pypol/polyhedra.py

   1 import functools
   2 import math
   3 import numbers
   4
   5 from . import islhelper
   6
   7 from .islhelper import mainctx, libisl
   8 from .geometry import GeometricObject, Point, Vector
   9 from .linexprs import Expression, Symbol, Rational
  10 from .domains import Domain
  11
  12
  13 __all__ = [
  14     'Polyhedron',
  15     'Lt', 'Le', 'Eq', 'Ne', 'Ge', 'Gt',
  16     'Empty', 'Universe',
  17 ]
  18
  19
  20 class Polyhedron(Domain):
  21
  22     __slots__ = (
  23         '_equalities',
  24         '_inequalities',
  25         '_constraints',
  26         '_symbols',
  27         '_dimension',
  28     )
  29
  30     def __new__(cls, equalities=None, inequalities=None):
  31         if isinstance(equalities, str):
  32             if inequalities is not None:
  33                 raise TypeError('too many arguments')
  34             return cls.fromstring(equalities)
  35         elif isinstance(equalities, GeometricObject):
  36             if inequalities is not None:
  37                 raise TypeError('too many arguments')
  38             return equalities.aspolyhedron()
  39         if equalities is None:
  40             equalities = []
  41         else:
  42             for i, equality in enumerate(equalities):
  43                 if not isinstance(equality, Expression):
  44                     raise TypeError('equalities must be linear expressions')
  45                 equalities[i] = equality.scaleint()
  46         if inequalities is None:
  47             inequalities = []
  48         else:
  49             for i, inequality in enumerate(inequalities):
  50                 if not isinstance(inequality, Expression):
  51                     raise TypeError('inequalities must be linear expressions')
  52                 inequalities[i] = inequality.scaleint()
  53         symbols = cls._xsymbols(equalities + inequalities)
  54         islbset = cls._toislbasicset(equalities, inequalities, symbols)
  55         return cls._fromislbasicset(islbset, symbols)
  56
  57     @property
  58     def equalities(self):
  59         return self._equalities
  60
  61     @property
  62     def inequalities(self):
  63         return self._inequalities
  64
  65     @property
  66     def constraints(self):
  67         return self._constraints
  68
  69     @property
  70     def polyhedra(self):
  71         return self,
  72
  73     def disjoint(self):
  74         return self
  75
  76     def isuniverse(self):
  77         islbset = self._toislbasicset(self.equalities, self.inequalities,
  78             self.symbols)
  79         universe = bool(libisl.isl_basic_set_is_universe(islbset))
  80         libisl.isl_basic_set_free(islbset)
  81         return universe
  82
  83     def aspolyhedron(self):
  84         return self
  85
  86     def __contains__(self, point):
  87         if not isinstance(point, Point):
  88             raise TypeError('point must be a Point instance')
  89         if self.symbols != point.symbols:
  90             raise ValueError('arguments must belong to the same space')
  91         for equality in self.equalities:
  92             if equality.subs(point.coordinates()) != 0:
  93                 return False
  94         for inequality in self.inequalities:
  95             if inequality.subs(point.coordinates()) < 0:
  96                 return False
  97         return True
  98
  99     def subs(self, symbol, expression=None):
 100         equalities = [equality.subs(symbol, expression)
 101             for equality in self.equalities]
 102         inequalities = [inequality.subs(symbol, expression)
 103             for inequality in self.inequalities]
 104         return Polyhedron(equalities, inequalities)
 105
 106     @classmethod
 107     def _fromislbasicset(cls, islbset, symbols):
 108         islconstraints = islhelper.isl_basic_set_constraints(islbset)
 109         equalities = []
 110         inequalities = []
 111         for islconstraint in islconstraints:
 112             constant = libisl.isl_constraint_get_constant_val(islconstraint)
 113             constant = islhelper.isl_val_to_int(constant)
 114             coefficients = {}
 115             for index, symbol in enumerate(symbols):
 116                 coefficient = libisl.isl_constraint_get_coefficient_val(islconstraint,
 117                     libisl.isl_dim_set, index)
 118                 coefficient = islhelper.isl_val_to_int(coefficient)
 119                 if coefficient != 0:
 120                     coefficients[symbol] = coefficient
 121             expression = Expression(coefficients, constant)
 122             if libisl.isl_constraint_is_equality(islconstraint):
 123                 equalities.append(expression)
 124             else:
 125                 inequalities.append(expression)
 126         libisl.isl_basic_set_free(islbset)
 127         self = object().__new__(Polyhedron)
 128         self._equalities = tuple(equalities)
 129         self._inequalities = tuple(inequalities)
 130         self._constraints = tuple(equalities + inequalities)
 131         self._symbols = cls._xsymbols(self._constraints)
 132         self._dimension = len(self._symbols)
 133         return self
 134
 135     @classmethod
 136     def _toislbasicset(cls, equalities, inequalities, symbols):
 137         dimension = len(symbols)
 138         indices = {symbol: index for index, symbol in enumerate(symbols)}
 139         islsp = libisl.isl_space_set_alloc(mainctx, 0, dimension)
 140         islbset = libisl.isl_basic_set_universe(libisl.isl_space_copy(islsp))
 141         islls = libisl.isl_local_space_from_space(islsp)
 142         for equality in equalities:
 143             isleq = libisl.isl_equality_alloc(libisl.isl_local_space_copy(islls))
 144             for symbol, coefficient in equality.coefficients():
 145                 islval = str(coefficient).encode()
 146                 islval = libisl.isl_val_read_from_str(mainctx, islval)
 147                 index = indices[symbol]
 148                 isleq = libisl.isl_constraint_set_coefficient_val(isleq,
 149                     libisl.isl_dim_set, index, islval)
 150             if equality.constant != 0:
 151                 islval = str(equality.constant).encode()
 152                 islval = libisl.isl_val_read_from_str(mainctx, islval)
 153                 isleq = libisl.isl_constraint_set_constant_val(isleq, islval)
 154             islbset = libisl.isl_basic_set_add_constraint(islbset, isleq)
 155         for inequality in inequalities:
 156             islin = libisl.isl_inequality_alloc(libisl.isl_local_space_copy(islls))
 157             for symbol, coefficient in inequality.coefficients():
 158                 islval = str(coefficient).encode()
 159                 islval = libisl.isl_val_read_from_str(mainctx, islval)
 160                 index = indices[symbol]
 161                 islin = libisl.isl_constraint_set_coefficient_val(islin,
 162                     libisl.isl_dim_set, index, islval)
 163             if inequality.constant != 0:
 164                 islval = str(inequality.constant).encode()
 165                 islval = libisl.isl_val_read_from_str(mainctx, islval)
 166                 islin = libisl.isl_constraint_set_constant_val(islin, islval)
 167             islbset = libisl.isl_basic_set_add_constraint(islbset, islin)
 168         return islbset
 169
 170     @classmethod
 171     def fromstring(cls, string):
 172         domain = Domain.fromstring(string)
 173         if not isinstance(domain, Polyhedron):
 174             raise ValueError('non-polyhedral expression: {!r}'.format(string))
 175         return domain
 176
 177     def __repr__(self):
 178         if self.isempty():
 179             return 'Empty'
 180         elif self.isuniverse():
 181             return 'Universe'
 182         else:
 183             strings = []
 184             for equality in self.equalities:
 185                 strings.append('0 == {}'.format(equality))
 186             for inequality in self.inequalities:
 187                 strings.append('0 <= {}'.format(inequality))
 188             if len(strings) == 1:
 189                 return strings[0]
 190             else:
 191                 return 'And({})'.format(', '.join(strings))
 192
 193     @classmethod
 194     def fromsympy(cls, expr):
 195         domain = Domain.fromsympy(expr)
 196         if not isinstance(domain, Polyhedron):
 197             raise ValueError('non-polyhedral expression: {!r}'.format(expr))
 198         return domain
 199
 200     def tosympy(self):
 201         import sympy
 202         constraints = []
 203         for equality in self.equalities:
 204             constraints.append(sympy.Eq(equality.tosympy(), 0))
 205         for inequality in self.inequalities:
 206             constraints.append(sympy.Ge(inequality.tosympy(), 0))
 207         return sympy.And(*constraints)
 208
 209     @classmethod
 210     def _polygon_inner_point(cls, points):
 211         symbols = points[0].symbols
 212         coordinates = {symbol: 0 for symbol in symbols}
 213         for point in points:
 214             for symbol, coordinate in point.coordinates():
 215                 coordinates[symbol] += coordinate
 216         for symbol in symbols:
 217             coordinates[symbol] /= len(points)
 218         return Point(coordinates)
 219
 220     @classmethod
 221     def _sort_polygon_2d(cls, points):
 222         if len(points) <= 3:
 223             return points
 224         o = cls._polygon_inner_point(points)
 225         angles = {}
 226         for m in points:
 227             om = Vector(o, m)
 228             dx, dy = (coordinate for symbol, coordinate in om.coordinates())
 229             angle = math.atan2(dy, dx)
 230             angles[m] = angle
 231         return sorted(points, key=angles.get)
 232
 233     @classmethod
 234     def _sort_polygon_3d(cls, points):
 235         if len(points) <= 3:
 236             return points
 237         o = cls._polygon_inner_point(points)
 238         a = points[0]
 239         oa = Vector(o, a)
 240         norm_oa = oa.norm()
 241         for b in points[1:]:
 242             ob = Vector(o, b)
 243             u = oa.cross(ob)
 244             if not u.isnull():
 245                 u = u.asunit()
 246                 break
 247         else:
 248             raise ValueError('degenerate polygon')
 249         angles = {a: 0.}
 250         for m in points[1:]:
 251             om = Vector(o, m)
 252             normprod = norm_oa * om.norm()
 253             cosinus = oa.dot(om) / normprod
 254             sinus = u.dot(oa.cross(om)) / normprod
 255             angle = math.acos(cosinus)
 256             angle = math.copysign(angle, sinus)
 257             angles[m] = angle
 258         return sorted(points, key=angles.get)
 259
 260     def faces(self):
 261         vertices = self.vertices()
 262         faces = []
 263         for constraint in self.constraints:
 264             face = []
 265             for vertex in vertices:
 266                 if constraint.subs(vertex.coordinates()) == 0:
 267                     face.append(vertex)
 268             faces.append(face)
 269         return faces
 270
 271     def plot(self):
 272         import matplotlib.pyplot as plt
 273         from matplotlib.path import Path
 274         import matplotlib.patches as patches
 275
 276         if len(self.symbols)> 3:
 277             raise TypeError
 278
 279         elif len(self.symbols) == 2:
 280             verts = self.vertices()
 281             points = []
 282             codes = [Path.MOVETO]
 283             for vert in verts:
 284                 pairs = ()
 285                 for sym in sorted(vert, key=Symbol.sortkey):
 286                     num = vert.get(sym)
 287                     pairs = pairs + (num,)
 288                 points.append(pairs)
 289             points.append((0.0, 0.0))
 290             num = len(points)
 291             while num > 2:
 292                 codes.append(Path.LINETO)
 293                 num = num - 1
 294             else:
 295                 codes.append(Path.CLOSEPOLY)
 296             path = Path(points, codes)
 297             fig = plt.figure()
 298             ax = fig.add_subplot(111)
 299             patch = patches.PathPatch(path, facecolor='blue', lw=2)
 300             ax.add_patch(patch)
 301             ax.set_xlim(-5,5)
 302             ax.set_ylim(-5,5)
 303             plt.show()
 304
 305         elif len(self.symbols)==3:
 306             return 0
 307
 308         return points
 309
 310
 311 def _polymorphic(func):
 312     @functools.wraps(func)
 313     def wrapper(left, right):
 314         if isinstance(left, numbers.Rational):
 315             left = Rational(left)
 316         elif not isinstance(left, Expression):
 317             raise TypeError('left must be a a rational number '
 318                 'or a linear expression')
 319         if isinstance(right, numbers.Rational):
 320             right = Rational(right)
 321         elif not isinstance(right, Expression):
 322             raise TypeError('right must be a a rational number '
 323                 'or a linear expression')
 324         return func(left, right)
 325     return wrapper
 326
 327 @_polymorphic
 328 def Lt(left, right):
 329     return Polyhedron([], [right - left - 1])
 330
 331 @_polymorphic
 332 def Le(left, right):
 333     return Polyhedron([], [right - left])
 334
 335 @_polymorphic
 336 def Eq(left, right):
 337     return Polyhedron([left - right], [])
 338
 339 @_polymorphic
 340 def Ne(left, right):
 341     return ~Eq(left, right)
 342
 343 @_polymorphic
 344 def Gt(left, right):
 345     return Polyhedron([], [left - right - 1])
 346
 347 @_polymorphic
 348 def Ge(left, right):
 349     return Polyhedron([], [left - right])
 350
 351
 352 Empty = Eq(1, 0)
 353
 354 Universe = Polyhedron([])