simplex::reduceBasis(IntMatrix &basis, unsigned level) {
1829  const Fraction epsilon(3, 4);
1830 
1831  if (level == basis.getNumRows() - 1)
1832  return;
1833 
1834  GBRSimplex gbrSimplex(*this);
1837  DynamicAPInt dualDenom;
1838 
1839  // Finds the value of u that minimizes width_i(b_{i+1} + u*b_i), caches the
1840  // duals from this computation, sets b_{i+1} to b_{i+1} + u*b_i, and returns
1841  // the new value of width_i(b_{i+1}).
1842  //
1843  // If dual_i is not an integer, the minimizing value must be either
1844  // floor(dual_i) or ceil(dual_i). We compute the expression for both and
1845  // choose the minimizing value.
1846  //
1847  // If dual_i is an integer, we don't need to perform these computations. We
1848  // know that in this case,
1849  // a) u = dual_i.
1850  // b) one can show that dual_j for j < i are the same duals we would have
1851  // gotten from computing width_i(b_{i + 1} + u*b_i), so the correct duals
1852  // are the ones already in the cache.
1853  // c) width_i(b_{i+1} + u*b_i) = min_{alpha} width_i(b_{i+1} + alpha * b_i),
1854  // which
1855  // one can show is equal to width_{i+1}(b_{i+1}). The latter value must
1856  // be in the cache, so we get it from there and return it.
1857  auto updateBasisWithUAndGetFCandidate = [&](unsigned i) -> Fraction {
1858  assert(i < level + dual.size() && "dual_i is not known!");
1859 
1860  DynamicAPInt u = floorDiv(dual[i - level], dualDenom);
1861  basis.addToRow(i, i + 1, u);
1862  if (dual[i - level] % dualDenom != 0) {
1863  SmallVector<DynamicAPInt, 8> candidateDual[2];
1864  DynamicAPInt candidateDualDenom[2];
1865  Fraction widthI[2];
1866 
1867  // Initially u is floor(dual) and basis reflects this.
1868  widthI[0] = gbrSimplex.computeWidthAndDuals(
1869  basis.getRow(i + 1), candidateDual[0], candidateDualDenom[0]);
1870 
1871  // Now try ceil(dual), i.e. floor(dual) + 1.
1872  ++u;
1873  basis.addToRow(i, i + 1, 1);
1874  widthI[1] = gbrSimplex.computeWidthAndDuals(
1875  basis.getRow(i + 1), candidateDual[1], candidateDualDenom[1]);
1876 
1877  unsigned j = widthI[0] < widthI[1] ? 0 : 1;
1878  if (j == 0)
1879  // Subtract 1 to go from u = ceil(dual) back to floor(dual).
1880  basis.addToRow(i, i + 1, -1);
1881 
1882  // width_i(b{i+1} + u*b_i) should be minimized at our value of u.
1883  // We assert that this holds by checking that the values of width_i at
1884  // u - 1 and u + 1 are greater than or equal to the value at u. If the
1885  // width is lesser at either of the adjacent values, then our computed
1886  // value of u is clearly not the minimizer. Otherwise by convexity the
1887  // computed value of u is really the minimizer.
1888 
1889  // Check the value at u - 1.
1890  assert(gbrSimplex.computeWidth(scaleAndAddForAssert(
1891  basis.getRow(i + 1), DynamicAPInt(-1), basis.getRow(i))) >=
1892  widthI[j] &&
1893  "Computed u value does not minimize the width!");
1894  // Check the value at u + 1.
1895  assert(gbrSimplex.computeWidth(scaleAndAddForAssert(
1896  basis.getRow(i + 1), DynamicAPInt(+1), basis.getRow(i))) >=
1897  widthI[j] &&
1898  "Computed u value does not minimize the width!");
1899 
1900  dual = std::move(candidateDual[j]);
1901  dualDenom = candidateDualDenom[j];
1902  return widthI[j];
1903  }
1904 
1905  assert(i + 1 - level < width.size() && "width_{i+1} wasn't saved");
1906  // f_i(b_{i+1} + dual*b_i) == width_{i+1}(b_{i+1}) when `dual` minimizes the
1907  // LHS. (note: the basis has already been updated, so b_{i+1} + dual*b_i in
1908  // the above expression is equal to basis.getRow(i+1) below.)
1909  assert(gbrSimplex.computeWidth(basis.getRow(i + 1)) ==
1910  width[i + 1 - level]);
1911  return width[i + 1 - level];
1912  };
1913 
1914  // In the ith iteration of the loop, gbrSimplex has constraints for directions
1915  // from `level` to i - 1.
1916  unsigned i = level;
1917  while (i < basis.getNumRows() - 1) {
1918  if (i >= level + width.size()) {
1919  // We don't even know the value of f_i(b_i), so let's find that first.
1920  // We have to do this first since later we assume that width already
1921  // contains values up to and including i.
1922 
1923  assert((i == 0 || i - 1 < level + width.size()) &&
1924  "We are at level i but we don't know the value of width_{i-1}");
1925 
1926  // We don't actually use these duals at all, but it doesn't matter
1927  // because this case should only occur when i is level, and there are no
1928  // duals in that case anyway.
1929  assert(i == level && "This case should only occur when i == level");
1930  width.emplace_back(
1931  gbrSimplex.computeWidthAndDuals(basis.getRow(i), dual, dualDenom));
1932  }
1933 
1934  if (i >= level + dual.size()) {
1935  assert(i + 1 >= level + width.size() &&
1936  "We don't know dual_i but we know width_{i+1}");
1937  // We don't know dual for our level, so let's find it.
1938  gbrSimplex.addEqualityForDirection(basis.getRow(i));
1939  width.emplace_back(gbrSimplex.computeWidthAndDuals(basis.getRow(i + 1),
1940  dual, dualDenom));
1941  gbrSimplex.removeLastEquality();
1942  }
1943 
1944  // This variable stores width_i(b_{i+1} + u*b_i).
1945  Fraction widthICandidate = updateBasisWithUAndGetFCandidate(i);
1946  if (widthICandidate < epsilon * width[i - level]) {
1947  basis.swapRows(i, i + 1);
1948  width[i - level] = widthICandidate;
1949  // The values of width_{i+1}(b_{i+1}) and higher may change after the
1950  // swap, so we remove the cached values here.
1951  width.resize(i - level + 1);
1952  if (i == level) {
1953  dual.clear();
1954  continue;
1955  }
1956 
1957  gbrSimplex.removeLastEquality();
1958  i--;
1959  continue;
1960  }
1961 
1962  // Invalidate duals since the higher level needs to recompute its own duals.
1963  dual.clear();
1964  gbrSimplex.addEqualityForDirection(basis.getRow(i));
1965  i++;
1966  }
1967 }
1968 
1969 /// Search for an integer sample point using a branch and bound algorithm.
1970 ///
1971 /// Each row in the basis matrix is a vector, and the set of basis vectors
1972 /// should span the space. Initially this is the identity matrix,
1973 /// i.e., the basis vectors are just the variables.
1974 ///
1975 /// In every level, a value is assigned to the level-th basis vector, as
1976 /// follows. Compute the minimum and maximum rational values of this direction.
1977 /// If only one integer point lies in this range, constrain the variable to
1978 /// have this value and recurse to the next variable.
1979 ///
1980 /// If the range has multiple values, perform generalized basis reduction via
1981 /// reduceBasis and then compute the bounds again. Now we try constraining
1982 /// this direction in the first value in this range and "recurse" to the next
1983 /// level. If we fail to find a sample, we try assigning the direction the next
1984 /// value in this range, and so on.
1985 ///
1986 /// If no integer sample is found from any of the assignments, or if the range
1987 /// contains no integer value, then of course the polytope is empty for the
1988 /// current assignment of the values in previous levels, so we return to
1989 /// the previous level.
1990 ///
1991 /// If we reach the last level where all the variables have been assigned values
1992 /// already, then we simply return the current sample point if it is integral,
1993 /// and go back to the previous level otherwise.
1994 ///
1995 /// To avoid potentially arbitrarily large recursion depths leading to stack
1996 /// overflows, this algorithm is implemented iteratively.
1997 std::optional<SmallVector<DynamicAPInt, 8>> Simplex::findIntegerSample() {
1998  if (empty)
1999  return {};
2000 
2001  unsigned nDims = var.size();
2002  IntMatrix basis = IntMatrix::identity(nDims);
2003 
2004  unsigned level = 0;
2005  // The snapshot just before constraining a direction to a value at each level.
2006  SmallVector<unsigned, 8> snapshotStack;
2007  // The maximum value in the range of the direction for each level.
2008  SmallVector<DynamicAPInt, 8> upperBoundStack;
2009  // The next value to try constraining the basis vector to at each level.
2010  SmallVector<DynamicAPInt, 8> nextValueStack;
2011 
2012  snapshotStack.reserve(basis.getNumRows());
2013  upperBoundStack.reserve(basis.getNumRows());
2014  nextValueStack.reserve(basis.getNumRows());
2015  while (level != -1u) {
2016  if (level == basis.getNumRows()) {
2017  // We've assigned values to all variables. Return if we have a sample,
2018  // or go back up to the previous level otherwise.
2019  if (auto maybeSample = getSamplePointIfIntegral())
2020  return maybeSample;
2021  level--;
2022  continue;
2023  }
2024 
2025  if (level >= upperBoundStack.size()) {
2026  // We haven't populated the stack values for this level yet, so we have
2027  // just come down a level ("recursed"). Find the lower and upper bounds.
2028  // If there is more than one integer point in the range, perform
2029  // generalized basis reduction.
2030  SmallVector<DynamicAPInt, 8> basisCoeffs =
2031  llvm::to_vector<8>(basis.getRow(level));
2032  basisCoeffs.emplace_back(0);
2033 
2034  auto [minRoundedUp, maxRoundedDown] = computeIntegerBounds(basisCoeffs);
2035 
2036  // We don't have any integer values in the range.
2037  // Pop the stack and return up a level.
2038  if (minRoundedUp.isEmpty() || maxRoundedDown.isEmpty()) {
2039  assert((minRoundedUp.isEmpty() && maxRoundedDown.isEmpty()) &&
2040  "If one bound is empty, both should be.");
2041  snapshotStack.pop_back();
2042  nextValueStack.pop_back();
2043  upperBoundStack.pop_back();
2044  level--;
2045  continue;
2046  }
2047 
2048  // We already checked the empty case above.
2049  assert((minRoundedUp.isBounded() && maxRoundedDown.isBounded()) &&
2050  "Polyhedron should be bounded!");
2051 
2052  // Heuristic: if the sample point is integral at this point, just return
2053  // it.
2054  if (auto maybeSample = getSamplePointIfIntegral())
2055  return *maybeSample;
2056 
2057  if (*minRoundedUp < *maxRoundedDown) {
2058  reduceBasis(basis, level);
2059  basisCoeffs = llvm::to_vector<8>(basis.getRow(level));
2060  basisCoeffs.emplace_back(0);
2061  std::tie(minRoundedUp, maxRoundedDown) =
2062  computeIntegerBounds(basisCoeffs);
2063  }
2064 
2065  snapshotStack.emplace_back(getSnapshot());
2066  // The smallest value in the range is the next value to try.
2067  // The values in the optionals are guaranteed to exist since we know the
2068  // polytope is bounded.
2069  nextValueStack.emplace_back(*minRoundedUp);
2070  upperBoundStack.emplace_back(*maxRoundedDown);
2071  }
2072 
2073  assert((snapshotStack.size() - 1 == level &&
2074  nextValueStack.size() - 1 == level &&
2075  upperBoundStack.size() - 1 == level) &&
2076  "Mismatched variable stack sizes!");
2077 
2078  // Whether we "recursed" or "returned" from a lower level, we rollback
2079  // to the snapshot of the starting state at this level. (in the "recursed"
2080  // case this has no effect)
2081  rollback(snapshotStack.back());
2082  DynamicAPInt nextValue = nextValueStack.back();
2083  ++nextValueStack.back();
2084  if (nextValue > upperBoundStack.back()) {
2085  // We have exhausted the range and found no solution. Pop the stack and
2086  // return up a level.
2087  snapshotStack.pop_back();
2088  nextValueStack.pop_back();
2089  upperBoundStack.pop_back();
2090  level--;
2091  continue;
2092  }
2093 
2094  // Try the next value in the range and "recurse" into the next level.
2095  SmallVector<DynamicAPInt, 8> basisCoeffs(basis.getRow(level).begin(),
2096  basis.getRow(level).end());
2097  basisCoeffs.emplace_back(-nextValue);
2098  addEquality(basisCoeffs);
2099  level++;
2100  }
2101 
2102  return {};
2103 }
2104 
2105 /// Compute the minimum and maximum integer values the expression can take. We
2106 /// compute each separately.
2107 std::pair<MaybeOptimum<DynamicAPInt>, MaybeOptimum<DynamicAPInt>>
2109  MaybeOptimum<DynamicAPInt> minRoundedUp(
2111  MaybeOptimum<DynamicAPInt> maxRoundedDown(
2113  return {minRoundedUp, maxRoundedDown};
2114 }
2115 
2117  assert(!isEmpty() && "cannot check for flatness of empty simplex!");
2118  auto upOpt = computeOptimum(Simplex::Direction::Up, coeffs);
2119  auto downOpt = computeOptimum(Simplex::Direction::Down, coeffs);
2120 
2121  if (!upOpt.isBounded())
2122  return false;
2123  if (!downOpt.isBounded())
2124  return false;
2125 
2126  return *upOpt == *downOpt;
2127 }
2128 
2129 void SimplexBase::print(raw_ostream &os) const {
2130  os << "rows = " << getNumRows() << ", columns = " << getNumColumns() << "\n";
2131  if (empty)
2132  os << "Simplex marked empty!\n";
2133  os << "var: ";
2134  for (unsigned i = 0; i < var.size(); ++i) {
2135  if (i > 0)
2136  os << ", ";
2137  var[i].print(os);
2138  }
2139  os << "\ncon: ";
2140  for (unsigned i = 0; i < con.size(); ++i) {
2141  if (i > 0)
2142  os << ", ";
2143  con[i].print(os);
2144  }
2145  os << '\n';
2146  for (unsigned row = 0, e = getNumRows(); row < e; ++row) {
2147  if (row > 0)
2148  os << ", ";
2149  os << "r" << row << ": " << rowUnknown[row];
2150  }
2151  os << '\n';
2152  os << "c0: denom, c1: const";
2153  for (unsigned col = 2, e = getNumColumns(); col < e; ++col)
2154  os << ", c" << col << ": " << colUnknown[col];
2155  os << '\n';
2156  PrintTableMetrics ptm = {0, 0, "-"};
2157  for (unsigned row = 0, numRows = getNumRows(); row < numRows; ++row)
2158  for (unsigned col = 0, numCols = getNumColumns(); col < numCols; ++col)
2159  updatePrintMetrics<DynamicAPInt>(tableau(row, col), ptm);
2160  unsigned MIN_SPACING = 1;
2161  for (unsigned row = 0, numRows = getNumRows(); row < numRows; ++row) {
2162  for (unsigned col = 0, numCols = getNumColumns(); col < numCols; ++col) {
2163  printWithPrintMetrics<DynamicAPInt>(os, tableau(row, col), MIN_SPACING,
2164  ptm);
2165  }
2166  os << '\n';
2167  }
2168  os << '\n';
2169 }
2170 
2171 void SimplexBase::dump() const { print(llvm::errs()); }
2172 
2174  if (isEmpty())
2175  return true;
2176 
2177  for (unsigned i = 0, e = rel.getNumInequalities(); i < e; ++i)
2179  return false;
2180 
2181  for (unsigned i = 0, e = rel.getNumEqualities(); i < e; ++i)
2182  if (!isRedundantEquality(rel.getEquality(i)))
2183  return false;
2184 
2185  return true;
2186 }
2187 
2188 /// Returns the type of the inequality with coefficients `coeffs`.
2189 /// Possible types are:
2190 /// Redundant The inequality is satisfied by all points in the polytope
2191 /// Cut The inequality is satisfied by some points, but not by others
2192 /// Separate The inequality is not satisfied by any point
2193 ///
2194 /// Internally, this computes the minimum and the maximum the inequality with
2195 /// coefficients `coeffs` can take. If the minimum is >= 0, the inequality holds
2196 /// for all points in the polytope, so it is redundant. If the minimum is <= 0
2197 /// and the maximum is >= 0, the points in between the minimum and the
2198 /// inequality do not satisfy it, the points in between the inequality and the
2199 /// maximum satisfy it. Hence, it is a cut inequality. If both are < 0, no
2200 /// points of the polytope satisfy the inequality, which means it is a separate
2201 /// inequality.
2204  if (minimum.isBounded() && *minimum >= Fraction(0, 1)) {
2205  return IneqType::Redundant;
2206  }
2208  if ((!minimum.isBounded() || *minimum <= Fraction(0, 1)) &&
2209  (!maximum.isBounded() || *maximum >= Fraction(0, 1))) {
2210  return IneqType::Cut;
2211  }
2212  return IneqType::Separate;
2213 }
2214 
2215 /// Checks whether the type of the inequality with coefficients `coeffs`
2216 /// is Redundant.
2218  assert(!empty &&
2219  "It is not meaningful to ask about redundancy in an empty set!");
2220  return findIneqType(coeffs) == IneqType::Redundant;
2221 }
2222 
2223 /// Check whether the equality given by `coeffs == 0` is redundant given
2224 /// the existing constraints. This is redundant when `coeffs` is already
2225 /// always zero under the existing constraints. `coeffs` is always zero
2226 /// when the minimum and maximum value that `coeffs` can take are both zero.
2228  assert(!empty &&
2229  "It is not meaningful to ask about redundancy in an empty set!");
2232  assert((!minimum.isEmpty() && !maximum.isEmpty()) &&
2233  "Optima should be non-empty for a non-empty set");
2234  return minimum.isBounded() && maximum.isBounded() &&
2235  *maximum == Fraction(0, 1) && *minimum == Fraction(0, 1);
2236 }
static Value max(ImplicitLocOpBuilder &builder, Value value, Value bound)
static bool isRangeDivisibleBy(ArrayRef< DynamicAPInt > range, const DynamicAPInt &divisor)
Definition: Simplex.cpp:372
const int nullIndex
Definition: Simplex.cpp:35
static LLVM_ATTRIBUTE_UNUSED SmallVector< DynamicAPInt, 8 > scaleAndAddForAssert(ArrayRef< DynamicAPInt > a, const DynamicAPInt &scale, ArrayRef< DynamicAPInt > b)
Definition: Simplex.cpp:40
static IntMatrix identity(unsigned dimension)
Return the identity matrix of the specified dimension.
Definition: Matrix.cpp:456
DynamicAPInt normalizeRow(unsigned row, unsigned nCols)
Divide the first nCols of the specified row by their GCD.
Definition: Matrix.cpp:549
An IntegerRelation represents the set of points from a PresburgerSpace that satisfy a list of affine ...
void truncate(const CountsSnapshot &counts)
ArrayRef< DynamicAPInt > getInequality(unsigned idx) const
void addLocalFloorDiv(ArrayRef< DynamicAPInt > dividend, const DynamicAPInt &divisor)
Adds a new local variable as the floordiv of an affine function of other variables,...
DivisionRepr getLocalReprs(std::vector< MaybeLocalRepr > *repr=nullptr) const
Returns a DivisonRepr representing the division representation of local variables in the constraint s...
void addInequality(ArrayRef< DynamicAPInt > inEq)
Adds an inequality (>= 0) from the coefficients specified in inEq.
ArrayRef< DynamicAPInt > getEquality(unsigned idx) const
void undoLastConstraint() final
Undo the addition of the last constraint.
Definition: Simplex.cpp:1213
LogicalResult moveRowUnknownToColumn(unsigned row)
Try to move the specified row to column orientation while preserving the lexicopositivity of the basi...
Definition: Simplex.cpp:777
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:281
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:793
void addInequality(ArrayRef< DynamicAPInt > coeffs) final
Add an inequality to the tableau.
Definition: Simplex.cpp:1592
unsigned getSnapshot()
Get a snapshot of the current state. This is used for rolling back.
Definition: Simplex.h:425
void appendSymbol()
Add new symbolic variables to the end of the list of variables.
Definition: Simplex.cpp:365
MaybeOptimum< SmallVector< Fraction, 8 > > findRationalLexMin()
Return the lexicographically minimum rational solution to the constraints.
Definition: Simplex.cpp:235
bool isSeparateInequality(ArrayRef< DynamicAPInt > coeffs)
Return whether the specified inequality is redundant/separate for the polytope.
Definition: Simplex.cpp:337
bool isRedundantInequality(ArrayRef< DynamicAPInt > coeffs)
Definition: Simplex.cpp:343
MaybeOptimum< SmallVector< DynamicAPInt, 8 > > findIntegerLexMin()
Return the lexicographically minimum integer solution to the constraints.
Definition: Simplex.cpp:306
unsigned getNumRows() const
Definition: Matrix.h:86
void swapColumns(unsigned column, unsigned otherColumn)
Swap the given columns.
Definition: Matrix.cpp:120
unsigned appendExtraRow()
Add an extra row at the bottom of the matrix and return its position.
Definition: Matrix.cpp:65
MutableArrayRef< T > getRow(unsigned row)
Get a [Mutable]ArrayRef corresponding to the specified row.
Definition: Matrix.cpp:130
void resizeVertically(unsigned newNRows)
Definition: Matrix.cpp:104
void swapRows(unsigned row, unsigned otherRow)
Swap the given rows.
Definition: Matrix.cpp:110
void resizeHorizontally(unsigned newNColumns)
Definition: Matrix.cpp:90
void reserveRows(unsigned rows)
Reserve enough space to resize to the specified number of rows without reallocations.
Definition: Matrix.cpp:60
void addToRow(unsigned sourceRow, unsigned targetRow, const T &scale)
Add scale multiples of the source row to the target row.
Definition: Matrix.cpp:299
bool isBounded() const
Definition: Utils.h:51
bool isUnbounded() const
Definition: Utils.h:52
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:170
void addPiece(const Piece &piece)
unsigned getNumOutputs() const
Definition: PWMAFunction.h:180
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:152
unsigned addZeroRow(bool makeRestricted=false)
Add a new row to the tableau and the associated data structures.
Definition: Simplex.cpp:107
bool isEmpty() const
Returns true if the tableau is empty (has conflicting constraints), false otherwise.
Definition: Simplex.cpp:1070
void appendVariable(unsigned count=1)
Add new variables to the end of the list of variables.
Definition: Simplex.cpp:1321
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:358
SmallVector< SmallVector< int, 8 >, 8 > savedBases
Holds a vector of bases.
Definition: Simplex.h:349
void intersectIntegerRelation(const IntegerRelation &rel)
Add all the constraints from the given IntegerRelation.
Definition: Simplex.cpp:1336
SmallVector< UndoLogEntry, 8 > undoLog
Holds a log of operations, used for rolling back to a previous state.
Definition: Simplex.h:344
bool usingBigM
Stores whether or not a big M column is present in the tableau.
Definition: Simplex.h:326
unsigned getSnapshot() const
Get a snapshot of the current state.
Definition: Simplex.cpp:1137
void print(raw_ostream &os) const
Print the tableau's internal state.
Definition: Simplex.cpp:2129
UndoLogEntry
Enum to denote operations that need to be undone during rollback.
Definition: Simplex.h:301
unsigned getNumRows() const
Definition: Simplex.h:322
const Unknown & unknownFromRow(unsigned row) const
Returns the unknown associated with row.
Definition: Simplex.cpp:87
SmallVector< int, 8 > colUnknown
Definition: Simplex.h:358
SmallVector< Unknown, 8 > var
Definition: Simplex.h:361
void addEquality(ArrayRef< DynamicAPInt > coeffs)
Add an equality to the tableau.
Definition: Simplex.cpp:1123
unsigned getSnapshotBasis()
Get a snapshot of the current state including the basis.
Definition: Simplex.cpp:1139
unsigned getNumFixedCols() const
Return the number of fixed columns, as described in the constructor above, this is the number of colu...
Definition: Simplex.h:321
SmallVector< Unknown, 8 > con
These hold information about each unknown.
Definition: Simplex.h:361
void markEmpty()
Mark the tableau as being empty.
Definition: Simplex.cpp:1093
bool empty
This is true if the tableau has been detected to be empty, false otherwise.
Definition: Simplex.h:341
void addDivisionVariable(ArrayRef< DynamicAPInt > coeffs, const DynamicAPInt &denom)
Append a new variable to the simplex and constrain it such that its only integer value is the floor d...
Definition: Simplex.cpp:1304
void swapColumns(unsigned i, unsigned j)
Definition: Simplex.cpp:1081
void removeLastConstraintRowOrientation()
Remove the last constraint, which must be in row orientation.
Definition: Simplex.cpp:1152
std::optional< unsigned > findAnyPivotRow(unsigned col)
Return any row that this column can be pivoted with, ignoring tableau consistency.
Definition: Simplex.cpp:1175
virtual void addInequality(ArrayRef< DynamicAPInt > coeffs)=0
Add an inequality to the tableau.
const Unknown & unknownFromColumn(unsigned col) const
Returns the unknown associated with col.
Definition: Simplex.cpp:82
void rollback(unsigned snapshot)
Rollback to a snapshot. This invalidates all later snapshots.
Definition: Simplex.cpp:1291
IntMatrix tableau
The matrix representing the tableau.
Definition: Simplex.h:337
void pivot(unsigned row, unsigned col)
Pivot the row with the column.
Definition: Simplex.cpp:951
void swapRows(unsigned i, unsigned j)
Swap the two rows/columns in the tableau and associated data structures.
Definition: Simplex.cpp:1072
void undo(UndoLogEntry entry)
Undo the operation represented by the log entry.
Definition: Simplex.cpp:1229
const Unknown & unknownFromIndex(int index) const
Returns the unknown associated with index.
Definition: Simplex.cpp:77
unsigned nSymbol
The number of parameters.
Definition: Simplex.h:334
unsigned nRedundant
The number of redundant rows in the tableau.
Definition: Simplex.h:330
unsigned addRow(ArrayRef< DynamicAPInt > coeffs, bool makeRestricted=false)
Add a new row to the tableau and the associated data structures.
Definition: Simplex.cpp:121
unsigned getNumVariables() const
Returns the number of variables in the tableau.
Definition: Simplex.cpp:1132
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:914
unsigned getNumColumns() const
Definition: Simplex.h:323
unsigned getNumConstraints() const
Returns the number of constraints in the tableau.
Definition: Simplex.cpp:1133
Takes a snapshot of the simplex state on construction and rolls back to the snapshot on destruction.
Definition: Simplex.h:874
The Simplex class uses the Normal pivot rule and supports integer emptiness checks as well as detecti...
Definition: Simplex.h:691
std::pair< MaybeOptimum< DynamicAPInt >, MaybeOptimum< DynamicAPInt > > computeIntegerBounds(ArrayRef< DynamicAPInt > coeffs)
Returns a (min, max) pair denoting the minimum and maximum integer values of the given expression.
Definition: Simplex.cpp:2108
bool isMarkedRedundant(unsigned constraintIndex) const
Returns whether the specified constraint has been marked as redundant.
Definition: Simplex.cpp:1409
std::optional< SmallVector< DynamicAPInt, 8 > > getSamplePointIfIntegral() const
Returns the current sample point if it is integral.
Definition: Simplex.cpp:1627
bool isFlatAlong(ArrayRef< DynamicAPInt > coeffs)
Check if the simplex takes only one rational value along the direction of coeffs.
Definition: Simplex.cpp:2116
bool isRedundantEquality(ArrayRef< DynamicAPInt > coeffs)
Check if the specified equality already holds in the polytope.
Definition: Simplex.cpp:2227
IneqType findIneqType(ArrayRef< DynamicAPInt > coeffs)
Returns the type of the inequality with coefficients coeffs.
Definition: Simplex.cpp:2202
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:1499
MaybeOptimum< Fraction > computeRowOptimum(Direction direction, unsigned row)
Compute the maximum or minimum value of the given row, depending on direction.
Definition: Simplex.cpp:1345
bool isRationalSubsetOf(const IntegerRelation &rel)
Returns true if this Simplex's polytope is a rational subset of rel.
Definition: Simplex.cpp:2173
bool isBoundedAlongConstraint(unsigned constraintIndex)
Returns whether the perpendicular of the specified constraint is a is a direction along which the pol...
Definition: Simplex.cpp:1399
bool isUnbounded()
Returns true if the polytope is unbounded, i.e., extends to infinity in some direction.
Definition: Simplex.cpp:1471
bool isRedundantInequality(ArrayRef< DynamicAPInt > coeffs)
Check if the specified inequality already holds in the polytope.
Definition: Simplex.cpp:2217
void addInequality(ArrayRef< DynamicAPInt > coeffs) final
Add an inequality to the tableau.
Definition: Simplex.cpp:1110
MaybeOptimum< Fraction > computeOptimum(Direction direction, ArrayRef< DynamicAPInt > coeffs)
Compute the maximum or minimum value of the given expression, depending on direction.
Definition: Simplex.cpp:1364
std::optional< SmallVector< Fraction, 8 > > getRationalSample() const
Returns the current sample point, which may contain non-integer (rational) coordinates.
Definition: Simplex.cpp:1571
std::optional< SmallVector< DynamicAPInt, 8 > > findIntegerSample()
Returns an integer sample point if one exists, or std::nullopt otherwise.
Definition: Simplex.cpp:1997
SymbolicLexOpt computeSymbolicIntegerLexMin()
The lexmin will be stored as a function lexopt from symbols to non-symbols in the result.
Definition: Simplex.cpp:537
Given a simplex for a polytope, construct a new simplex whose variables are identified with a pair of...
Definition: Simplex.cpp:1655
Fraction computeWidthAndDuals(ArrayRef< DynamicAPInt > dir, SmallVectorImpl< DynamicAPInt > &dual, DynamicAPInt &dualDenom)
Compute max(dotProduct(dir, x - y)) and save the dual variables for only the direction equalities to ...
Definition: Simplex.cpp:1682
void removeLastEquality()
Remove the last equality that was added through addEqualityForDirection.
Definition: Simplex.cpp:1742
Fraction computeWidth(ArrayRef< DynamicAPInt > dir)
Compute max(dotProduct(dir, x - y)).
Definition: Simplex.cpp:1673
GBRSimplex(const Simplex &originalSimplex)
Definition: Simplex.cpp:1659
void addEqualityForDirection(ArrayRef< DynamicAPInt > dir)
Add an equality dotProduct(dir, x - y) == 0.
Definition: Simplex.cpp:1666
SmallVector< AffineExpr, 4 > concat(ArrayRef< AffineExpr > a, ArrayRef< AffineExpr > b)
Return the vector that is the concatenation of a and b.
Definition: LinalgOps.cpp:2276
void normalizeDiv(MutableArrayRef< DynamicAPInt > num, DynamicAPInt &denom)
Normalize the given (numerator, denominator) pair by dividing out the common factors between them.
Definition: Utils.cpp:361
DynamicAPInt floor(const Fraction &f)
Definition: Fraction.h:77
DynamicAPInt ceil(const Fraction &f)
Definition: Fraction.h:79
DynamicAPInt normalizeRange(MutableArrayRef< DynamicAPInt > range)
Divide the range by its gcd and return the gcd.
Definition: Utils.cpp:352
SmallVector< DynamicAPInt, 8 > getComplementIneq(ArrayRef< DynamicAPInt > ineq)
Return the complement of the given inequality.
Definition: Utils.cpp:382
Include the generated interface declarations.
A class to represent fractions.
Definition: Fraction.h:29
DynamicAPInt getAsInteger() const
Definition: Fraction.h:51
The struct CountsSnapshot stores the count of each VarKind, and also of each constraint type.
Example usage: Print .12, 3.4, 56.7 preAlign = ".", minSpacing = 1, .12 .12 3.4 3....
Definition: Utils.h:303
An Unknown is either a variable or a constraint.
Definition: Simplex.h:234
Represents the result of a symbolic lexicographic optimization computation.
Definition: Simplex.h:529
PWMAFunction lexopt
This maps assignments of symbols to the corresponding lexopt.
Definition: Simplex.h:537
PresburgerSet unboundedDomain
Contains all assignments to the symbols that made the lexopt unbounded.
Definition: Simplex.h:541
Eliminates variable at the specified position using Fourier-Motzkin variable elimination.