X-Git-Url: https://scm.cri.ensmp.fr/git/linpy.git/blobdiff_plain/ba15f3f33f837b1291f74bc94081e99b860d3228..6d08d8c0a84c1ffa31f5eb16e33b340727f46175:/linpy/polyhedra.py diff --git a/linpy/polyhedra.py b/linpy/polyhedra.py index b486be1..f802151 100644 --- a/linpy/polyhedra.py +++ b/linpy/polyhedra.py @@ -37,14 +37,14 @@ __all__ = [ 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. + 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', ) @@ -57,28 +57,31 @@ class Polyhedron(Domain): 0 <= x <= 2, 0 <= y <= 2 can be constructed with: >>> x, y = symbols('x y') - >>> square = Polyhedron([], [x, 2 - x, y, 2 - 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') - >>> square = (0 <= x) & (x <= 2) & (0 <= y) & (y <= 2) - >>> square = Le(0, x, 2) & Le(0, y, 2) + >>> 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. - >>> square = Polyhedron('0 <= x <= 2, 0 <= y <= 2') + >>> 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: - >>> square = Polyhedron('0 <= x <= 2, 0 <= y <= 2') - >>> square2 = Polyhedron('2 <= x <= 4, 2 <= y <= 4') - >>> Polyhedron(square | square2) + >>> 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: @@ -88,22 +91,20 @@ 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): + sc_equalities = [] + if equalities is not None: + for equality in equalities: if not isinstance(equality, LinExpr): raise TypeError('equalities must be linear expressions') - equalities[i] = equality.scaleint() - if inequalities is None: - inequalities = [] - else: - for i, inequality in enumerate(inequalities): + sc_equalities.append(equality.scaleint()) + sc_inequalities = [] + if inequalities is not None: + for inequality in inequalities: if not isinstance(inequality, LinExpr): raise TypeError('inequalities must be linear expressions') - inequalities[i] = inequality.scaleint() - symbols = cls._xsymbols(equalities + inequalities) - islbset = cls._toislbasicset(equalities, inequalities, symbols) + sc_inequalities.append(inequality.scaleint()) + symbols = cls._xsymbols(sc_equalities + sc_inequalities) + islbset = cls._toislbasicset(sc_equalities, sc_inequalities, symbols) return cls._fromislbasicset(islbset, symbols) @property @@ -128,7 +129,7 @@ class Polyhedron(Domain): 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): @@ -147,6 +148,15 @@ class Polyhedron(Domain): def aspolyhedron(self): return self + def convex_union(self, *others): + """ + Return the convex union of two or more polyhedra. + """ + 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): raise TypeError('point must be a Point instance') @@ -167,7 +177,11 @@ class Polyhedron(Domain): 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) @@ -176,11 +190,14 @@ class Polyhedron(Domain): def widen(self, other): """ Compute the standard widening of two polyhedra, Ã  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]): @@ -219,8 +236,7 @@ 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 @@ -269,9 +285,33 @@ 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: @@ -307,13 +347,10 @@ class EmptyType(Polyhedron): The empty polyhedron, whose set of constraints is not satisfiable. """ - __slots__ = Polyhedron.__slots__ - def __new__(cls): self = object().__new__(cls) self._equalities = (Rational(1),) self._inequalities = () - self._constraints = self._equalities self._symbols = () self._dimension = 0 return self @@ -338,13 +375,10 @@ class UniverseType(Polyhedron): i.e. is empty. """ - __slots__ = Polyhedron.__slots__ - def __new__(cls): self = object().__new__(cls) self._equalities = () self._inequalities = () - self._constraints = () self._symbols = () self._dimension = () return self @@ -358,63 +392,77 @@ class UniverseType(Polyhedron): Universe = UniverseType() -def _polymorphic(func): +def _pseudoconstructor(func): @functools.wraps(func) - def wrapper(left, right): - if not isinstance(left, LinExpr): - 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, LinExpr): - 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(expr1, expr2, *exprs): + exprs = (expr1, expr2) + exprs + for expr in exprs: + if not isinstance(expr, LinExpr): + if isinstance(expr, numbers.Rational): + expr = Rational(expr) + else: + raise TypeError('arguments must be rational numbers ' + 'or linear expressions') + return func(*exprs) return wrapper -@_polymorphic -def Lt(left, right): +@_pseudoconstructor +def Lt(*exprs): """ Create the polyhedron with constraints expr1 < expr2 < expr3 ... """ - return Polyhedron([], [right - left - 1]) + inequalities = [] + for left, right in zip(exprs, exprs[1:]): + inequalities.append(right - left - 1) + return Polyhedron([], inequalities) -@_polymorphic -def Le(left, right): +@_pseudoconstructor +def Le(*exprs): """ Create the polyhedron with constraints expr1 <= expr2 <= expr3 ... """ - return Polyhedron([], [right - left]) + inequalities = [] + for left, right in zip(exprs, exprs[1:]): + inequalities.append(right - left) + return Polyhedron([], inequalities) -@_polymorphic -def Eq(left, right): +@_pseudoconstructor +def Eq(*exprs): """ Create the polyhedron with constraints expr1 == expr2 == expr3 ... """ - return Polyhedron([left - right], []) + equalities = [] + for left, right in zip(exprs, exprs[1:]): + equalities.append(left - right) + return Polyhedron(equalities, []) -@_polymorphic -def Ne(left, right): +@_pseudoconstructor +def Ne(*exprs): """ Create the domain such that expr1 != expr2 != expr3 ... The result is a - Domain, not a Polyhedron. + Domain object, not a Polyhedron. """ - return ~Eq(left, right) + domain = Universe + for left, right in zip(exprs, exprs[1:]): + domain &= ~Eq(left, right) + return domain -@_polymorphic -def Gt(left, right): +@_pseudoconstructor +def Ge(*exprs): """ - Create the polyhedron with constraints expr1 > expr2 > expr3 ... + Create the polyhedron with constraints expr1 >= expr2 >= expr3 ... """ - return Polyhedron([], [left - right - 1]) + inequalities = [] + for left, right in zip(exprs, exprs[1:]): + inequalities.append(left - right) + return Polyhedron([], inequalities) -@_polymorphic -def Ge(left, right): +@_pseudoconstructor +def Gt(*exprs): """ - Create the polyhedron with constraints expr1 >= expr2 >= expr3 ... + Create the polyhedron with constraints expr1 > expr2 > expr3 ... """ - return Polyhedron([], [left - right]) + inequalities = [] + for left, right in zip(exprs, exprs[1:]): + inequalities.append(left - right - 1) + return Polyhedron([], inequalities)