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constraint.py
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constraint.py
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#!/usr/bin/python
#
# Copyright (c) 2005-2014 - Gustavo Niemeyer <gustavo@niemeyer.net>
#
# All rights reserved.
#
# Redistribution and use in source and binary forms, with or without
# modification, are permitted provided that the following conditions are met:
#
# 1. Redistributions of source code must retain the above copyright notice, this
# list of conditions and the following disclaimer.
# 2. Redistributions in binary form must reproduce the above copyright notice,
# this list of conditions and the following disclaimer in the documentation
# and/or other materials provided with the distribution.
#
# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
# AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
# IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
# DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE
# FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
# SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
# CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
# OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
# OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
from __future__ import absolute_import, division, print_function
import random
import copy
__all__ = [
"Problem",
"Variable",
"Domain",
"Unassigned",
"Solver",
"BacktrackingSolver",
"RecursiveBacktrackingSolver",
"MinConflictsSolver",
"Constraint",
"FunctionConstraint",
"AllDifferentConstraint",
"AllEqualConstraint",
"MaxSumConstraint",
"ExactSumConstraint",
"MinSumConstraint",
"InSetConstraint",
"NotInSetConstraint",
"SomeInSetConstraint",
"SomeNotInSetConstraint",
]
class Problem(object):
"""
Class used to define a problem and retrieve solutions
"""
def __init__(self, solver=None):
"""
@param solver: Problem solver used to find solutions
(default is L{BacktrackingSolver})
@type solver: instance of a L{Solver} subclass
"""
self._solver = solver or BacktrackingSolver()
self._constraints = []
self._variables = {}
def reset(self):
"""
Reset the current problem definition
Example:
>>> problem = Problem()
>>> problem.addVariable("a", [1, 2])
>>> problem.reset()
>>> problem.getSolution()
>>>
"""
del self._constraints[:]
self._variables.clear()
def setSolver(self, solver):
"""
Change the problem solver currently in use
Example:
>>> solver = BacktrackingSolver()
>>> problem = Problem(solver)
>>> problem.getSolver() is solver
True
@param solver: New problem solver
@type solver: instance of a C{Solver} subclass
"""
self._solver = solver
def getSolver(self):
"""
Obtain the problem solver currently in use
Example:
>>> solver = BacktrackingSolver()
>>> problem = Problem(solver)
>>> problem.getSolver() is solver
True
@return: Solver currently in use
@rtype: instance of a L{Solver} subclass
"""
return self._solver
def addVariable(self, variable, domain):
"""
Add a variable to the problem
Example:
>>> problem = Problem()
>>> problem.addVariable("a", [1, 2])
>>> problem.getSolution() in ({'a': 1}, {'a': 2})
True
@param variable: Object representing a problem variable
@type variable: hashable object
@param domain: Set of items defining the possible values that
the given variable may assume
@type domain: list, tuple, or instance of C{Domain}
"""
if variable in self._variables:
msg = "Tried to insert duplicated variable %s" % repr(variable)
raise ValueError(msg)
if isinstance(domain, Domain):
domain = copy.deepcopy(domain)
elif hasattr(domain, "__getitem__"):
domain = Domain(domain)
else:
msg = "Domains must be instances of subclasses of the Domain class"
raise TypeError(msg)
if not domain:
raise ValueError("Domain is empty")
self._variables[variable] = domain
def addVariables(self, variables, domain):
"""
Add one or more variables to the problem
Example:
>>> problem = Problem()
>>> problem.addVariables(["a", "b"], [1, 2, 3])
>>> solutions = problem.getSolutions()
>>> len(solutions)
9
>>> {'a': 3, 'b': 1} in solutions
True
@param variables: Any object containing a sequence of objects
represeting problem variables
@type variables: sequence of hashable objects
@param domain: Set of items defining the possible values that
the given variables may assume
@type domain: list, tuple, or instance of C{Domain}
"""
for variable in variables:
self.addVariable(variable, domain)
def addConstraint(self, constraint, variables=None):
"""
Add a constraint to the problem
Example:
>>> problem = Problem()
>>> problem.addVariables(["a", "b"], [1, 2, 3])
>>> problem.addConstraint(lambda a, b: b == a+1, ["a", "b"])
>>> solutions = problem.getSolutions()
>>>
@param constraint: Constraint to be included in the problem
@type constraint: instance a L{Constraint} subclass or a
function to be wrapped by L{FunctionConstraint}
@param variables: Variables affected by the constraint (default to
all variables). Depending on the constraint type
the order may be important.
@type variables: set or sequence of variables
"""
if not isinstance(constraint, Constraint):
if callable(constraint):
constraint = FunctionConstraint(constraint)
else:
msg = "Constraints must be instances of subclasses " "of the Constraint class"
raise ValueError(msg)
self._constraints.append((constraint, variables))
def getSolution(self):
"""
Find and return a solution to the problem
Example:
>>> problem = Problem()
>>> problem.getSolution() is None
True
>>> problem.addVariables(["a"], [42])
>>> problem.getSolution()
{'a': 42}
@return: Solution for the problem
@rtype: dictionary mapping variables to values
"""
domains, constraints, vconstraints = self._getArgs()
if not domains:
return None
return self._solver.getSolution(domains, constraints, vconstraints)
def getSolutions(self):
"""
Find and return all solutions to the problem
Example:
>>> problem = Problem()
>>> problem.getSolutions() == []
True
>>> problem.addVariables(["a"], [42])
>>> problem.getSolutions()
[{'a': 42}]
@return: All solutions for the problem
@rtype: list of dictionaries mapping variables to values
"""
domains, constraints, vconstraints = self._getArgs()
if not domains:
return []
return self._solver.getSolutions(domains, constraints, vconstraints)
def getSolutionIter(self):
"""
Return an iterator to the solutions of the problem
Example:
>>> problem = Problem()
>>> list(problem.getSolutionIter()) == []
True
>>> problem.addVariables(["a"], [42])
>>> iter = problem.getSolutionIter()
>>> next(iter)
{'a': 42}
>>> next(iter)
Traceback (most recent call last):
File "<stdin>", line 1, in ?
StopIteration
"""
domains, constraints, vconstraints = self._getArgs()
if not domains:
return iter(())
return self._solver.getSolutionIter(domains, constraints, vconstraints)
def _getArgs(self):
domains = self._variables.copy()
allvariables = domains.keys()
constraints = []
for constraint, variables in self._constraints:
if not variables:
variables = list(allvariables)
constraints.append((constraint, variables))
vconstraints = {}
for variable in domains:
vconstraints[variable] = []
for constraint, variables in constraints:
for variable in variables:
vconstraints[variable].append((constraint, variables))
for constraint, variables in constraints[:]:
constraint.preProcess(variables, domains, constraints, vconstraints)
for domain in domains.values():
domain.resetState()
if not domain:
return None, None, None
# doArc8(getArcs(domains, constraints), domains, {})
return domains, constraints, vconstraints
# ----------------------------------------------------------------------
# Solvers
# ----------------------------------------------------------------------
def getArcs(domains, constraints):
"""
Return a dictionary mapping pairs (arcs) of constrained variables
@attention: Currently unused.
"""
arcs = {}
for x in constraints:
constraint, variables = x
if len(variables) == 2:
variable1, variable2 = variables
arcs.setdefault(variable1, {}).setdefault(variable2, []).append(x)
arcs.setdefault(variable2, {}).setdefault(variable1, []).append(x)
return arcs
def doArc8(arcs, domains, assignments):
"""
Perform the ARC-8 arc checking algorithm and prune domains
@attention: Currently unused.
"""
check = dict.fromkeys(domains, True)
while check:
variable, _ = check.popitem()
if variable not in arcs or variable in assignments:
continue
domain = domains[variable]
arcsvariable = arcs[variable]
for othervariable in arcsvariable:
arcconstraints = arcsvariable[othervariable]
if othervariable in assignments:
otherdomain = [assignments[othervariable]]
else:
otherdomain = domains[othervariable]
if domain:
# changed = False
for value in domain[:]:
assignments[variable] = value
if otherdomain:
for othervalue in otherdomain:
assignments[othervariable] = othervalue
for constraint, variables in arcconstraints:
if not constraint(
variables, domains, assignments, True
):
break
else:
# All constraints passed. Value is safe.
break
else:
# All othervalues failed. Kill value.
domain.hideValue(value)
# changed = True
del assignments[othervariable]
del assignments[variable]
# if changed:
# check.update(dict.fromkeys(arcsvariable))
if not domain:
return False
return True
class Solver(object):
"""Abstract base class for solvers
"""
def getSolution(self, domains, constraints, vconstraints):
"""
Return one solution for the given problem
@param domains: Dictionary mapping variables to their domains
@type domains: dict
@param constraints: List of pairs of (constraint, variables)
@type constraints: list
@param vconstraints: Dictionary mapping variables to a list of
constraints affecting the given variables.
@type vconstraints: dict
"""
msg = "%s is an abstract class" % self.__class__.__name__
raise NotImplementedError(msg)
def getSolutions(self, domains, constraints, vconstraints):
"""
Return all solutions for the given problem
@param domains: Dictionary mapping variables to domains
@type domains: dict
@param constraints: List of pairs of (constraint, variables)
@type constraints: list
@param vconstraints: Dictionary mapping variables to a list of
constraints affecting the given variables.
@type vconstraints: dict
"""
msg = "%s provides only a single solution" % self.__class__.__name__
raise NotImplementedError(msg)
def getSolutionIter(self, domains, constraints, vconstraints):
"""
Return an iterator for the solutions of the given problem
@param domains: Dictionary mapping variables to domains
@type domains: dict
@param constraints: List of pairs of (constraint, variables)
@type constraints: list
@param vconstraints: Dictionary mapping variables to a list of
constraints affecting the given variables.
@type vconstraints: dict
"""
msg = "%s doesn't provide iteration" % self.__class__.__name__
raise NotImplementedError(msg)
class BacktrackingSolver(Solver):
"""
Problem solver with backtracking capabilities
Examples:
>>> result = [[('a', 1), ('b', 2)],
... [('a', 1), ('b', 3)],
... [('a', 2), ('b', 3)]]
>>> problem = Problem(BacktrackingSolver())
>>> problem.addVariables(["a", "b"], [1, 2, 3])
>>> problem.addConstraint(lambda a, b: b > a, ["a", "b"])
>>> solution = problem.getSolution()
>>> sorted(solution.items()) in result
True
>>> for solution in problem.getSolutionIter():
... sorted(solution.items()) in result
True
True
True
>>> for solution in problem.getSolutions():
... sorted(solution.items()) in result
True
True
True
"""
def __init__(self, forwardcheck=True):
"""
@param forwardcheck: If false forward checking will not be requested
to constraints while looking for solutions
(default is true)
@type forwardcheck: bool
"""
self._forwardcheck = forwardcheck
def getSolutionIter(self, domains, constraints, vconstraints):
forwardcheck = self._forwardcheck
assignments = {}
queue = []
while True:
# Mix the Degree and Minimum Remaing Values (MRV) heuristics
lst = [
(-len(vconstraints[variable]), len(domains[variable]), variable)
for variable in domains
]
lst.sort()
for item in lst:
if item[-1] not in assignments:
# Found unassigned variable
variable = item[-1]
values = domains[variable][:]
if forwardcheck:
pushdomains = [
domains[x]
for x in domains
if x not in assignments and x != variable
]
else:
pushdomains = None
break
else:
# No unassigned variables. We've got a solution. Go back
# to last variable, if there's one.
yield assignments.copy()
if not queue:
return
variable, values, pushdomains = queue.pop()
if pushdomains:
for domain in pushdomains:
domain.popState()
while True:
# We have a variable. Do we have any values left?
if not values:
# No. Go back to last variable, if there's one.
del assignments[variable]
while queue:
variable, values, pushdomains = queue.pop()
if pushdomains:
for domain in pushdomains:
domain.popState()
if values:
break
del assignments[variable]
else:
return
# Got a value. Check it.
assignments[variable] = values.pop()
if pushdomains:
for domain in pushdomains:
domain.pushState()
for constraint, variables in vconstraints[variable]:
if not constraint(variables, domains, assignments, pushdomains):
# Value is not good.
break
else:
break
if pushdomains:
for domain in pushdomains:
domain.popState()
# Push state before looking for next variable.
queue.append((variable, values, pushdomains))
raise RuntimeError("Can't happen")
def getSolution(self, domains, constraints, vconstraints):
iter = self.getSolutionIter(domains, constraints, vconstraints)
try:
return next(iter)
except StopIteration:
return None
def getSolutions(self, domains, constraints, vconstraints):
return list(self.getSolutionIter(domains, constraints, vconstraints))
class RecursiveBacktrackingSolver(Solver):
"""
Recursive problem solver with backtracking capabilities
Examples:
>>> result = [[('a', 1), ('b', 2)],
... [('a', 1), ('b', 3)],
... [('a', 2), ('b', 3)]]
>>> problem = Problem(RecursiveBacktrackingSolver())
>>> problem.addVariables(["a", "b"], [1, 2, 3])
>>> problem.addConstraint(lambda a, b: b > a, ["a", "b"])
>>> solution = problem.getSolution()
>>> sorted(solution.items()) in result
True
>>> for solution in problem.getSolutions():
... sorted(solution.items()) in result
True
True
True
>>> problem.getSolutionIter()
Traceback (most recent call last):
...
NotImplementedError: RecursiveBacktrackingSolver doesn't provide iteration
"""
def __init__(self, forwardcheck=True):
"""
@param forwardcheck: If false forward checking will not be requested
to constraints while looking for solutions
(default is true)
@type forwardcheck: bool
"""
self._forwardcheck = forwardcheck
def recursiveBacktracking(
self, solutions, domains, vconstraints, assignments, single
):
# Mix the Degree and Minimum Remaing Values (MRV) heuristics
lst = [
(-len(vconstraints[variable]), len(domains[variable]), variable)
for variable in domains
]
lst.sort()
for item in lst:
if item[-1] not in assignments:
# Found an unassigned variable. Let's go.
break
else:
# No unassigned variables. We've got a solution.
solutions.append(assignments.copy())
return solutions
variable = item[-1]
assignments[variable] = None
forwardcheck = self._forwardcheck
if forwardcheck:
pushdomains = [domains[x] for x in domains if x not in assignments]
else:
pushdomains = None
for value in domains[variable]:
assignments[variable] = value
if pushdomains:
for domain in pushdomains:
domain.pushState()
for constraint, variables in vconstraints[variable]:
if not constraint(variables, domains, assignments, pushdomains):
# Value is not good.
break
else:
# Value is good. Recurse and get next variable.
self.recursiveBacktracking(
solutions, domains, vconstraints, assignments, single
)
if solutions and single:
return solutions
if pushdomains:
for domain in pushdomains:
domain.popState()
del assignments[variable]
return solutions
def getSolution(self, domains, constraints, vconstraints):
solutions = self.recursiveBacktracking([], domains, vconstraints, {}, True)
return solutions and solutions[0] or None
def getSolutions(self, domains, constraints, vconstraints):
return self.recursiveBacktracking([], domains, vconstraints, {}, False)
class MinConflictsSolver(Solver):
"""
Problem solver based on the minimum conflicts theory
Examples:
>>> result = [[('a', 1), ('b', 2)],
... [('a', 1), ('b', 3)],
... [('a', 2), ('b', 3)]]
>>> problem = Problem(MinConflictsSolver())
>>> problem.addVariables(["a", "b"], [1, 2, 3])
>>> problem.addConstraint(lambda a, b: b > a, ["a", "b"])
>>> solution = problem.getSolution()
>>> sorted(solution.items()) in result
True
>>> problem.getSolutions()
Traceback (most recent call last):
...
NotImplementedError: MinConflictsSolver provides only a single solution
>>> problem.getSolutionIter()
Traceback (most recent call last):
...
NotImplementedError: MinConflictsSolver doesn't provide iteration
"""
def __init__(self, steps=1000):
"""
@param steps: Maximum number of steps to perform before giving up
when looking for a solution (default is 1000)
@type steps: int
"""
self._steps = steps
def getSolution(self, domains, constraints, vconstraints):
assignments = {}
# Initial assignment
for variable in domains:
assignments[variable] = random.choice(domains[variable])
for _ in range(self._steps):
conflicted = False
lst = list(domains.keys())
random.shuffle(lst)
for variable in lst:
# Check if variable is not in conflict
for constraint, variables in vconstraints[variable]:
if not constraint(variables, domains, assignments):
break
else:
continue
# Variable has conflicts. Find values with less conflicts.
mincount = len(vconstraints[variable])
minvalues = []
for value in domains[variable]:
assignments[variable] = value
count = 0
for constraint, variables in vconstraints[variable]:
if not constraint(variables, domains, assignments):
count += 1
if count == mincount:
minvalues.append(value)
elif count < mincount:
mincount = count
del minvalues[:]
minvalues.append(value)
# Pick a random one from these values.
assignments[variable] = random.choice(minvalues)
conflicted = True
if not conflicted:
return assignments
return None
# ----------------------------------------------------------------------
# Variables
# ----------------------------------------------------------------------
class Variable(object):
"""
Helper class for variable definition
Using this class is optional, since any hashable object,
including plain strings and integers, may be used as variables.
"""
def __init__(self, name):
"""
@param name: Generic variable name for problem-specific purposes
@type name: string
"""
self.name = name
def __repr__(self):
return self.name
Unassigned = Variable("Unassigned") #: Helper object instance representing unassigned values
# ----------------------------------------------------------------------
# Domains
# ----------------------------------------------------------------------
class Domain(list):
"""
Class used to control possible values for variables
When list or tuples are used as domains, they are automatically
converted to an instance of that class.
"""
def __init__(self, set):
"""
@param set: Set of values that the given variables may assume
@type set: set of objects comparable by equality
"""
list.__init__(self, set)
self._hidden = []
self._states = []
def resetState(self):
"""
Reset to the original domain state, including all possible values
"""
self.extend(self._hidden)
del self._hidden[:]
del self._states[:]
def pushState(self):
"""
Save current domain state
Variables hidden after that call are restored when that state
is popped from the stack.
"""
self._states.append(len(self))
def popState(self):
"""
Restore domain state from the top of the stack
Variables hidden since the last popped state are then available
again.
"""
diff = self._states.pop() - len(self)
if diff:
self.extend(self._hidden[-diff:])
del self._hidden[-diff:]
def hideValue(self, value):
"""
Hide the given value from the domain
After that call the given value won't be seen as a possible value
on that domain anymore. The hidden value will be restored when the
previous saved state is popped.
@param value: Object currently available in the domain
"""
list.remove(self, value)
self._hidden.append(value)
# ----------------------------------------------------------------------
# Constraints
# ----------------------------------------------------------------------
class Constraint(object):
"""
Abstract base class for constraints
"""
def __call__(self, variables, domains, assignments, forwardcheck=False):
"""
Perform the constraint checking
If the forwardcheck parameter is not false, besides telling if
the constraint is currently broken or not, the constraint
implementation may choose to hide values from the domains of
unassigned variables to prevent them from being used, and thus
prune the search space.
@param variables: Variables affected by that constraint, in the
same order provided by the user
@type variables: sequence
@param domains: Dictionary mapping variables to their domains
@type domains: dict
@param assignments: Dictionary mapping assigned variables to their
current assumed value
@type assignments: dict
@param forwardcheck: Boolean value stating whether forward checking
should be performed or not
@return: Boolean value stating if this constraint is currently
broken or not
@rtype: bool
"""
return True
def preProcess(self, variables, domains, constraints, vconstraints):
"""
Preprocess variable domains
This method is called before starting to look for solutions,
and is used to prune domains with specific constraint logic
when possible. For instance, any constraints with a single
variable may be applied on all possible values and removed,
since they may act on individual values even without further
knowledge about other assignments.
@param variables: Variables affected by that constraint, in the
same order provided by the user
@type variables: sequence
@param domains: Dictionary mapping variables to their domains
@type domains: dict
@param constraints: List of pairs of (constraint, variables)
@type constraints: list
@param vconstraints: Dictionary mapping variables to a list of
constraints affecting the given variables.
@type vconstraints: dict
"""
if len(variables) == 1:
variable = variables[0]
domain = domains[variable]
for value in domain[:]:
if not self(variables, domains, {variable: value}):
domain.remove(value)
constraints.remove((self, variables))
vconstraints[variable].remove((self, variables))
def forwardCheck(self, variables, domains, assignments, _unassigned=Unassigned):
"""
Helper method for generic forward checking
Currently, this method acts only when there's a single
unassigned variable.
@param variables: Variables affected by that constraint, in the
same order provided by the user
@type variables: sequence
@param domains: Dictionary mapping variables to their domains
@type domains: dict
@param assignments: Dictionary mapping assigned variables to their
current assumed value
@type assignments: dict
@return: Boolean value stating if this constraint is currently
broken or not
@rtype: bool
"""
unassignedvariable = _unassigned
for variable in variables:
if variable not in assignments:
if unassignedvariable is _unassigned:
unassignedvariable = variable
else:
break
else:
if unassignedvariable is not _unassigned:
# Remove from the unassigned variable domain's all
# values which break our variable's constraints.
domain = domains[unassignedvariable]
if domain:
for value in domain[:]:
assignments[unassignedvariable] = value
if not self(variables, domains, assignments):
domain.hideValue(value)
del assignments[unassignedvariable]
if not domain:
return False
return True
class FunctionConstraint(Constraint):
"""
Constraint which wraps a function defining the constraint logic
Examples:
>>> problem = Problem()
>>> problem.addVariables(["a", "b"], [1, 2])
>>> def func(a, b):
... return b > a
>>> problem.addConstraint(func, ["a", "b"])
>>> problem.getSolution()
{'a': 1, 'b': 2}
>>> problem = Problem()
>>> problem.addVariables(["a", "b"], [1, 2])
>>> def func(a, b):
... return b > a
>>> problem.addConstraint(FunctionConstraint(func), ["a", "b"])
>>> problem.getSolution()
{'a': 1, 'b': 2}
"""
def __init__(self, func, assigned=True):
"""
@param func: Function wrapped and queried for constraint logic
@type func: callable object
@param assigned: Whether the function may receive unassigned
variables or not
@type assigned: bool
"""
self._func = func
self._assigned = assigned
def __call__(
self,
variables,
domains,
assignments,
forwardcheck=False,
_unassigned=Unassigned,
):
parms = [assignments.get(x, _unassigned) for x in variables]
missing = parms.count(_unassigned)
if missing:
return (self._assigned or self._func(*parms)) and (
not forwardcheck or
missing != 1 or
self.forwardCheck(variables, domains, assignments)
)
return self._func(*parms)
class AllDifferentConstraint(Constraint):
"""
Constraint enforcing that values of all given variables are different
Example:
>>> problem = Problem()
>>> problem.addVariables(["a", "b"], [1, 2])
>>> problem.addConstraint(AllDifferentConstraint())
>>> sorted(sorted(x.items()) for x in problem.getSolutions())
[[('a', 1), ('b', 2)], [('a', 2), ('b', 1)]]
"""
def __call__(
self,
variables,
domains,
assignments,
forwardcheck=False,
_unassigned=Unassigned,
):
seen = {}
for variable in variables:
value = assignments.get(variable, _unassigned)
if value is not _unassigned:
if value in seen:
return False
seen[value] = True
if forwardcheck:
for variable in variables:
if variable not in assignments:
domain = domains[variable]
for value in seen:
if value in domain:
domain.hideValue(value)
if not domain: