X-Git-Url: https://svn.cri.ensmp.fr/git/linpy.git/blobdiff_plain/7b93cea1daf2889e9ee10ca9c22a1b5124404937..refs/heads/master:/linpy/polyhedra.py diff --git a/linpy/polyhedra.py b/linpy/polyhedra.py index e9226f2..9e740a4 100644 --- a/linpy/polyhedra.py +++ b/linpy/polyhedra.py @@ -16,35 +16,79 @@ # along with LinPy. If not, see . import functools -import math import numbers from . import islhelper -from .islhelper import mainctx, libisl -from .geometry import GeometricObject, Point -from .linexprs import Expression, Rational from .domains import Domain +from .geometry import GeometricObject, Point +from .islhelper import libisl, mainctx +from .linexprs import LinExpr, Rational __all__ = [ + 'Empty', + 'Eq', + 'Ge', + 'Gt', + 'Le', + 'Lt', + 'Ne', 'Polyhedron', - 'Lt', 'Le', 'Eq', 'Ne', 'Ge', 'Gt', - 'Empty', 'Universe', + 'Universe', ] class Polyhedron(Domain): + """ + A convex polyhedron (or simply "polyhedron") is the space defined by a + system of linear equalities and inequalities. This space can be unbounded. + A Z-polyhedron (simply called "polyhedron" in LinPy) is the set of integer + points in a convex polyhedron. + """ __slots__ = ( '_equalities', '_inequalities', - '_constraints', '_symbols', '_dimension', ) def __new__(cls, equalities=None, inequalities=None): + """ + Return a polyhedron from two sequences of linear expressions: + equalities is a list of expressions equal to 0, and inequalities is a + list of expressions greater or equal to 0. For example, the polyhedron + 0 <= x <= 2, 0 <= y <= 2 can be constructed with: + + >>> x, y = symbols('x y') + >>> square1 = Polyhedron([], [x, 2 - x, y, 2 - y]) + >>> square1 + And(0 <= x, x <= 2, 0 <= y, y <= 2) + + It may be easier to use comparison operators LinExpr.__lt__(), + LinExpr.__le__(), LinExpr.__ge__(), LinExpr.__gt__(), or + functions Lt(), Le(), Eq(), Ge() and Gt(), using one of the following + instructions: + + >>> x, y = symbols('x y') + >>> square1 = (0 <= x) & (x <= 2) & (0 <= y) & (y <= 2) + >>> square1 = Le(0, x, 2) & Le(0, y, 2) + + It is also possible to build a polyhedron from a string. + + >>> square1 = Polyhedron('0 <= x <= 2, 0 <= y <= 2') + + Finally, a polyhedron can be constructed from a GeometricObject + instance, calling the GeometricObject.aspolyedron() method. This way, + it is possible to compute the polyhedral hull of a Domain instance, + i.e., the convex hull of two polyhedra: + + >>> square1 = Polyhedron('0 <= x <= 2, 0 <= y <= 2') + >>> square2 = Polyhedron('1 <= x <= 3, 1 <= y <= 3') + >>> Polyhedron(square1 | square2) + And(0 <= x, 0 <= y, x <= y + 2, y <= x + 2, x <= 3, y <= 3) + """ if isinstance(equalities, str): if inequalities is not None: raise TypeError('too many arguments') @@ -53,70 +97,79 @@ class Polyhedron(Domain): if inequalities is not None: raise TypeError('too many arguments') return equalities.aspolyhedron() - if equalities is None: - equalities = [] - else: - for i, equality in enumerate(equalities): - if not isinstance(equality, Expression): - raise TypeError('equalities must be linear expressions') - equalities[i] = equality.scaleint() - if inequalities is None: - inequalities = [] - else: - for i, inequality in enumerate(inequalities): - if not isinstance(inequality, Expression): - raise TypeError('inequalities must be linear expressions') - inequalities[i] = inequality.scaleint() - symbols = cls._xsymbols(equalities + inequalities) - islbset = cls._toislbasicset(equalities, inequalities, symbols) + sc_equalities = [] + if equalities is not None: + for equality in equalities: + if isinstance(equality, LinExpr): + sc_equalities.append(equality.scaleint()) + elif isinstance(equality, numbers.Rational): + sc_equalities.append(Rational(equality).scaleint()) + else: + raise TypeError('equalities must be linear expressions ' + 'or rational numbers') + sc_inequalities = [] + if inequalities is not None: + for inequality in inequalities: + if isinstance(inequality, LinExpr): + sc_inequalities.append(inequality.scaleint()) + elif isinstance(inequality, numbers.Rational): + sc_inequalities.append(Rational(inequality).scaleint()) + else: + raise TypeError('inequalities must be linear expressions ' + 'or rational numbers') + symbols = cls._xsymbols(sc_equalities + sc_inequalities) + islbset = cls._toislbasicset(sc_equalities, sc_inequalities, symbols) return cls._fromislbasicset(islbset, symbols) @property def equalities(self): """ - Return a list of the equalities in a set. + The tuple of equalities. This is a list of LinExpr instances that are + equal to 0 in the polyhedron. """ return self._equalities @property def inequalities(self): """ - Return a list of the inequalities in a set. + The tuple of inequalities. This is a list of LinExpr instances that are + greater or equal to 0 in the polyhedron. """ return self._inequalities @property def constraints(self): """ - Return ta list of the constraints of a set. + The tuple of constraints, i.e., equalities and inequalities. This is + semantically equivalent to: equalities + inequalities. """ - return self._constraints + return self._equalities + self._inequalities @property def polyhedra(self): return self, - def disjoint(self): - """ - Return a set as disjoint. - """ + def make_disjoint(self): return self def isuniverse(self): - """ - Return true if a set is the Universe set. - """ islbset = self._toislbasicset(self.equalities, self.inequalities, - self.symbols) + self.symbols) universe = bool(libisl.isl_basic_set_is_universe(islbset)) libisl.isl_basic_set_free(islbset) return universe def aspolyhedron(self): + return self + + def convex_union(self, *others): """ - Return polyhedral hull of a set. + Return the convex union of two or more polyhedra. """ - return self + for other in others: + if not isinstance(other, Polyhedron): + raise TypeError('arguments must be Polyhedron instances') + return Polyhedron(self.union(*others)) def __contains__(self, point): if not isinstance(point, Point): @@ -132,27 +185,33 @@ class Polyhedron(Domain): return True def subs(self, symbol, expression=None): - """ - Subsitute the given value into an expression and return the resulting - expression. - """ equalities = [equality.subs(symbol, expression) - for equality in self.equalities] + for equality in self.equalities] inequalities = [inequality.subs(symbol, expression) - for inequality in self.inequalities] + for inequality in self.inequalities] return Polyhedron(equalities, inequalities) - def _asinequalities(self): + def asinequalities(self): + """ + Express the polyhedron using inequalities, given as a list of + expressions greater or equal to 0. + """ inequalities = list(self.equalities) inequalities.extend([-expression for expression in self.equalities]) inequalities.extend(self.inequalities) return inequalities def widen(self, other): + """ + Compute the standard widening of two polyhedra, à la Halbwachs. + + In its current implementation, this method is slow and should not be + used on large polyhedra. + """ if not isinstance(other, Polyhedron): - raise ValueError('argument must be a Polyhedron instance') - inequalities1 = self._asinequalities() - inequalities2 = other._asinequalities() + raise TypeError('argument must be a Polyhedron instance') + inequalities1 = self.asinequalities() + inequalities2 = other.asinequalities() inequalities = [] for inequality1 in inequalities1: if other <= Polyhedron(inequalities=[inequality1]): @@ -169,6 +228,10 @@ class Polyhedron(Domain): @classmethod def _fromislbasicset(cls, islbset, symbols): + if bool(libisl.isl_basic_set_is_empty(islbset)): + return Empty + if bool(libisl.isl_basic_set_is_universe(islbset)): + return Universe islconstraints = islhelper.isl_basic_set_constraints(islbset) equalities = [] inequalities = [] @@ -177,12 +240,12 @@ class Polyhedron(Domain): constant = islhelper.isl_val_to_int(constant) coefficients = {} for index, symbol in enumerate(symbols): - coefficient = libisl.isl_constraint_get_coefficient_val(islconstraint, - libisl.isl_dim_set, index) + coefficient = libisl.isl_constraint_get_coefficient_val( + islconstraint, libisl.isl_dim_set, index) coefficient = islhelper.isl_val_to_int(coefficient) if coefficient != 0: coefficients[symbol] = coefficient - expression = Expression(coefficients, constant) + expression = LinExpr(coefficients, constant) if libisl.isl_constraint_is_equality(islconstraint): equalities.append(expression) else: @@ -191,8 +254,7 @@ class Polyhedron(Domain): self = object().__new__(Polyhedron) self._equalities = tuple(equalities) self._inequalities = tuple(inequalities) - self._constraints = tuple(equalities + inequalities) - self._symbols = cls._xsymbols(self._constraints) + self._symbols = cls._xsymbols(self.constraints) self._dimension = len(self._symbols) return self @@ -204,26 +266,28 @@ class Polyhedron(Domain): islbset = libisl.isl_basic_set_universe(libisl.isl_space_copy(islsp)) islls = libisl.isl_local_space_from_space(islsp) for equality in equalities: - isleq = libisl.isl_equality_alloc(libisl.isl_local_space_copy(islls)) + isleq = libisl.isl_equality_alloc( + libisl.isl_local_space_copy(islls)) for symbol, coefficient in equality.coefficients(): islval = str(coefficient).encode() islval = libisl.isl_val_read_from_str(mainctx, islval) index = indices[symbol] - isleq = libisl.isl_constraint_set_coefficient_val(isleq, - libisl.isl_dim_set, index, islval) + isleq = libisl.isl_constraint_set_coefficient_val( + isleq, libisl.isl_dim_set, index, islval) if equality.constant != 0: islval = str(equality.constant).encode() islval = libisl.isl_val_read_from_str(mainctx, islval) isleq = libisl.isl_constraint_set_constant_val(isleq, islval) islbset = libisl.isl_basic_set_add_constraint(islbset, isleq) for inequality in inequalities: - islin = libisl.isl_inequality_alloc(libisl.isl_local_space_copy(islls)) + islin = libisl.isl_inequality_alloc( + libisl.isl_local_space_copy(islls)) for symbol, coefficient in inequality.coefficients(): islval = str(coefficient).encode() islval = libisl.isl_val_read_from_str(mainctx, islval) index = indices[symbol] - islin = libisl.isl_constraint_set_coefficient_val(islin, - libisl.isl_dim_set, index, islval) + islin = libisl.isl_constraint_set_coefficient_val( + islin, libisl.isl_dim_set, index, islval) if inequality.constant != 0: islval = str(inequality.constant).encode() islval = libisl.isl_val_read_from_str(mainctx, islval) @@ -241,37 +305,47 @@ class Polyhedron(Domain): def __repr__(self): strings = [] for equality in self.equalities: - strings.append('Eq({}, 0)'.format(equality)) + left, right, swap = 0, 0, False + for i, (symbol, coefficient) in enumerate(equality.coefficients()): + if coefficient > 0: + left += coefficient * symbol + else: + right -= coefficient * symbol + if i == 0: + swap = True + if equality.constant > 0: + left += equality.constant + else: + right -= equality.constant + if swap: + left, right = right, left + strings.append('{} == {}'.format(left, right)) for inequality in self.inequalities: - strings.append('Ge({}, 0)'.format(inequality)) + left, right = 0, 0 + for symbol, coefficient in inequality.coefficients(): + if coefficient < 0: + left -= coefficient * symbol + else: + right += coefficient * symbol + if inequality.constant < 0: + left -= inequality.constant + else: + right += inequality.constant + strings.append('{} <= {}'.format(left, right)) if len(strings) == 1: return strings[0] else: return 'And({})'.format(', '.join(strings)) - - def _repr_latex_(self): - strings = [] - for equality in self.equalities: - strings.append('{} = 0'.format(equality._repr_latex_().strip('$'))) - for inequality in self.inequalities: - strings.append('{} \\ge 0'.format(inequality._repr_latex_().strip('$'))) - return '$${}$$'.format(' \\wedge '.join(strings)) - @classmethod - def fromsympy(cls, expr): - """ - Convert a sympy object to an expression. - """ - domain = Domain.fromsympy(expr) + def fromsympy(cls, expression): + domain = Domain.fromsympy(expression) if not isinstance(domain, Polyhedron): - raise ValueError('non-polyhedral expression: {!r}'.format(expr)) + raise ValueError('non-polyhedral expression: {!r}'.format( + expression)) return domain def tosympy(self): - """ - Return an expression as a sympy object. - """ import sympy constraints = [] for equality in self.equalities: @@ -282,14 +356,14 @@ class Polyhedron(Domain): class EmptyType(Polyhedron): - - __slots__ = Polyhedron.__slots__ + """ + The empty polyhedron, whose set of constraints is not satisfiable. + """ def __new__(cls): self = object().__new__(cls) self._equalities = (Rational(1),) self._inequalities = () - self._constraints = self._equalities self._symbols = () self._dimension = 0 return self @@ -302,21 +376,19 @@ class EmptyType(Polyhedron): def __repr__(self): return 'Empty' - def _repr_latex_(self): - return '$$\\emptyset$$' - Empty = EmptyType() class UniverseType(Polyhedron): - - __slots__ = Polyhedron.__slots__ + """ + The universe polyhedron, whose set of constraints is always satisfiable, + i.e. is empty. + """ def __new__(cls): self = object().__new__(cls) self._equalities = () self._inequalities = () - self._constraints = () self._symbols = () self._dimension = () return self @@ -324,68 +396,86 @@ class UniverseType(Polyhedron): def __repr__(self): return 'Universe' - def _repr_latex_(self): - return '$$\\Omega$$' - Universe = UniverseType() -def _polymorphic(func): +def _pseudoconstructor(func): @functools.wraps(func) - def wrapper(left, right): - if not isinstance(left, Expression): - if isinstance(left, numbers.Rational): - left = Rational(left) - else: - raise TypeError('left must be a a rational number ' - 'or a linear expression') - if not isinstance(right, Expression): - if isinstance(right, numbers.Rational): - right = Rational(right) - else: - raise TypeError('right must be a a rational number ' - 'or a linear expression') - return func(left, right) + def wrapper(expression1, expression2, *expressions): + expressions = (expression1, expression2) + expressions + for expression in expressions: + if not isinstance(expression, LinExpr): + if isinstance(expression, numbers.Rational): + expression = Rational(expression) + else: + raise TypeError('arguments must be rational numbers ' + 'or linear expressions') + return func(*expressions) return wrapper -@_polymorphic -def Lt(left, right): + +@_pseudoconstructor +def Lt(*expressions): """ - Assert first set is less than the second set. + Create the polyhedron with constraints expr1 < expr2 < expr3 ... """ - return Polyhedron([], [right - left - 1]) + inequalities = [] + for left, right in zip(expressions, expressions[1:]): + inequalities.append(right - left - 1) + return Polyhedron([], inequalities) + -@_polymorphic -def Le(left, right): +@_pseudoconstructor +def Le(*expressions): """ - Assert first set is less than or equal to the second set. + Create the polyhedron with constraints expr1 <= expr2 <= expr3 ... """ - return Polyhedron([], [right - left]) + inequalities = [] + for left, right in zip(expressions, expressions[1:]): + inequalities.append(right - left) + return Polyhedron([], inequalities) + -@_polymorphic -def Eq(left, right): +@_pseudoconstructor +def Eq(*expressions): """ - Assert first set is equal to the second set. + Create the polyhedron with constraints expr1 == expr2 == expr3 ... """ - return Polyhedron([left - right], []) + equalities = [] + for left, right in zip(expressions, expressions[1:]): + equalities.append(left - right) + return Polyhedron(equalities, []) -@_polymorphic -def Ne(left, right): + +@_pseudoconstructor +def Ne(*expressions): """ - Assert first set is not equal to the second set. + Create the domain such that expr1 != expr2 != expr3 ... The result is a + Domain object, not a Polyhedron. """ - return ~Eq(left, right) + domain = Universe + for left, right in zip(expressions, expressions[1:]): + domain &= ~Eq(left, right) + return domain + -@_polymorphic -def Gt(left, right): +@_pseudoconstructor +def Ge(*expressions): """ - Assert first set is greater than the second set. + Create the polyhedron with constraints expr1 >= expr2 >= expr3 ... """ - return Polyhedron([], [left - right - 1]) + inequalities = [] + for left, right in zip(expressions, expressions[1:]): + inequalities.append(left - right) + return Polyhedron([], inequalities) + -@_polymorphic -def Ge(left, right): +@_pseudoconstructor +def Gt(*expressions): """ - Assert first set is greater than or equal to the second set. + Create the polyhedron with constraints expr1 > expr2 > expr3 ... """ - return Polyhedron([], [left - right]) + inequalities = [] + for left, right in zip(expressions, expressions[1:]): + inequalities.append(left - right - 1) + return Polyhedron([], inequalities)