utils.py

# Copyright 2018 D-Wave Systems Inc.
#
#    Licensed under the Apache License, Version 2.0 (the "License");
#    you may not use this file except in compliance with the License.
#    You may obtain a copy of the License at
#
#        http://www.apache.org/licenses/LICENSE-2.0
#
#    Unless required by applicable law or agreed to in writing, software
#    distributed under the License is distributed on an "AS IS" BASIS,
#    WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
#    See the License for the specific language governing permissions and
#    limitations under the License.

from typing import Hashable, FrozenSet, List, Mapping, Optional, Tuple, Union
from numbers import Number

import itertools
import warnings

from collections import Counter

from collections import defaultdict
from functools import partial

import numpy as np

import dimod
from dimod.binary_quadratic_model import BinaryQuadraticModel
from dimod.constrained import ConstrainedQuadraticModel
from dimod.higherorder.polynomial import BinaryPolynomial
from dimod.sampleset import as_samples
from dimod.typing import Bias, Polynomial, SamplesLike, Variable
from dimod.vartypes import as_vartype, Vartype

__all__ = ['make_quadratic', 'make_quadratic_cqm', 'reduce_binary_polynomial']


def _spin_product(variables):
    """A BQM with a gap of 1 that represents the product of two spin variables.

    Note that spin-product requires an auxiliary variable.

    Args:
        variables (list):
            multiplier, multiplicand, product, aux

    Returns:
        :obj:`.BinaryQuadraticModel`

    """
    multiplier, multiplicand, product, aux = variables

    return BinaryQuadraticModel({multiplier: -.5,
                                 multiplicand: -.5,
                                 product: -.5,
                                 aux: -1.},
                                {(multiplier, multiplicand): .5,
                                 (multiplier, product): .5,
                                 (multiplier, aux): 1.,
                                 (multiplicand, product): .5,
                                 (multiplicand, aux): 1.,
                                 (product, aux): 1.},
                                2.,
                                Vartype.SPIN)

def _new_product(variables, u, v):
    # make a new product variable not in variables, then add it
    p = '{}*{}'.format(u, v)
    while p in variables:
        p = '_' + p
    variables.add(p)
    return p


def _new_aux(variables, u, v):
    # make a new auxiliary variable not in variables, then add it
    aux = 'aux{},{}'.format(u, v)
    while aux in variables:
        aux = '_' + aux
    variables.add(aux)
    return aux


def _decrement_count(idx, que, pair):
    count = len(idx[pair])
    que_count = que[count]
    que_count.remove(pair)
    if not que_count:
        del que[count]
    if count > 1:
        que[count - 1].add(pair)


def _remove_old(idx, term, pair):
    idx_pair = idx[pair]
    del idx_pair[term]
    if not idx_pair:
        del idx[pair]


def reduce_binary_polynomial(poly: BinaryPolynomial) -> Tuple[
        List[Tuple[FrozenSet[Hashable], Number]],
        List[Tuple[FrozenSet[Hashable], Hashable]]
    ]:
    """Reduce a binary polynomial to linear and quadratic terms, plus constraints.

    Introduces auxillary variables and constraints to reduce the polynomial
    to linear and quadratic terms.

    Args:
        poly: a binary polynomial that might have higher order terms.

    Returns:
        Two-tuple of a list of terms and their biases, as tuples, and a list of
        the original and auxiliary variables, as a tuple.

    Example:
        >>> poly = dimod.BinaryPolynomial({(0,): -1, (1,): 1, (2,): 1.5, (0, 1): -1, (0, 1, 2): -2}, dimod.BINARY)
        >>> dimod.reduce_binary_polynomial(poly)           # doctest:+SKIP
        ([(frozenset({0}), -1),
          (frozenset({1}), 1),
          (frozenset({2}), 1.5),
          (frozenset({0, 1}), -1),
          (frozenset({'0*1', 2}), -2)],
         [(frozenset({0, 1}), '0*1')])
    """

    variables = poly.variables
    constraints = []

    reduced_terms = []
    idx = defaultdict(dict)
    for item in poly.items():
        term, bias = item
        if len(term) <= 2:
            reduced_terms.append(item)
        else:
            for pair in itertools.combinations(term, 2):
                idx[frozenset(pair)][term] = bias

    que = defaultdict(set)
    for pair, terms in idx.items():
        que[len(terms)].add(pair)

    while idx:
        new_pairs = set()
        most = max(que)
        que_most = que[most]
        pair = que_most.pop()
        if not que_most:
            del que[most]
        terms = idx.pop(pair)

        prod_var = _new_product(variables, *pair)
        constraints.append((pair, prod_var))
        prod_var_set = {prod_var}

        for old_term, bias in terms.items():
            common_subterm = (old_term - pair)
            new_term = common_subterm | prod_var_set

            for old_pair in map(frozenset, itertools.product(pair, common_subterm)):
                _decrement_count(idx, que, old_pair)
                _remove_old(idx, old_term, old_pair)

            for common_pair in map(frozenset, itertools.combinations(common_subterm, 2)):
                idx[common_pair][new_term] = bias
                _remove_old(idx, old_term, common_pair)

            if len(new_term) > 2:
                for new_pair in (frozenset((prod_var, v)) for v in common_subterm):
                    idx[new_pair][new_term] = bias
                    new_pairs.add(new_pair)
            else:
                reduced_terms.append((new_term, bias))

        for new_pair in new_pairs:
            que[len(idx[new_pair])].add(new_pair)

    return reduced_terms, constraints


def _init_quadratic_model(qm, vartype, qm_factory):
    if vartype is None:
        if qm is None:
            raise ValueError("one of vartype or qm must be provided")
        else:
            vartype = qm.vartype
    else:
        vartype = as_vartype(vartype)  # handle other vartype inputs
        if qm is None:
            qm = qm_factory(vartype)
        else:
            qm = qm.change_vartype(vartype, inplace=False)

    # for backwards compatibility, add an info field
    if not hasattr(qm, 'info'):
        qm.info = {}
    qm.info['reduction'] = {}
    return qm, vartype

def _init_binary_polynomial(poly, vartype):
    if not (isinstance(poly, BinaryPolynomial) and (vartype in (poly.vartype, None))):
        poly = BinaryPolynomial(poly, vartype=vartype)
    return poly

def _init_objective(bqm, reduced_terms):
    for term, bias in reduced_terms:
        if len(term) == 2:
            bqm.add_interaction(*term , bias)
        elif len(term) == 1:
            bqm.add_variable(*term, bias)
        elif len(term) == 0:
            bqm.offset += bias
        else:
            # still has higher order terms, this shouldn't happen
            msg = ('Internal error: not all higher-order terms were reduced. '
                   'Please file a bug report.')
            raise RuntimeError(msg)

def make_quadratic_cqm(poly: Union[Polynomial, BinaryPolynomial],
                       vartype: Optional[Vartype] = None,
                       cqm: Optional[ConstrainedQuadraticModel] = None
                       ) -> ConstrainedQuadraticModel:
    """Create a constrained quadratic model from a higher order polynomial.

    Args:
        poly:
            Either a polynomial, as a dict of form `{term: bias, ...}`, where `term`
            is a tuple of one or more variables and `bias` the associated bias,
            or a :class:`.BinaryPolynomial`.

        vartype:
            Variable type for the binary quadratic model. Accepted input values:

            * :class:`~dimod.Vartype.SPIN`, ``'SPIN'``, ``{-1, 1}``
            * :class:`~dimod.Vartype.BINARY`, ``'BINARY'``, ``{0, 1}``

            If ``poly`` is a :class:`.BinaryPolynomial` , ``vartype`` is not required.

        cqm:
            Terms of the reduced polynomial are added to this constrained quadratic
            model. If not provided, a new constrained quadratic model is created.

    Examples:

        >>> poly = {(0,): -1, (1,): 1, (2,): 1.5, (0, 1): -1, (0, 1, 2): -2}
        >>> cqm = dimod.make_quadratic_cqm(poly, dimod.SPIN)

    """
    if not (vartype or isinstance(poly, BinaryPolynomial)):
        raise ValueError("can not infer vartype")
    cqm = cqm or ConstrainedQuadraticModel()
    vartype = vartype or poly.vartype
    poly = _init_binary_polynomial(poly, vartype)
    reduced_terms, constraints = reduce_binary_polynomial(poly)

    def var(x):
        return BinaryQuadraticModel({x: 1.0}, {}, 0.0, vartype)

    for (u, v), p in constraints:
        cqm.add_constraint( var(u)*var(v) - var(p)  == 0, label = f"'{u}'*'{v}' == '{p}'")

    obj = BinaryQuadraticModel(vartype=vartype)
    _init_objective(obj, reduced_terms)
    cqm.set_objective(obj + cqm.objective)

    return cqm


def make_quadratic(poly: Union[Polynomial, BinaryPolynomial], strength: float,
                   vartype: Optional[Vartype] = None,
                   bqm: Optional[BinaryQuadraticModel] = None) -> BinaryQuadraticModel:
    """Create a binary quadratic model from a higher order polynomial.

    Args:
        poly:
            Either a polynomial, as a dict of form `{term: bias, ...}`, where `term`
            is a tuple of one or more variables and `bias` the associated bias,
            or a :class:`.BinaryPolynomial`.

        strength:
            Energy penalty for violating the product constraint.
            Insufficient strength can result in the binary quadratic model not
            having the same minimizations as the polynomial.

        vartype:
            Variable type for the binary quadratic model. Accepted input values:

            * :class:`~dimod.Vartype.SPIN`, ``'SPIN'``, ``{-1, 1}``
            * :class:`~dimod.Vartype.BINARY`, ``'BINARY'``, ``{0, 1}``

            If ``bqm`` is provided, ``vartype`` is not required.

        bqm:
            Terms of the reduced polynomial are added to this binary quadratic model.
            If not provided, a new binary quadratic model is created.

    Examples:

        >>> poly = {(0,): -1, (1,): 1, (2,): 1.5, (0, 1): -1, (0, 1, 2): -2}
        >>> bqm = dimod.make_quadratic(poly, 5.0, dimod.SPIN)

    """
    from dimod.generators import and_gate

    bqm, vartype = _init_quadratic_model(bqm, vartype, BinaryQuadraticModel)
    poly = _init_binary_polynomial(poly, vartype)

    variables = set().union(*poly)
    reduced_terms, constraints = reduce_binary_polynomial(poly)

    for (u, v), p in constraints:

        # add a constraint enforcing the relationship between p == u*v
        if vartype is Vartype.BINARY:
            constraint = and_gate(u, v, p)
            bqm.info['reduction'][(u, v)] = {'product': p}
        elif vartype is Vartype.SPIN:
            aux = _new_aux(variables, u, v)  # need an aux in SPIN-space
            constraint = _spin_product([u, v, p, aux])
            bqm.info['reduction'][(u, v)] = {'product': p, 'auxiliary': aux}
        else:
            raise RuntimeError("unknown vartype: {!r}".format(vartype))

        # scale constraint and update the polynomial with it
        constraint.scale(strength)
        for v, bias in constraint.linear.items():
            bqm.add_variable(v, bias)
        for uv, bias in constraint.quadratic.items():
            bqm.add_interaction(*uv, bias)
        bqm.offset += constraint.offset

    _init_objective(bqm, reduced_terms)

    return bqm


def poly_energy(sample_like: SamplesLike,
                poly: Union[Polynomial, BinaryPolynomial]) -> float:
    """Calculate energy of a sample from a higher order polynomial.

    Args:
         sample_like:
            A raw sample. `samples-like` is an extension of NumPy's
            array_like structure. See :func:`.as_samples`.

        poly (dict):
            Either a polynomial, as a dict of form `{term: bias, ...}`, where `term`
            is a tuple of one or more variables and `bias` the associated bias,
            or a :class:`.BinaryPolynomial`.
            Variable labeling/indexing here must match that of ``sample_like``

    Returns: Energy of the sample.

    Examples:
        >>> poly = dimod.BinaryPolynomial({'a': -1, ('a', 'b'): 1, ('a', 'b', 'c'): -1},
        ...                               dimod.BINARY)
        >>> sample = {'a': 1, 'b': 1, 'c': 0}
        >>> dimod.poly_energy(sample, poly)
        0.0

    """
    return BinaryPolynomial(poly, 'SPIN').energy(sample_like)


def poly_energies(samples_like: SamplesLike,
                  poly: Union[Polynomial, BinaryPolynomial]) -> np.ndarray:
    """Calculates energy of samples from a higher order polynomial.

    Args:
        samples_like:
            A collection of raw samples. `samples-like` is an extension of
            NumPy's array_like structure. See :func:`.as_samples`.

        poly:
            Either a polynomial, as a dict of form `{term: bias, ...}`, where `term`
            is a tuple of one or more variables and `bias` the associated bias,
            or a :class:`.BinaryPolynomial`. Variable labeling/indexing here must
            match that of ``samples_like``.

    Returns: Energies of the samples.

    Examples:
        >>> poly = dimod.BinaryPolynomial({'a': -1, ('a', 'b'): 1, ('a', 'b', 'c'): -1},
        ...                               dimod.BINARY)
        >>> samples = [{'a': 1, 'b': 1, 'c': 0},
        ...            {'a': 1, 'b': 1, 'c': 1}]
        >>> dimod.poly_energies(samples, poly)
        array([ 0., -1.])

    """
    return BinaryPolynomial(poly, 'SPIN').energies(samples_like)