linpy/polyhedra.py

   1 # Copyright 2014 MINES ParisTech
   2 #
   3 # This file is part of LinPy.
   4 #
   5 # LinPy is free software: you can redistribute it and/or modify
   6 # it under the terms of the GNU General Public License as published by
   7 # the Free Software Foundation, either version 3 of the License, or
   8 # (at your option) any later version.
   9 #
  10 # LinPy is distributed in the hope that it will be useful,
  11 # but WITHOUT ANY WARRANTY; without even the implied warranty of
  12 # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  13 # GNU General Public License for more details.
  14 #
  15 # You should have received a copy of the GNU General Public License
  16 # along with LinPy.  If not, see <http://www.gnu.org/licenses/>.
  17
  18 import functools
  19 import math
  20 import numbers
  21
  22 from . import islhelper
  23
  24 from .islhelper import mainctx, libisl
  25 from .geometry import GeometricObject, Point
  26 from .linexprs import LinExpr, Rational
  27 from .domains import Domain
  28
  29
  30 __all__ = [
  31     'Polyhedron',
  32     'Lt', 'Le', 'Eq', 'Ne', 'Ge', 'Gt',
  33     'Empty', 'Universe',
  34 ]
  35
  36
  37 class Polyhedron(Domain):
  38     """
  39     A convex polyhedron (or simply "polyhedron") is the space defined by a
  40     system of linear equalities and inequalities. This space can be unbounded. A
  41     Z-polyhedron (simply called "polyhedron" in LinPy) is the set of integer
  42     points in a convex polyhedron.
  43     """
  44
  45     __slots__ = (
  46         '_equalities',
  47         '_inequalities',
  48         '_symbols',
  49         '_dimension',
  50     )
  51
  52     def __new__(cls, equalities=None, inequalities=None):
  53         """
  54         Return a polyhedron from two sequences of linear expressions: equalities
  55         is a list of expressions equal to 0, and inequalities is a list of
  56         expressions greater or equal to 0. For example, the polyhedron
  57         0 <= x <= 2, 0 <= y <= 2 can be constructed with:
  58
  59         >>> x, y = symbols('x y')
  60         >>> square1 = Polyhedron([], [x, 2 - x, y, 2 - y])
  61         >>> square1
  62         And(0 <= x, x <= 2, 0 <= y, y <= 2)
  63
  64         It may be easier to use comparison operators LinExpr.__lt__(),
  65         LinExpr.__le__(), LinExpr.__ge__(), LinExpr.__gt__(), or functions Lt(),
  66         Le(), Eq(), Ge() and Gt(), using one of the following instructions:
  67
  68         >>> x, y = symbols('x y')
  69         >>> square1 = (0 <= x) & (x <= 2) & (0 <= y) & (y <= 2)
  70         >>> square1 = Le(0, x, 2) & Le(0, y, 2)
  71
  72         It is also possible to build a polyhedron from a string.
  73
  74         >>> square1 = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
  75
  76         Finally, a polyhedron can be constructed from a GeometricObject
  77         instance, calling the GeometricObject.aspolyedron() method. This way, it
  78         is possible to compute the polyhedral hull of a Domain instance, i.e.,
  79         the convex hull of two polyhedra:
  80
  81         >>> square1 = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
  82         >>> square2 = Polyhedron('1 <= x <= 3, 1 <= y <= 3')
  83         >>> Polyhedron(square1 | square2)
  84         And(0 <= x, 0 <= y, x <= y + 2, y <= x + 2, x <= 3, y <= 3)
  85         """
  86         if isinstance(equalities, str):
  87             if inequalities is not None:
  88                 raise TypeError('too many arguments')
  89             return cls.fromstring(equalities)
  90         elif isinstance(equalities, GeometricObject):
  91             if inequalities is not None:
  92                 raise TypeError('too many arguments')
  93             return equalities.aspolyhedron()
  94         sc_equalities = []
  95         if equalities is not None:
  96             for equality in equalities:
  97                 if isinstance(equality, LinExpr):
  98                     sc_equalities.append(equality.scaleint())
  99                 elif isinstance(equality, numbers.Rational):
 100                     sc_equalities.append(Rational(equality).scaleint())
 101                 else:
 102                     raise TypeError('equalities must be linear expressions '
 103                         'or rational numbers')
 104         sc_inequalities = []
 105         if inequalities is not None:
 106             for inequality in inequalities:
 107                 if isinstance(inequality, LinExpr):
 108                     sc_inequalities.append(inequality.scaleint())
 109                 elif isinstance(inequality, numbers.Rational):
 110                     sc_inequalities.append(Rational(inequality).scaleint())
 111                 else:
 112                     raise TypeError('inequalities must be linear expressions '
 113                         'or rational numbers')
 114         symbols = cls._xsymbols(sc_equalities + sc_inequalities)
 115         islbset = cls._toislbasicset(sc_equalities, sc_inequalities, symbols)
 116         return cls._fromislbasicset(islbset, symbols)
 117
 118     @property
 119     def equalities(self):
 120         """
 121         The tuple of equalities. This is a list of LinExpr instances that are
 122         equal to 0 in the polyhedron.
 123         """
 124         return self._equalities
 125
 126     @property
 127     def inequalities(self):
 128         """
 129         The tuple of inequalities. This is a list of LinExpr instances that are
 130         greater or equal to 0 in the polyhedron.
 131         """
 132         return self._inequalities
 133
 134     @property
 135     def constraints(self):
 136         """
 137         The tuple of constraints, i.e., equalities and inequalities. This is
 138         semantically equivalent to: equalities + inequalities.
 139         """
 140         return self._equalities + self._inequalities
 141
 142     @property
 143     def polyhedra(self):
 144         return self,
 145
 146     def make_disjoint(self):
 147         return self
 148
 149     def isuniverse(self):
 150         islbset = self._toislbasicset(self.equalities, self.inequalities,
 151             self.symbols)
 152         universe = bool(libisl.isl_basic_set_is_universe(islbset))
 153         libisl.isl_basic_set_free(islbset)
 154         return universe
 155
 156     def aspolyhedron(self):
 157         return self
 158
 159     def convex_union(self, *others):
 160         """
 161         Return the convex union of two or more polyhedra.
 162         """
 163         for other in others:
 164             if not isinstance(other, Polyhedron):
 165                 raise TypeError('arguments must be Polyhedron instances')
 166         return Polyhedron(self.union(*others))
 167
 168     def __contains__(self, point):
 169         if not isinstance(point, Point):
 170             raise TypeError('point must be a Point instance')
 171         if self.symbols != point.symbols:
 172             raise ValueError('arguments must belong to the same space')
 173         for equality in self.equalities:
 174             if equality.subs(point.coordinates()) != 0:
 175                 return False
 176         for inequality in self.inequalities:
 177             if inequality.subs(point.coordinates()) < 0:
 178                 return False
 179         return True
 180
 181     def subs(self, symbol, expression=None):
 182         equalities = [equality.subs(symbol, expression)
 183             for equality in self.equalities]
 184         inequalities = [inequality.subs(symbol, expression)
 185             for inequality in self.inequalities]
 186         return Polyhedron(equalities, inequalities)
 187
 188     def asinequalities(self):
 189         """
 190         Express the polyhedron using inequalities, given as a list of
 191         expressions greater or equal to 0.
 192         """
 193         inequalities = list(self.equalities)
 194         inequalities.extend([-expression for expression in self.equalities])
 195         inequalities.extend(self.inequalities)
 196         return inequalities
 197
 198     def widen(self, other):
 199         """
 200         Compute the standard widening of two polyhedra, à la Halbwachs.
 201
 202         In its current implementation, this method is slow and should not be
 203         used on large polyhedra.
 204         """
 205         if not isinstance(other, Polyhedron):
 206             raise TypeError('argument must be a Polyhedron instance')
 207         inequalities1 = self.asinequalities()
 208         inequalities2 = other.asinequalities()
 209         inequalities = []
 210         for inequality1 in inequalities1:
 211             if other <= Polyhedron(inequalities=[inequality1]):
 212                 inequalities.append(inequality1)
 213         for inequality2 in inequalities2:
 214             for i in range(len(inequalities1)):
 215                 inequalities3 = inequalities1[:i] + inequalities[i + 1:]
 216                 inequalities3.append(inequality2)
 217                 polyhedron3 = Polyhedron(inequalities=inequalities3)
 218                 if self == polyhedron3:
 219                     inequalities.append(inequality2)
 220                     break
 221         return Polyhedron(inequalities=inequalities)
 222
 223     @classmethod
 224     def _fromislbasicset(cls, islbset, symbols):
 225         if bool(libisl.isl_basic_set_is_empty(islbset)):
 226             return Empty
 227         if bool(libisl.isl_basic_set_is_universe(islbset)):
 228             return Universe
 229         islconstraints = islhelper.isl_basic_set_constraints(islbset)
 230         equalities = []
 231         inequalities = []
 232         for islconstraint in islconstraints:
 233             constant = libisl.isl_constraint_get_constant_val(islconstraint)
 234             constant = islhelper.isl_val_to_int(constant)
 235             coefficients = {}
 236             for index, symbol in enumerate(symbols):
 237                 coefficient = libisl.isl_constraint_get_coefficient_val(islconstraint,
 238                     libisl.isl_dim_set, index)
 239                 coefficient = islhelper.isl_val_to_int(coefficient)
 240                 if coefficient != 0:
 241                     coefficients[symbol] = coefficient
 242             expression = LinExpr(coefficients, constant)
 243             if libisl.isl_constraint_is_equality(islconstraint):
 244                 equalities.append(expression)
 245             else:
 246                 inequalities.append(expression)
 247         libisl.isl_basic_set_free(islbset)
 248         self = object().__new__(Polyhedron)
 249         self._equalities = tuple(equalities)
 250         self._inequalities = tuple(inequalities)
 251         self._symbols = cls._xsymbols(self.constraints)
 252         self._dimension = len(self._symbols)
 253         return self
 254
 255     @classmethod
 256     def _toislbasicset(cls, equalities, inequalities, symbols):
 257         dimension = len(symbols)
 258         indices = {symbol: index for index, symbol in enumerate(symbols)}
 259         islsp = libisl.isl_space_set_alloc(mainctx, 0, dimension)
 260         islbset = libisl.isl_basic_set_universe(libisl.isl_space_copy(islsp))
 261         islls = libisl.isl_local_space_from_space(islsp)
 262         for equality in equalities:
 263             isleq = libisl.isl_equality_alloc(libisl.isl_local_space_copy(islls))
 264             for symbol, coefficient in equality.coefficients():
 265                 islval = str(coefficient).encode()
 266                 islval = libisl.isl_val_read_from_str(mainctx, islval)
 267                 index = indices[symbol]
 268                 isleq = libisl.isl_constraint_set_coefficient_val(isleq,
 269                     libisl.isl_dim_set, index, islval)
 270             if equality.constant != 0:
 271                 islval = str(equality.constant).encode()
 272                 islval = libisl.isl_val_read_from_str(mainctx, islval)
 273                 isleq = libisl.isl_constraint_set_constant_val(isleq, islval)
 274             islbset = libisl.isl_basic_set_add_constraint(islbset, isleq)
 275         for inequality in inequalities:
 276             islin = libisl.isl_inequality_alloc(libisl.isl_local_space_copy(islls))
 277             for symbol, coefficient in inequality.coefficients():
 278                 islval = str(coefficient).encode()
 279                 islval = libisl.isl_val_read_from_str(mainctx, islval)
 280                 index = indices[symbol]
 281                 islin = libisl.isl_constraint_set_coefficient_val(islin,
 282                     libisl.isl_dim_set, index, islval)
 283             if inequality.constant != 0:
 284                 islval = str(inequality.constant).encode()
 285                 islval = libisl.isl_val_read_from_str(mainctx, islval)
 286                 islin = libisl.isl_constraint_set_constant_val(islin, islval)
 287             islbset = libisl.isl_basic_set_add_constraint(islbset, islin)
 288         return islbset
 289
 290     @classmethod
 291     def fromstring(cls, string):
 292         domain = Domain.fromstring(string)
 293         if not isinstance(domain, Polyhedron):
 294             raise ValueError('non-polyhedral expression: {!r}'.format(string))
 295         return domain
 296
 297     def __repr__(self):
 298         strings = []
 299         for equality in self.equalities:
 300             left, right, swap = 0, 0, False
 301             for i, (symbol, coefficient) in enumerate(equality.coefficients()):
 302                 if coefficient > 0:
 303                     left += coefficient * symbol
 304                 else:
 305                     right -= coefficient * symbol
 306                     if i == 0:
 307                         swap = True
 308             if equality.constant > 0:
 309                 left += equality.constant
 310             else:
 311                 right -= equality.constant
 312             if swap:
 313                 left, right = right, left
 314             strings.append('{} == {}'.format(left, right))
 315         for inequality in self.inequalities:
 316             left, right = 0, 0
 317             for symbol, coefficient in inequality.coefficients():
 318                 if coefficient < 0:
 319                     left -= coefficient * symbol
 320                 else:
 321                     right += coefficient * symbol
 322             if inequality.constant < 0:
 323                 left -= inequality.constant
 324             else:
 325                 right += inequality.constant
 326             strings.append('{} <= {}'.format(left, right))
 327         if len(strings) == 1:
 328             return strings[0]
 329         else:
 330             return 'And({})'.format(', '.join(strings))
 331
 332     @classmethod
 333     def fromsympy(cls, expression):
 334         domain = Domain.fromsympy(expression)
 335         if not isinstance(domain, Polyhedron):
 336             raise ValueError('non-polyhedral expression: {!r}'.format(expression))
 337         return domain
 338
 339     def tosympy(self):
 340         import sympy
 341         constraints = []
 342         for equality in self.equalities:
 343             constraints.append(sympy.Eq(equality.tosympy(), 0))
 344         for inequality in self.inequalities:
 345             constraints.append(sympy.Ge(inequality.tosympy(), 0))
 346         return sympy.And(*constraints)
 347
 348
 349 class EmptyType(Polyhedron):
 350     """
 351     The empty polyhedron, whose set of constraints is not satisfiable.
 352     """
 353
 354     def __new__(cls):
 355         self = object().__new__(cls)
 356         self._equalities = (Rational(1),)
 357         self._inequalities = ()
 358         self._symbols = ()
 359         self._dimension = 0
 360         return self
 361
 362     def widen(self, other):
 363         if not isinstance(other, Polyhedron):
 364             raise ValueError('argument must be a Polyhedron instance')
 365         return other
 366
 367     def __repr__(self):
 368         return 'Empty'
 369
 370 Empty = EmptyType()
 371
 372
 373 class UniverseType(Polyhedron):
 374     """
 375     The universe polyhedron, whose set of constraints is always satisfiable,
 376     i.e. is empty.
 377     """
 378
 379     def __new__(cls):
 380         self = object().__new__(cls)
 381         self._equalities = ()
 382         self._inequalities = ()
 383         self._symbols = ()
 384         self._dimension = ()
 385         return self
 386
 387     def __repr__(self):
 388         return 'Universe'
 389
 390 Universe = UniverseType()
 391
 392
 393 def _pseudoconstructor(func):
 394     @functools.wraps(func)
 395     def wrapper(expression1, expression2, *expressions):
 396         expressions = (expression1, expression2) + expressions
 397         for expression in expressions:
 398             if not isinstance(expression, LinExpr):
 399                 if isinstance(expression, numbers.Rational):
 400                     expression = Rational(expression)
 401                 else:
 402                     raise TypeError('arguments must be rational numbers '
 403                         'or linear expressions')
 404         return func(*expressions)
 405     return wrapper
 406
 407 @_pseudoconstructor
 408 def Lt(*expressions):
 409     """
 410     Create the polyhedron with constraints expr1 < expr2 < expr3 ...
 411     """
 412     inequalities = []
 413     for left, right in zip(expressions, expressions[1:]):
 414         inequalities.append(right - left - 1)
 415     return Polyhedron([], inequalities)
 416
 417 @_pseudoconstructor
 418 def Le(*expressions):
 419     """
 420     Create the polyhedron with constraints expr1 <= expr2 <= expr3 ...
 421     """
 422     inequalities = []
 423     for left, right in zip(expressions, expressions[1:]):
 424         inequalities.append(right - left)
 425     return Polyhedron([], inequalities)
 426
 427 @_pseudoconstructor
 428 def Eq(*expressions):
 429     """
 430     Create the polyhedron with constraints expr1 == expr2 == expr3 ...
 431     """
 432     equalities = []
 433     for left, right in zip(expressions, expressions[1:]):
 434         equalities.append(left - right)
 435     return Polyhedron(equalities, [])
 436
 437 @_pseudoconstructor
 438 def Ne(*expressions):
 439     """
 440     Create the domain such that expr1 != expr2 != expr3 ... The result is a
 441     Domain object, not a Polyhedron.
 442     """
 443     domain = Universe
 444     for left, right in zip(expressions, expressions[1:]):
 445         domain &= ~Eq(left, right)
 446     return domain
 447
 448 @_pseudoconstructor
 449 def Ge(*expressions):
 450     """
 451     Create the polyhedron with constraints expr1 >= expr2 >= expr3 ...
 452     """
 453     inequalities = []
 454     for left, right in zip(expressions, expressions[1:]):
 455         inequalities.append(left - right)
 456     return Polyhedron([], inequalities)
 457
 458 @_pseudoconstructor
 459 def Gt(*expressions):
 460     """
 461     Create the polyhedron with constraints expr1 > expr2 > expr3 ...
 462     """
 463     inequalities = []
 464     for left, right in zip(expressions, expressions[1:]):
 465         inequalities.append(left - right - 1)
 466     return Polyhedron([], inequalities)