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__ = (
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:
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')
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)
"""
if not isinstance(other, Polyhedron):
raise TypeError('argument must be a Polyhedron instance')
- inequalities1 = self._asinequalities()
- inequalities2 = other._asinequalities()
+ inequalities1 = self.asinequalities()
+ inequalities2 = other.asinequalities()
inequalities = []
for inequality1 in inequalities1:
if other <= Polyhedron(inequalities=[inequality1]):
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):
domain = Domain.fromsympy(expr)
def __repr__(self):
return 'Empty'
- def _repr_latex_(self):
- return '$$\\emptyset$$'
-
Empty = EmptyType()
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, 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 Ge(left, right):
+@_pseudoconstructor
+def Ge(*exprs):
"""
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)
+ return Polyhedron([], inequalities)
-@_polymorphic
-def Gt(left, right):
+@_pseudoconstructor
+def Gt(*exprs):
"""
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 - 1)
+ return Polyhedron([], inequalities)
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