Minor improvements in diamonds.py example
[linpy.git] / pypol / polyhedra.py
index 787e965..aabe0fd 100644 (file)
@@ -1,12 +1,12 @@
-import ast
 import functools
+import math
 import numbers
-import re
 
 from . import islhelper
 
 from .islhelper import mainctx, libisl
-from .linexprs import Expression, Constant
+from .geometry import GeometricObject, Point, Vector
+from .linexprs import Expression, Symbol, Rational
 from .domains import Domain
 
 
@@ -32,28 +32,24 @@ class Polyhedron(Domain):
             if inequalities is not None:
                 raise TypeError('too many arguments')
             return cls.fromstring(equalities)
-        elif isinstance(equalities, Polyhedron):
+        elif isinstance(equalities, GeometricObject):
             if inequalities is not None:
                 raise TypeError('too many arguments')
-            return equalities
-        elif isinstance(equalities, Domain):
-            if inequalities is not None:
-                raise TypeError('too many arguments')
-            return equalities.polyhedral_hull()
+            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._toint()
+                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._toint()
+                inequalities[i] = inequality.scaleint()
         symbols = cls._xsymbols(equalities + inequalities)
         islbset = cls._toislbasicset(equalities, inequalities, symbols)
         return cls._fromislbasicset(islbset, symbols)
@@ -75,30 +71,59 @@ class Polyhedron(Domain):
         return self,
 
     def disjoint(self):
+        """
+        Return this set as disjoint.
+        """
         return self
 
     def isuniverse(self):
+        """
+        Return true if this set is the Universe set.
+        """
         islbset = self._toislbasicset(self.equalities, self.inequalities,
             self.symbols)
         universe = bool(libisl.isl_basic_set_is_universe(islbset))
         libisl.isl_basic_set_free(islbset)
         return universe
 
-    def polyhedral_hull(self):
+    def aspolyhedron(self):
+        """
+        Return polyhedral hull of this set.
+        """
         return self
 
+    def __contains__(self, point):
+        if not isinstance(point, Point):
+            raise TypeError('point must be a Point instance')
+        if self.symbols != point.symbols:
+            raise ValueError('arguments must belong to the same space')
+        for equality in self.equalities:
+            if equality.subs(point.coordinates()) != 0:
+                return False
+        for inequality in self.inequalities:
+            if inequality.subs(point.coordinates()) < 0:
+                return False
+        return True
+
+    def subs(self, symbol, expression=None):
+        equalities = [equality.subs(symbol, expression)
+            for equality in self.equalities]
+        inequalities = [inequality.subs(symbol, expression)
+            for inequality in self.inequalities]
+        return Polyhedron(equalities, inequalities)
+
     @classmethod
     def _fromislbasicset(cls, islbset, symbols):
         islconstraints = islhelper.isl_basic_set_constraints(islbset)
         equalities = []
         inequalities = []
         for islconstraint in islconstraints:
-            islpr = libisl.isl_printer_to_str(mainctx)
             constant = libisl.isl_constraint_get_constant_val(islconstraint)
             constant = islhelper.isl_val_to_int(constant)
             coefficients = {}
-            for dim, symbol in enumerate(symbols):
-                coefficient = libisl.isl_constraint_get_coefficient_val(islconstraint, libisl.isl_dim_set, dim)
+            for index, symbol in enumerate(symbols):
+                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
@@ -119,85 +144,44 @@ class Polyhedron(Domain):
     @classmethod
     def _toislbasicset(cls, equalities, inequalities, symbols):
         dimension = len(symbols)
+        indices = {symbol: index for index, symbol in enumerate(symbols)}
         islsp = libisl.isl_space_set_alloc(mainctx, 0, dimension)
         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))
             for symbol, coefficient in equality.coefficients():
-                val = str(coefficient).encode()
-                val = libisl.isl_val_read_from_str(mainctx, val)
-                sid = symbols.index(symbol)
+                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, sid, val)
+                    libisl.isl_dim_set, index, islval)
             if equality.constant != 0:
-                val = str(equality.constant).encode()
-                val = libisl.isl_val_read_from_str(mainctx, val)
-                isleq = libisl.isl_constraint_set_constant_val(isleq, val)
+                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))
             for symbol, coefficient in inequality.coefficients():
-                val = str(coefficient).encode()
-                val = libisl.isl_val_read_from_str(mainctx, val)
-                sid = symbols.index(symbol)
+                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, sid, val)
+                    libisl.isl_dim_set, index, islval)
             if inequality.constant != 0:
-                val = str(inequality.constant).encode()
-                val = libisl.isl_val_read_from_str(mainctx, val)
-                islin = libisl.isl_constraint_set_constant_val(islin, val)
+                islval = str(inequality.constant).encode()
+                islval = libisl.isl_val_read_from_str(mainctx, islval)
+                islin = libisl.isl_constraint_set_constant_val(islin, islval)
             islbset = libisl.isl_basic_set_add_constraint(islbset, islin)
         return islbset
 
-    @classmethod
-    def _fromast(cls, node):
-        if isinstance(node, ast.Module) and len(node.body) == 1:
-            return cls._fromast(node.body[0])
-        elif isinstance(node, ast.Expr):
-            return cls._fromast(node.value)
-        elif isinstance(node, ast.BinOp) and isinstance(node.op, ast.BitAnd):
-            equalities1, inequalities1 = cls._fromast(node.left)
-            equalities2, inequalities2 = cls._fromast(node.right)
-            equalities = equalities1 + equalities2
-            inequalities = inequalities1 + inequalities2
-            return equalities, inequalities
-        elif isinstance(node, ast.Compare):
-            equalities = []
-            inequalities = []
-            left = Expression._fromast(node.left)
-            for i in range(len(node.ops)):
-                op = node.ops[i]
-                right = Expression._fromast(node.comparators[i])
-                if isinstance(op, ast.Lt):
-                    inequalities.append(right - left - 1)
-                elif isinstance(op, ast.LtE):
-                    inequalities.append(right - left)
-                elif isinstance(op, ast.Eq):
-                    equalities.append(left - right)
-                elif isinstance(op, ast.GtE):
-                    inequalities.append(left - right)
-                elif isinstance(op, ast.Gt):
-                    inequalities.append(left - right - 1)
-                else:
-                    break
-                left = right
-            else:
-                return equalities, inequalities
-        raise SyntaxError('invalid syntax')
-
     @classmethod
     def fromstring(cls, string):
-        string = string.strip()
-        string = re.sub(r'^\{\s*|\s*\}$', '', string)
-        string = re.sub(r'([^<=>])=([^<=>])', r'\1==\2', string)
-        string = re.sub(r'(\d+|\))\s*([^\W\d_]\w*|\()', r'\1*\2', string)
-        tokens = re.split(r',|;|and|&&|/\\|∧', string, flags=re.I)
-        tokens = ['({})'.format(token) for token in tokens]
-        string = ' & '.join(tokens)
-        tree = ast.parse(string, 'eval')
-        equalities, inequalities = cls._fromast(tree)
-        return cls(equalities, inequalities)
+        domain = Domain.fromstring(string)
+        if not isinstance(domain, Polyhedron):
+            raise ValueError('non-polyhedral expression: {!r}'.format(string))
+        return domain
 
     def __repr__(self):
         if self.isempty():
@@ -215,38 +199,25 @@ class Polyhedron(Domain):
             else:
                 return 'And({})'.format(', '.join(strings))
 
-    @classmethod
-    def _fromsympy(cls, expr):
-        import sympy
-        equalities = []
-        inequalities = []
-        if expr.func == sympy.And:
-            for arg in expr.args:
-                arg_eqs, arg_ins = cls._fromsympy(arg)
-                equalities.extend(arg_eqs)
-                inequalities.extend(arg_ins)
-        elif expr.func == sympy.Eq:
-            expr = Expression.fromsympy(expr.args[0] - expr.args[1])
-            equalities.append(expr)
+    def _repr_latex_(self):
+        if self.isempty():
+            return '$\\emptyset$'
+        elif self.isuniverse():
+            return '$\\Omega$'
         else:
-            if expr.func == sympy.Lt:
-                expr = Expression.fromsympy(expr.args[1] - expr.args[0] - 1)
-            elif expr.func == sympy.Le:
-                expr = Expression.fromsympy(expr.args[1] - expr.args[0])
-            elif expr.func == sympy.Ge:
-                expr = Expression.fromsympy(expr.args[0] - expr.args[1])
-            elif expr.func == sympy.Gt:
-                expr = Expression.fromsympy(expr.args[0] - expr.args[1] - 1)
-            else:
-                raise ValueError('non-polyhedral expression: {!r}'.format(expr))
-            inequalities.append(expr)
-        return equalities, inequalities
+            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):
-        import sympy
-        equalities, inequalities = cls._fromsympy(expr)
-        return cls(equalities, inequalities)
+        domain = Domain.fromsympy(expr)
+        if not isinstance(domain, Polyhedron):
+            raise ValueError('non-polyhedral expression: {!r}'.format(expr))
+        return domain
 
     def tosympy(self):
         import sympy
@@ -257,45 +228,187 @@ class Polyhedron(Domain):
             constraints.append(sympy.Ge(inequality.tosympy(), 0))
         return sympy.And(*constraints)
 
+    @classmethod
+    def _polygon_inner_point(cls, points):
+        symbols = points[0].symbols
+        coordinates = {symbol: 0 for symbol in symbols}
+        for point in points:
+            for symbol, coordinate in point.coordinates():
+                coordinates[symbol] += coordinate
+        for symbol in symbols:
+            coordinates[symbol] /= len(points)
+        return Point(coordinates)
+
+    @classmethod
+    def _sort_polygon_2d(cls, points):
+        if len(points) <= 3:
+            return points
+        o = cls._polygon_inner_point(points)
+        angles = {}
+        for m in points:
+            om = Vector(o, m)
+            dx, dy = (coordinate for symbol, coordinate in om.coordinates())
+            angle = math.atan2(dy, dx)
+            angles[m] = angle
+        return sorted(points, key=angles.get)
+
+    @classmethod
+    def _sort_polygon_3d(cls, points):
+        if len(points) <= 3:
+            return points
+        o = cls._polygon_inner_point(points)
+        a = points[0]
+        oa = Vector(o, a)
+        norm_oa = oa.norm()
+        for b in points[1:]:
+            ob = Vector(o, b)
+            u = oa.cross(ob)
+            if not u.isnull():
+                u = u.asunit()
+                break
+        else:
+            raise ValueError('degenerate polygon')
+        angles = {a: 0.}
+        for m in points[1:]:
+            om = Vector(o, m)
+            normprod = norm_oa * om.norm()
+            cosinus = max(oa.dot(om) / normprod, -1.)
+            sinus = u.dot(oa.cross(om)) / normprod
+            angle = math.acos(cosinus)
+            angle = math.copysign(angle, sinus)
+            angles[m] = angle
+        return sorted(points, key=angles.get)
+
+    def faces(self):
+        vertices = self.vertices()
+        faces = []
+        for constraint in self.constraints:
+            face = []
+            for vertex in vertices:
+                if constraint.subs(vertex.coordinates()) == 0:
+                    face.append(vertex)
+            faces.append(face)
+        return faces
+
+    def _plot_2d(self, plot=None, **kwargs):
+        import matplotlib.pyplot as plt
+        from matplotlib.patches import Polygon
+        vertices = self._sort_polygon_2d(self.vertices())
+        xys = [tuple(vertex.values()) for vertex in vertices]
+        if plot is None:
+            fig = plt.figure()
+            plot = fig.add_subplot(1, 1, 1)
+        xmin, xmax = plot.get_xlim()
+        ymin, ymax = plot.get_xlim()
+        xs, ys = zip(*xys)
+        xmin, xmax = min(xmin, float(min(xs))), max(xmax, float(max(xs)))
+        ymin, ymax = min(ymin, float(min(ys))), max(ymax, float(max(ys)))
+        plot.set_xlim(xmin, xmax)
+        plot.set_ylim(ymin, ymax)
+        plot.add_patch(Polygon(xys, closed=True, **kwargs))
+        return plot
+
+    def _plot_3d(self, plot=None, **kwargs):
+        import matplotlib.pyplot as plt
+        from mpl_toolkits.mplot3d import Axes3D
+        from mpl_toolkits.mplot3d.art3d import Poly3DCollection
+        if plot is None:
+            fig = plt.figure()
+            axes = Axes3D(fig)
+        else:
+            axes = plot
+        xmin, xmax = axes.get_xlim()
+        ymin, ymax = axes.get_xlim()
+        zmin, zmax = axes.get_xlim()
+        poly_xyzs = []
+        for vertices in self.faces():
+            if len(vertices) == 0:
+                continue
+            vertices = Polyhedron._sort_polygon_3d(vertices)
+            vertices.append(vertices[0])
+            face_xyzs = [tuple(vertex.values()) for vertex in vertices]
+            xs, ys, zs = zip(*face_xyzs)
+            xmin, xmax = min(xmin, float(min(xs))), max(xmax, float(max(xs)))
+            ymin, ymax = min(ymin, float(min(ys))), max(ymax, float(max(ys)))
+            zmin, zmax = min(zmin, float(min(zs))), max(zmax, float(max(zs)))
+            poly_xyzs.append(face_xyzs)
+        collection = Poly3DCollection(poly_xyzs, **kwargs)
+        axes.add_collection3d(collection)
+        axes.set_xlim(xmin, xmax)
+        axes.set_ylim(ymin, ymax)
+        axes.set_zlim(zmin, zmax)
+        return axes
+
+    def plot(self, plot=None, **kwargs):
+        """
+        Display 3D plot of set.
+        """
+        if self.dimension == 2:
+            return self._plot_2d(plot=plot, **kwargs)
+        elif self.dimension == 3:
+            return self._plot_3d(plot=plot, **kwargs)
+        else:
+            raise ValueError('polyhedron must be 2 or 3-dimensional')
+
 
 def _polymorphic(func):
     @functools.wraps(func)
     def wrapper(left, right):
-        if isinstance(left, numbers.Rational):
-            left = Constant(left)
-        elif not isinstance(left, Expression):
-            raise TypeError('left must be a a rational number '
-                'or a linear expression')
-        if isinstance(right, numbers.Rational):
-            right = Constant(right)
-        elif not isinstance(right, Expression):
-            raise TypeError('right must be a a rational number '
-                'or a linear expression')
+        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)
     return wrapper
 
 @_polymorphic
 def Lt(left, right):
+    """
+    Return true if the first set is less than the second.
+    """
     return Polyhedron([], [right - left - 1])
 
 @_polymorphic
 def Le(left, right):
+    """
+    Return true the first set is less than or equal to the second.
+    """
     return Polyhedron([], [right - left])
 
 @_polymorphic
 def Eq(left, right):
+    """
+    Return true if the sets are equal.
+    """
     return Polyhedron([left - right], [])
 
 @_polymorphic
 def Ne(left, right):
+    """
+    Return true if the sets are NOT equal.
+    """
     return ~Eq(left, right)
 
 @_polymorphic
 def Gt(left, right):
+    """
+    Return true if the first set is greater than the second set.
+    """
     return Polyhedron([], [left - right - 1])
 
 @_polymorphic
 def Ge(left, right):
+    """
+    Return true if the first set is greater than or equal the second set.
+    """
     return Polyhedron([], [left - right])