Extract/Improve Polyhedron class

loki/analyse/util_polyhedron.py

@@ -0,0 +1,354 @@
+# (C) Copyright 2018- ECMWF.
+# This software is licensed under the terms of the Apache Licence Version 2.0
+# which can be obtained at http://www.apache.org/licenses/LICENSE-2.0.
+# In applying this licence, ECMWF does not waive the privileges and immunities
+# granted to it by virtue of its status as an intergovernmental organisation
+# nor does it submit to any jurisdiction.
+
+from typing import List
+import numpy as np
+from loki.ir import Loop
+from loki.expression import (
+    symbols as sym,
+    FindVariables,
+    simplify,
+    is_constant,
+    accumulate_polynomial_terms,
+)
+from loki.tools import as_tuple
+
+__all__ = ["Polyhedron"]
+
+
+class Polyhedron:
+    """
+    Halfspace representation of a (convex) polyhedron.
+
+    A polyhedron `P c R^d` is described by a set of inequalities, in matrix form
+    ```
+    P = { x=[x1,...,xd]^T c R^d | Ax <= b }
+    ```
+    with n-by-d matrix `A` and d-dimensional right hand side `b`.
+
+    In loop transformations, polyhedrons are used to represent iteration spaces of
+    d-deep loop nests.
+
+
+    Parameters
+    ----------
+    A : numpy.array
+        The representation matrix A.
+    b : numpy.array
+        The right hand-side vector b.
+    variables : list, optional
+        List of variables representing the dimensions in the polyhedron.
+
+    Attributes
+    ----------
+    A : numpy.array
+        The representation matrix A.
+    b : numpy.array
+        The right hand-side vector b.
+    variables : list, optional, default = None
+    """
+
+    def __init__(self, A, b, variables=None):
+        A = np.array(A, dtype=np.dtype(int))
+        b = np.array(b, dtype=np.dtype(int))
+        assert A.ndim == 2 and b.ndim == 1
+        assert A.shape[0] == b.shape[0]
+        self.A = A
+        self.b = b
+
+        self.variables = None
+        self.variable_names = None
+        if variables is not None:
+            assert len(variables) == A.shape[1]
+            self.variables = variables
+            self.variable_names = [v.name.lower() for v in self.variables]
+
+    def __str__(self):
+        str_A = "[" + ", ".join([str(row) for row in self.A]) + "]"
+        str_b = f"[{', '.join(map(str, self.b))}]"
+        str_variable_names = (
+            f"[{', '.join(map(str, self.variable_names))}]"
+            if self.variable_names
+            else "[]"
+        )
+
+        return f"Polyhedron(\n\tA={str_A}, \n\tb={str_b}, \n\tvariables={str_variable_names}\n)"
+
+    def __repr__(self):
+        return str(self)
+
+    def _has_satisfiable_constant_restrictions(self):
+        """
+        Check whether the constant restrictions of the polyhedron are satisfiable.
+
+        This method checks if 0 <= b, assuming that A x = 0.
+
+        Returns:
+        bool: True if all constant restrictions are satisfiable, False otherwise.
+
+        """
+
+        return (0 <= self.b).all()
+
+    def is_empty(self):
+        """
+        Determine whether a polyhedron is empty.
+
+        A polyhedron is considered empty under the following conditions:
+        1. It contains no inequalities.
+        2. It spans no space, which is a nontrivial problem. The simplest case is when it has an empty
+        matrix A and does not satisfy the constant restrictions 0 <= b.
+
+        Notes
+        -----
+        An empty polyhedron implies that it has no valid solutions or feasible points within its boundaries.
+        This function is expected to be called only for polyhedrons with an empty matrix.
+
+        Returns
+        -------
+        bool
+            True if the polyhedron is empty; False if it is not.
+        """
+        if self.A.size == 0:
+            return self.b.size == 0 or not self._has_satisfiable_constant_restrictions()
+
+        raise RuntimeError(
+            """
+            Checking if a polyhedron with a non-empty matrix spans no space is a nontrivial problem.
+            This function is expected to be only called upon polyhedrons with an empty matrix!
+            """
+        )
+
+    def variable_to_index(self, variable):
+        if self.variable_names is None:
+            raise RuntimeError("No variables list associated with polyhedron.")
+        if isinstance(variable, sym.TypedSymbol):
+            variable = variable.name.lower()
+        assert isinstance(variable, str)
+        return self.variable_names.index(variable)
+
+    @staticmethod
+    def _to_literal(value):
+        if value < 0:
+            return sym.Product((-1, sym.IntLiteral(abs(value))))
+        return sym.IntLiteral(value)
+
+    def lower_bounds(self, index_or_variable, ignore_variables=None):
+        """
+        Return all lower bounds imposed on a variable.
+
+        The lower bounds for the variable `j` are given by the index set:
+        ```
+        L = {i | A_ij < 0, i in {0, ..., d-1}}
+        ```
+        Parameters
+        ----------
+        index_or_variable : int or str or sym.Array or sym.Scalar
+            The index, name, or expression symbol for which the lower bounds are produced.
+        ignore_variables : list or None, optional
+            A list of variable names, indices, or symbols for which constraints should be ignored
+            if they depend on one of them.
+
+        Returns
+        -------
+        list
+            The bounds for the specified variable.
+        """
+        if isinstance(index_or_variable, int):
+            j = index_or_variable
+        else:
+            j = self.variable_to_index(index_or_variable)
+
+        if ignore_variables:
+            ignore_variables = [
+                i if isinstance(i, int) else self.variable_to_index(i)
+                for i in ignore_variables
+            ]
+
+        bounds = []
+        for i in range(self.A.shape[0]):
+            if self.A[i, j] < 0:
+                if ignore_variables and any(
+                    self.A[i, k] != 0 for k in ignore_variables
+                ):
+                    # Skip constraint that depends on any of the ignored variables
+                    continue
+
+                components = [
+                    self._to_literal(self.A[i, k]) * self.variables[k]
+                    for k in range(self.A.shape[1])
+                    if k != j and self.A[i, k] != 0
+                ]
+                if not components:
+                    lhs = sym.IntLiteral(0)
+                elif len(components) == 1:
+                    lhs = components[0]
+                else:
+                    lhs = sym.Sum(as_tuple(components))
+                bounds += [
+                    simplify(
+                        sym.Quotient(
+                            self._to_literal(self.b[i]) - lhs,
+                            self._to_literal(self.A[i, j]),
+                        )
+                    )
+                ]
+        return bounds
+
+    def upper_bounds(self, index_or_variable, ignore_variables=None):
+        """
+        Return all upper bounds imposed on a variable.
+
+        The upper bounds for the variable `j` are given by the index set:
+        ```
+        U = {i | A_ij > 0, i in {0, ..., d-1}}
+        ```
+        Parameters
+        ----------
+        index_or_variable : int or str or sym.Array or sym.Scalar
+            The index, name, or expression symbol for which the upper bounds are produced.
+        ignore_variables : list or None, optional
+            A list of variable names, indices, or symbols for which constraints should be ignored
+            if they depend on one of them.
+
+        Returns
+        -------
+        list
+            The bounds for the specified variable.
+        """
+        if isinstance(index_or_variable, int):
+            j = index_or_variable
+        else:
+            j = self.variable_to_index(index_or_variable)
+
+        if ignore_variables:
+            ignore_variables = [
+                i if isinstance(i, int) else self.variable_to_index(i)
+                for i in ignore_variables
+            ]
+
+        bounds = []
+        for i in range(self.A.shape[0]):
+            if self.A[i, j] > 0:
+                if ignore_variables and any(
+                    self.A[i, k] != 0 for k in ignore_variables
+                ):
+                    # Skip constraint that depends on any of the ignored variables
+                    continue
+
+                components = [
+                    self._to_literal(self.A[i, k]) * self.variables[k]
+                    for k in range(self.A.shape[1])
+                    if k != j and self.A[i, k] != 0
+                ]
+                if not components:
+                    lhs = sym.IntLiteral(0)
+                elif len(components) == 1:
+                    lhs = components[0]
+                else:
+                    lhs = sym.Sum(as_tuple(components))
+                bounds += [
+                    simplify(
+                        sym.Quotient(
+                            self._to_literal(self.b[i]) - lhs,
+                            self._to_literal(self.A[i, j]),
+                        )
+                    )
+                ]
+        return bounds
+
+    @staticmethod
+    def generate_entries_for_lower_bound(bound, variables, index):
+        """
+        Helper routine to generate matrix and right-hand side entries for a given lower bound.
+
+        Note that this routine can only handle affine bounds, which means expressions that are
+        constant or can be reduced to a linear polynomial.
+
+        Upper bounds can be derived from this by multiplying the left-hand side and right-hand side
+        with -1.
+
+        Parameters
+        ----------
+        bound : int or str or sym.Array or sym.Scalar
+            The expression representing the lower bound.
+        variables : list of str
+            The list of variable names.
+        index : int
+            The index of the variable constrained by this bound.
+
+        Returns
+        -------
+        tuple(np.array, np.array)
+            The pair ``(lhs, rhs)`` of the row in the matrix inequality, where `lhs` is the left-hand side
+            and `rhs` is the right-hand side.
+        """
+        supported_types = (sym.TypedSymbol, sym.MetaSymbol, sym.Sum, sym.Product)
+        if not (is_constant(bound) or isinstance(bound, supported_types)):
+            raise ValueError(f"Cannot derive inequality from bound {str(bound)}")
+        summands = accumulate_polynomial_terms(bound)
+        b = -summands.pop(1, 0)  # Constant term or 0
+        A = np.zeros([1, len(variables)], dtype=np.dtype(int))
+        A[0, index] = -1
+        for base, coef in summands.items():
+            if not len(base) == 1:
+                raise ValueError(f"Non-affine bound {str(bound)}")
+            A[0, variables.index(base[0].name.lower())] = coef
+        return A, b
+
+    @classmethod
+    def from_loop_ranges(cls, loop_variables, loop_ranges):
+        """
+        Create polyhedron from a list of loop ranges and associated variables.
+        """
+        assert len(loop_ranges) == len(loop_variables)
+
+        # Add any variables that are not loop variables to the vector of variables
+        variables = list(loop_variables)
+        variable_names = [v.name.lower() for v in variables]
+        for v in sorted(
+            FindVariables().visit(loop_ranges), key=lambda v: v.name.lower()
+        ):
+            if v.name.lower() not in variable_names:
+                variables += [v]
+                variable_names += [v.name.lower()]
+
+        n = 2 * len(loop_ranges)
+        d = len(variables)
+        A = np.zeros([n, d], dtype=np.dtype(int))
+        b = np.zeros([n], dtype=np.dtype(int))
+
+        for i, (loop_variable, loop_range) in enumerate(
+            zip(loop_variables, loop_ranges)
+        ):
+            assert loop_range.step is None or loop_range.step == "1"
+            j = variables.index(loop_variable.name.lower())
+
+            # Create inequality from lower bound
+            lhs, rhs = cls.generate_entries_for_lower_bound(
+                loop_range.start, variable_names, j
+            )
+            A[2 * i, :] = lhs
+            b[2 * i] = rhs
+
+            # Create inequality from upper bound
+            lhs, rhs = cls.generate_entries_for_lower_bound(
+                loop_range.stop, variable_names, j
+            )
+            A[2 * i + 1, :] = -lhs
+            b[2 * i + 1] = -rhs
+
+        return cls(A, b, variables)
+
+    @classmethod
+    def from_nested_loops(cls, nested_loops: List[Loop]):
+        """
+        Helper function, for creating a polyhedron from a list of loops.
+        """
+        return cls.from_loop_ranges(
+            [l.variable for l in nested_loops], [l.bounds for l in nested_loops]
+        )