Rename example squares.py into tutorial.py

[linpy.git] / linpy / polyhedra.py

diff --git a/linpy/polyhedra.py b/linpy/polyhedra.py
index 9fdf6e7..1ccbe9c 100644 (file)
--- a/linpy/polyhedra.py
+++ b/linpy/polyhedra.py
@@ -37,8 +37,9 @@ __all__ = [
 class Polyhedron(Domain):
     """
     A convex polyhedron (or simply "polyhedron") is the space defined by a
 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__ = (
     """
 
     __slots__ = (


@@ -56,28 +57,31 @@ class Polyhedron(Domain):
         0 <= x <= 2, 0 <= y <= 2 can be constructed with:
 
         >>> x, y = symbols('x y')
         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')
 
         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.
 
 
         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:
 
 
         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:
         """
         if isinstance(equalities, str):
             if inequalities is not None:


@@ -144,6 +148,15 @@ class Polyhedron(Domain):
     def aspolyhedron(self):
         return self
 
     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')
     def __contains__(self, point):
         if not isinstance(point, Point):
             raise TypeError('point must be a Point instance')


@@ -164,7 +177,11 @@ class Polyhedron(Domain):
             for inequality in self.inequalities]
         return Polyhedron(equalities, 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)
         inequalities = list(self.equalities)
         inequalities.extend([-expression for expression in self.equalities])
         inequalities.extend(self.inequalities)


@@ -178,9 +195,9 @@ class Polyhedron(Domain):
         used on large polyhedra.
         """
         if not isinstance(other, Polyhedron):
         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]):
         inequalities = []
         for inequality1 in inequalities1:
             if other <= Polyhedron(inequalities=[inequality1]):


@@ -268,27 +285,43 @@ class Polyhedron(Domain):
     def __repr__(self):
         strings = []
         for equality in self.equalities:
     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:
         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))
 
         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
     @classmethod

-    def fromsympy(cls, expr):
-        domain = Domain.fromsympy(expr)
+    def fromsympy(cls, expression):
+        domain = Domain.fromsympy(expression)

         if not isinstance(domain, Polyhedron):
         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 domain
 
     def tosympy(self):


@@ -306,8 +339,6 @@ class EmptyType(Polyhedron):
     The empty polyhedron, whose set of constraints is not satisfiable.
     """
 
     The empty polyhedron, whose set of constraints is not satisfiable.
     """
 

-    __slots__ = Polyhedron.__slots__
-

     def __new__(cls):
         self = object().__new__(cls)
         self._equalities = (Rational(1),)
     def __new__(cls):
         self = object().__new__(cls)
         self._equalities = (Rational(1),)


@@ -324,9 +355,6 @@ class EmptyType(Polyhedron):
     def __repr__(self):
         return 'Empty'
 
     def __repr__(self):
         return 'Empty'
 

-    def _repr_latex_(self):
-        return '$$\\emptyset$$'
-

 Empty = EmptyType()
 
 
 Empty = EmptyType()
 
 


@@ -336,8 +364,6 @@ class UniverseType(Polyhedron):
     i.e. is empty.
     """
 
     i.e. is empty.
     """
 

-    __slots__ = Polyhedron.__slots__
-

     def __new__(cls):
         self = object().__new__(cls)
         self._equalities = ()
     def __new__(cls):
         self = object().__new__(cls)
         self._equalities = ()


@@ -349,69 +375,80 @@ class UniverseType(Polyhedron):
     def __repr__(self):
         return 'Universe'
 
     def __repr__(self):
         return 'Universe'
 

-    def _repr_latex_(self):
-        return '$$\\Omega$$'
-

 Universe = UniverseType()
 
 
 Universe = UniverseType()
 
 

-def _polymorphic(func):
+def _pseudoconstructor(func):

     @functools.wraps(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(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
 
     return wrapper
 

-@_polymorphic
-def Lt(left, right):
+@_pseudoconstructor
+def Lt(*expressions):

     """
     Create the polyhedron with constraints expr1 < expr2 < expr3 ...
     """
     """
     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):

     """
     Create the polyhedron with constraints expr1 <= expr2 <= expr3 ...
     """
     """
     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):

     """
     Create the polyhedron with constraints expr1 == expr2 == expr3 ...
     """
     """
     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):

     """
     Create the domain such that expr1 != expr2 != expr3 ... The result is a
     """
     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(expressions, expressions[1:]):
+        domain &= ~Eq(left, right)
+    return domain

 
 

-@_polymorphic
-def Gt(left, right):
+@_pseudoconstructor
+def Ge(*expressions):

     """
     """

-    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(expressions, expressions[1:]):
+        inequalities.append(left - right)
+    return Polyhedron([], inequalities)

 
 

-@_polymorphic
-def Ge(left, right):
+@_pseudoconstructor
+def Gt(*expressions):

     """
     """

-    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(expressions, expressions[1:]):
+        inequalities.append(left - right - 1)
+    return Polyhedron([], inequalities)