search.py

"""Search (Chapters 3-4)

The way to use this code is to subclass Problem to create a class of problems,
then create problem instances and solve them with calls to the various search
functions."""

from utils import (
    is_in, argmin, argmax, argmax_random_tie, probability, weighted_sampler,
    memoize, print_table, open_data, Stack, FIFOQueue, PriorityQueue, name,
    distance
)

from collections import defaultdict
import math
import random
import sys
import bisect
from operator import itemgetter

infinity = float('inf')

# ______________________________________________________________________________


class Problem(object):

    """The abstract class for a formal problem. You should subclass
    this and implement the methods actions and result, and possibly
    __init__, goal_test, and path_cost. Then you will create instances
    of your subclass and solve them with the various search functions."""

    def __init__(self, initial, goal=None):
        """The constructor specifies the initial state, and possibly a goal
        state, if there is a unique goal. Your subclass's constructor can add
        other arguments."""
        self.initial = initial
        self.goal = goal

    def actions(self, state):
        """Return the actions that can be executed in the given
        state. The result would typically be a list, but if there are
        many actions, consider yielding them one at a time in an
        iterator, rather than building them all at once."""
        raise NotImplementedError

    def result(self, state, action):
        """Return the state that results from executing the given
        action in the given state. The action must be one of
        self.actions(state)."""
        raise NotImplementedError

    def goal_test(self, state):
        """Return True if the state is a goal. The default method compares the
        state to self.goal or checks for state in self.goal if it is a
        list, as specified in the constructor. Override this method if
        checking against a single self.goal is not enough."""
        if isinstance(self.goal, list):
            return is_in(state, self.goal)
        else:
            return state == self.goal

    def path_cost(self, c, state1, action, state2):
        """Return the cost of a solution path that arrives at state2 from
        state1 via action, assuming cost c to get up to state1. If the problem
        is such that the path doesn't matter, this function will only look at
        state2.  If the path does matter, it will consider c and maybe state1
        and action. The default method costs 1 for every step in the path."""
        return c + 1

    def value(self, state):
        """For optimization problems, each state has a value.  Hill-climbing
        and related algorithms try to maximize this value."""
        raise NotImplementedError
# ______________________________________________________________________________


class Node:

    """A node in a search tree. Contains a pointer to the parent (the node
    that this is a successor of) and to the actual state for this node. Note
    that if a state is arrived at by two paths, then there are two nodes with
    the same state.  Also includes the action that got us to this state, and
    the total path_cost (also known as g) to reach the node.  Other functions
    may add an f and h value; see best_first_graph_search and astar_search for
    an explanation of how the f and h values are handled. You will not need to
    subclass this class."""

    def __init__(self, state, parent=None, action=None, path_cost=0):
        """Create a search tree Node, derived from a parent by an action."""
        self.state = state
        self.parent = parent
        self.action = action
        self.path_cost = path_cost
        self.depth = 0
        if parent:
            self.depth = parent.depth + 1

    def __repr__(self):
        return "<Node {}>".format(self.state)

    def __lt__(self, node):
        return self.state < node.state

    def expand(self, problem):
        """List the nodes reachable in one step from this node."""
        return [self.child_node(problem, action)
                for action in problem.actions(self.state)]

    def child_node(self, problem, action):
        """[Figure 3.10]"""
        next = problem.result(self.state, action)
        return Node(next, self, action,
                    problem.path_cost(self.path_cost, self.state,
                                      action, next))

    def solution(self):
        """Return the sequence of actions to go from the root to this node."""
        return [node.action for node in self.path()[1:]]

    def path(self):
        """Return a list of nodes forming the path from the root to this node."""
        node, path_back = self, []
        while node:
            path_back.append(node)
            node = node.parent
        return list(reversed(path_back))

    # We want for a queue of nodes in breadth_first_search or
    # astar_search to have no duplicated states, so we treat nodes
    # with the same state as equal. [Problem: this may not be what you
    # want in other contexts.]

    def __eq__(self, other):
        return isinstance(other, Node) and self.state == other.state

    def __hash__(self):
        return hash(self.state)

# ______________________________________________________________________________


class SimpleProblemSolvingAgentProgram:

    """Abstract framework for a problem-solving agent. [Figure 3.1]"""

    def __init__(self, initial_state=None):
        """State is an sbstract representation of the state
        of the world, and seq is the list of actions required
        to get to a particular state from the initial state(root)."""
        self.state = initial_state
        self.seq = []

    def __call__(self, percept):
        """[Figure 3.1] Formulate a goal and problem, then
        search for a sequence of actions to solve it."""
        self.state = self.update_state(self.state, percept)
        if not self.seq:
            goal = self.formulate_goal(self.state)
            problem = self.formulate_problem(self.state, goal)
            self.seq = self.search(problem)
            if not self.seq:
                return None
        return self.seq.pop(0)

    def update_state(self, percept):
        raise NotImplementedError

    def formulate_goal(self, state):
        raise NotImplementedError

    def formulate_problem(self, state, goal):
        raise NotImplementedError

    def search(self, problem):
        raise NotImplementedError

# ______________________________________________________________________________
# Uninformed Search algorithms


def tree_search(problem, frontier):
    """Search through the successors of a problem to find a goal.
    The argument frontier should be an empty queue.
    Don't worry about repeated paths to a state. [Figure 3.7]"""
    frontier.append(Node(problem.initial))
    while frontier:
        node = frontier.pop()
        if problem.goal_test(node.state):
            return node
        frontier.extend(node.expand(problem))
    return None


def graph_search(problem, frontier):
    """Search through the successors of a problem to find a goal.
    The argument frontier should be an empty queue.
    If two paths reach a state, only use the first one. [Figure 3.7]"""
    frontier.append(Node(problem.initial))
    explored = set()
    while frontier:
        node = frontier.pop()
        if problem.goal_test(node.state):
            return node
        explored.add(node.state)
        frontier.extend(child for child in node.expand(problem)
                        if child.state not in explored and
                        child not in frontier)
    return None


def breadth_first_tree_search(problem):
    """Search the shallowest nodes in the search tree first."""
    return tree_search(problem, FIFOQueue())


def depth_first_tree_search(problem):
    """Search the deepest nodes in the search tree first."""
    return tree_search(problem, Stack())


def depth_first_graph_search(problem):
    """Search the deepest nodes in the search tree first."""
    return graph_search(problem, Stack())


def breadth_first_search(problem):
    """[Figure 3.11]"""
    node = Node(problem.initial)
    if problem.goal_test(node.state):
        return node
    frontier = FIFOQueue()
    frontier.append(node)
    explored = set()
    while frontier:
        node = frontier.pop()
        explored.add(node.state)
        for child in node.expand(problem):
            if child.state not in explored and child not in frontier:
                if problem.goal_test(child.state):
                    return child
                frontier.append(child)
    return None


def best_first_graph_search(problem, f):
    """Search the nodes with the lowest f scores first.
    You specify the function f(node) that you want to minimize; for example,
    if f is a heuristic estimate to the goal, then we have greedy best
    first search; if f is node.depth then we have breadth-first search.
    There is a subtlety: the line "f = memoize(f, 'f')" means that the f
    values will be cached on the nodes as they are computed. So after doing
    a best first search you can examine the f values of the path returned."""
    f = memoize(f, 'f')
    node = Node(problem.initial)
    if problem.goal_test(node.state):
        return node
    frontier = PriorityQueue(min, f)
    frontier.append(node)
    explored = set()
    while frontier:
        node = frontier.pop()
        if problem.goal_test(node.state):
            return node
        explored.add(node.state)
        for child in node.expand(problem):
            if child.state not in explored and child not in frontier:
                frontier.append(child)
            elif child in frontier:
                incumbent = frontier[child]
                if f(child) < f(incumbent):
                    del frontier[incumbent]
                    frontier.append(child)
    return None


def uniform_cost_search(problem):
    """[Figure 3.14]"""
    return best_first_graph_search(problem, lambda node: node.path_cost)


def depth_limited_search(problem, limit=50):
    """[Figure 3.17]"""
    def recursive_dls(node, problem, limit):
        if problem.goal_test(node.state):
            return node
        elif limit == 0:
            return 'cutoff'
        else:
            cutoff_occurred = False
            for child in node.expand(problem):
                result = recursive_dls(child, problem, limit - 1)
                if result == 'cutoff':
                    cutoff_occurred = True
                elif result is not None:
                    return result
            return 'cutoff' if cutoff_occurred else None

    # Body of depth_limited_search:
    return recursive_dls(Node(problem.initial), problem, limit)


def iterative_deepening_search(problem):
    """[Figure 3.18]"""
    for depth in range(sys.maxsize):
        result = depth_limited_search(problem, depth)
        if result != 'cutoff':
            return result

# ______________________________________________________________________________
# Bidirectional Search
# Pseudocode from https://webdocs.cs.ualberta.ca/%7Eholte/Publications/MM-AAAI2016.pdf

def bidirectional_search(problem):
    e = problem.find_min_edge()
    gF, gB = {problem.initial : 0}, {problem.goal : 0}
    openF, openB = [problem.initial], [problem.goal]
    closedF, closedB = [], []
    U = infinity


    def extend(U, open_dir, open_other, g_dir, g_other, closed_dir):
        """Extend search in given direction"""
        n = find_key(C, open_dir, g_dir)

        open_dir.remove(n)
        closed_dir.append(n)

        for c in problem.actions(n):
            if c in open_dir or c in closed_dir:
                if g_dir[c] <= problem.path_cost(g_dir[n], n, None, c):
                    continue

                open_dir.remove(c)

            g_dir[c] = problem.path_cost(g_dir[n], n, None, c)
            open_dir.append(c)

            if c in open_other:
                U = min(U, g_dir[c] + g_other[c])

        return U, open_dir, closed_dir, g_dir


    def find_min(open_dir, g):
        """Finds minimum priority, g and f values in open_dir"""
        m, m_f = infinity, infinity
        for n in open_dir:
            f = g[n] + problem.h(n)
            pr = max(f, 2*g[n])
            m = min(m, pr)
            m_f = min(m_f, f)

        return m, m_f, min(g.values())


    def find_key(pr_min, open_dir, g):
        """Finds key in open_dir with value equal to pr_min
        and minimum g value."""
        m = infinity
        state = -1
        for n in open_dir:
            pr = max(g[n] + problem.h(n), 2*g[n])
            if pr == pr_min:
                if g[n] < m:
                    m = g[n]
                    state = n

        return state


    while openF and openB:
        pr_min_f, f_min_f, g_min_f = find_min(openF, gF)
        pr_min_b, f_min_b, g_min_b = find_min(openB, gB)
        C = min(pr_min_f, pr_min_b)

        if U <= max(C, f_min_f, f_min_b, g_min_f + g_min_b + e):
            return U

        if C == pr_min_f:
            # Extend forward
            U, openF, closedF, gF = extend(U, openF, openB, gF, gB, closedF)
        else:
            # Extend backward
            U, openB, closedB, gB = extend(U, openB, openF, gB, gF, closedB)

    return infinity

# ______________________________________________________________________________
# Informed (Heuristic) Search


greedy_best_first_graph_search = best_first_graph_search
# Greedy best-first search is accomplished by specifying f(n) = h(n).


def astar_search(problem, h=None):
    """A* search is best-first graph search with f(n) = g(n)+h(n).
    You need to specify the h function when you call astar_search, or
    else in your Problem subclass."""
    h = memoize(h or problem.h, 'h')
    return best_first_graph_search(problem, lambda n: n.path_cost + h(n))

# ______________________________________________________________________________
# Other search algorithms


def recursive_best_first_search(problem, h=None):
    """[Figure 3.26]"""
    h = memoize(h or problem.h, 'h')

    def RBFS(problem, node, flimit):
        if problem.goal_test(node.state):
            return node, 0   # (The second value is immaterial)
        successors = node.expand(problem)
        if len(successors) == 0:
            return None, infinity
        for s in successors:
            s.f = max(s.path_cost + h(s), node.f)
        while True:
            # Order by lowest f value
            successors.sort(key=lambda x: x.f)
            best = successors[0]
            if best.f > flimit:
                return None, best.f
            if len(successors) > 1:
                alternative = successors[1].f
            else:
                alternative = infinity
            result, best.f = RBFS(problem, best, min(flimit, alternative))
            if result is not None:
                return result, best.f

    node = Node(problem.initial)
    node.f = h(node)
    result, bestf = RBFS(problem, node, infinity)
    return result


def hill_climbing(problem):
    """From the initial node, keep choosing the neighbor with highest value,
    stopping when no neighbor is better. [Figure 4.2]"""
    current = Node(problem.initial)
    while True:
        neighbors = current.expand(problem)
        if not neighbors:
            break
        neighbor = argmax_random_tie(neighbors,
                                     key=lambda node: problem.value(node.state))
        if problem.value(neighbor.state) <= problem.value(current.state):
            break
        current = neighbor
    return current.state


def exp_schedule(k=20, lam=0.005, limit=100):
    """One possible schedule function for simulated annealing"""
    return lambda t: (k * math.exp(-lam * t) if t < limit else 0)


def simulated_annealing(problem, schedule=exp_schedule()):
    """[Figure 4.5] CAUTION: This differs from the pseudocode as it
    returns a state instead of a Node."""
    current = Node(problem.initial)
    for t in range(sys.maxsize):
        T = schedule(t)
        if T == 0:
            return current.state
        neighbors = current.expand(problem)
        if not neighbors:
            return current.state
        next = random.choice(neighbors)
        delta_e = problem.value(next.state) - problem.value(current.state)
        if delta_e > 0 or probability(math.exp(delta_e / T)):
            current = next


def and_or_graph_search(problem):
    """[Figure 4.11]Used when the environment is nondeterministic and completely observable.
    Contains OR nodes where the agent is free to choose any action.
    After every action there is an AND node which contains all possible states
    the agent may reach due to stochastic nature of environment.
    The agent must be able to handle all possible states of the AND node (as it
    may end up in any of them).
    Returns a conditional plan to reach goal state,
    or failure if the former is not possible."""

    # functions used by and_or_search
    def or_search(state, problem, path):
        """returns a plan as a list of actions"""
        if problem.goal_test(state):
            return []
        if state in path:
            return None
        for action in problem.actions(state):
            plan = and_search(problem.result(state, action),
                              problem, path + [state, ])
            if plan is not None:
                return [action, plan]

    def and_search(states, problem, path):
        """Returns plan in form of dictionary where we take action plan[s] if we reach state s."""
        plan = {}
        for s in states:
            plan[s] = or_search(s, problem, path)
            if plan[s] is None:
                return None
        return plan

    # body of and or search
    return or_search(problem.initial, problem, [])


class PeakFindingProblem(Problem):
    """Problem of finding the highest peak in a limited grid"""

    def __init__(self, initial, grid):
        """The grid is a 2 dimensional array/list whose state is specified by tuple of indices"""
        Problem.__init__(self, initial)
        self.grid = grid
        self.n = len(grid)
        assert self.n > 0
        self.m = len(grid[0])
        assert self.m > 0

    def actions(self, state):
        """Allows movement in only 4 directions"""
        # TODO: Add flag to allow diagonal motion
        allowed_actions = []
        if state[0] > 0:
            allowed_actions.append('N')
        if state[0] < self.n - 1:
            allowed_actions.append('S')
        if state[1] > 0:
            allowed_actions.append('W')
        if state[1] < self.m - 1:
            allowed_actions.append('E')
        return allowed_actions

    def result(self, state, action):
        """Moves in the direction specified by action"""
        x, y = state
        x = x + (1 if action == 'S' else (-1 if action == 'N' else 0))
        y = y + (1 if action == 'E' else (-1 if action == 'W' else 0))
        return (x, y)

    def value(self, state):
        """Value of a state is the value it is the index to"""
        x, y = state
        assert 0 <= x < self.n
        assert 0 <= y < self.m
        return self.grid[x][y]


class OnlineDFSAgent:

    """[Figure 4.21] The abstract class for an OnlineDFSAgent. Override
    update_state method to convert percept to state. While initializing
    the subclass a problem needs to be provided which is an instance of
    a subclass of the Problem class."""

    def __init__(self, problem):
        self.problem = problem
        self.s = None
        self.a = None
        self.untried = defaultdict(list)
        self.unbacktracked = defaultdict(list)
        self.result = {}

    def __call__(self, percept):
        s1 = self.update_state(percept)
        if self.problem.goal_test(s1):
            self.a = None
        else:
            if s1 not in self.untried.keys():
                self.untried[s1] = self.problem.actions(s1)
            if self.s is not None:
                if s1 != self.result[(self.s, self.a)]:
                    self.result[(self.s, self.a)] = s1
                    self.unbacktracked[s1].insert(0, self.s)
            if len(self.untried[s1]) == 0:
                if len(self.unbacktracked[s1]) == 0:
                    self.a = None
                else:
                    # else a <- an action b such that result[s', b] = POP(unbacktracked[s'])
                    unbacktracked_pop = self.unbacktracked[s1].pop(0)
                    for (s, b) in self.result.keys():
                        if self.result[(s, b)] == unbacktracked_pop:
                            self.a = b
                            break
            else:
                self.a = self.untried[s1].pop(0)
        self.s = s1
        return self.a

    def update_state(self, percept):
        """To be overridden in most cases. The default case
        assumes the percept to be of type state."""
        return percept

# ______________________________________________________________________________


class OnlineSearchProblem(Problem):
    """
    A problem which is solved by an agent executing
    actions, rather than by just computation.
    Carried in a deterministic and a fully observable environment."""

    def __init__(self, initial, goal, graph):
        self.initial = initial
        self.goal = goal
        self.graph = graph

    def actions(self, state):
        return self.graph.dict[state].keys()

    def output(self, state, action):
        return self.graph.dict[state][action]

    def h(self, state):
        """Returns least possible cost to reach a goal for the given state."""
        return self.graph.least_costs[state]

    def c(self, s, a, s1):
        """Returns a cost estimate for an agent to move from state 's' to state 's1'."""
        return 1

    def update_state(self, percept):
        raise NotImplementedError

    def goal_test(self, state):
        if state == self.goal:
            return True
        return False


class LRTAStarAgent:

    """ [Figure 4.24]
    Abstract class for LRTA*-Agent. A problem needs to be
    provided which is an instance of a subclass of Problem Class.

    Takes a OnlineSearchProblem [Figure 4.23] as a problem.
    """

    def __init__(self, problem):
        self.problem = problem
        # self.result = {}      # no need as we are using problem.result
        self.H = {}
        self.s = None
        self.a = None

    def __call__(self, s1):     # as of now s1 is a state rather than a percept
        if self.problem.goal_test(s1):
            self.a = None
            return self.a
        else:
            if s1 not in self.H:
                self.H[s1] = self.problem.h(s1)
            if self.s is not None:
                # self.result[(self.s, self.a)] = s1    # no need as we are using problem.output

                # minimum cost for action b in problem.actions(s)
                self.H[self.s] = min(self.LRTA_cost(self.s, b, self.problem.output(self.s, b),
                                     self.H) for b in self.problem.actions(self.s))

            # an action b in problem.actions(s1) that minimizes costs
            self.a = argmin(self.problem.actions(s1),
                            key=lambda b: self.LRTA_cost(s1, b, self.problem.output(s1, b), self.H))

            self.s = s1
            return self.a

    def LRTA_cost(self, s, a, s1, H):
        """Returns cost to move from state 's' to state 's1' plus
        estimated cost to get to goal from s1."""
        print(s, a, s1)
        if s1 is None:
            return self.problem.h(s)
        else:
            # sometimes we need to get H[s1] which we haven't yet added to H
            # to replace this try, except: we can initialize H with values from problem.h
            try:
                return self.problem.c(s, a, s1) + self.H[s1]
            except:
                return self.problem.c(s, a, s1) + self.problem.h(s1)

# ______________________________________________________________________________
# Genetic Algorithm


def genetic_search(problem, fitness_fn, ngen=1000, pmut=0.1, n=20):
    """Call genetic_algorithm on the appropriate parts of a problem.
    This requires the problem to have states that can mate and mutate,
    plus a value method that scores states."""

    # NOTE: This is not tested and might not work.
    # TODO: Use this function to make Problems work with genetic_algorithm.

    s = problem.initial_state
    states = [problem.result(s, a) for a in problem.actions(s)]
    random.shuffle(states)
    return genetic_algorithm(states[:n], problem.value, ngen, pmut)


def genetic_algorithm(population, fitness_fn, gene_pool=[0, 1], f_thres=None, ngen=1000, pmut=0.1):
    """[Figure 4.8]"""
    for i in range(ngen):
        population = [mutate(recombine(*select(2, population, fitness_fn)), gene_pool, pmut)
                      for i in range(len(population))]

        fittest_individual = fitness_threshold(fitness_fn, f_thres, population)
        if fittest_individual:
            return fittest_individual


    return argmax(population, key=fitness_fn)


def fitness_threshold(fitness_fn, f_thres, population):
    if not f_thres:
        return None

    fittest_individual = argmax(population, key=fitness_fn)
    if fitness_fn(fittest_individual) >= f_thres:
        return fittest_individual

    return None



def init_population(pop_number, gene_pool, state_length):
    """Initializes population for genetic algorithm
    pop_number  :  Number of individuals in population
    gene_pool   :  List of possible values for individuals
    state_length:  The length of each individual"""
    g = len(gene_pool)
    population = []
    for i in range(pop_number):
        new_individual = [gene_pool[random.randrange(0, g)] for j in range(state_length)]
        population.append(new_individual)

    return population


def select(r, population, fitness_fn):
    fitnesses = map(fitness_fn, population)
    sampler = weighted_sampler(population, fitnesses)
    return [sampler() for i in range(r)]


def recombine(x, y):
    n = len(x)
    c = random.randrange(0, n)
    return x[:c] + y[c:]


def mutate(x, gene_pool, pmut):
    if random.uniform(0, 1) >= pmut:
        return x

    n = len(x)
    g = len(gene_pool)
    c = random.randrange(0, n)
    r = random.randrange(0, g)

    new_gene = gene_pool[r]
    return x[:c] + [new_gene] + x[c+1:]

# _____________________________________________________________________________
# The remainder of this file implements examples for the search algorithms.

# ______________________________________________________________________________
# Graphs and Graph Problems


class Graph:

    """A graph connects nodes (verticies) by edges (links).  Each edge can also
    have a length associated with it.  The constructor call is something like:
        g = Graph({'A': {'B': 1, 'C': 2})
    this makes a graph with 3 nodes, A, B, and C, with an edge of length 1 from
    A to B,  and an edge of length 2 from A to C.  You can also do:
        g = Graph({'A': {'B': 1, 'C': 2}, directed=False)
    This makes an undirected graph, so inverse links are also added. The graph
    stays undirected; if you add more links with g.connect('B', 'C', 3), then
    inverse link is also added.  You can use g.nodes() to get a list of nodes,
    g.get('A') to get a dict of links out of A, and g.get('A', 'B') to get the
    length of the link from A to B.  'Lengths' can actually be any object at
    all, and nodes can be any hashable object."""

    def __init__(self, dict=None, directed=True):
        self.dict = dict or {}
        self.directed = directed
        if not directed:
            self.make_undirected()

    def make_undirected(self):
        """Make a digraph into an undirected graph by adding symmetric edges."""
        for a in list(self.dict.keys()):
            for (b, dist) in self.dict[a].items():
                self.connect1(b, a, dist)

    def connect(self, A, B, distance=1):
        """Add a link from A and B of given distance, and also add the inverse
        link if the graph is undirected."""
        self.connect1(A, B, distance)
        if not self.directed:
            self.connect1(B, A, distance)

    def connect1(self, A, B, distance):
        """Add a link from A to B of given distance, in one direction only."""
        self.dict.setdefault(A, {})[B] = distance

    def get(self, a, b=None):
        """Return a link distance or a dict of {node: distance} entries.
        .get(a,b) returns the distance or None;
        .get(a) returns a dict of {node: distance} entries, possibly {}."""
        links = self.dict.setdefault(a, {})
        if b is None:
            return links
        else:
            return links.get(b)

    def nodes(self):
        """Return a list of nodes in the graph."""
        return list(self.dict.keys())


def UndirectedGraph(dict=None):
    """Build a Graph where every edge (including future ones) goes both ways."""
    return Graph(dict=dict, directed=False)


def RandomGraph(nodes=list(range(10)), min_links=2, width=400, height=300,
                curvature=lambda: random.uniform(1.1, 1.5)):
    """Construct a random graph, with the specified nodes, and random links.
    The nodes are laid out randomly on a (width x height) rectangle.
    Then each node is connected to the min_links nearest neighbors.
    Because inverse links are added, some nodes will have more connections.
    The distance between nodes is the hypotenuse times curvature(),
    where curvature() defaults to a random number between 1.1 and 1.5."""
    g = UndirectedGraph()
    g.locations = {}
    # Build the cities
    for node in nodes:
        g.locations[node] = (random.randrange(width), random.randrange(height))
    # Build roads from each city to at least min_links nearest neighbors.
    for i in range(min_links):
        for node in nodes:
            if len(g.get(node)) < min_links:
                here = g.locations[node]

                def distance_to_node(n):
                    if n is node or g.get(node, n):
                        return infinity
                    return distance(g.locations[n], here)
                neighbor = argmin(nodes, key=distance_to_node)
                d = distance(g.locations[neighbor], here) * curvature()
                g.connect(node, neighbor, int(d))
    return g


""" [Figure 3.2]
Simplified road map of Romania
"""
romania_map = UndirectedGraph(dict(
    Arad=dict(Zerind=75, Sibiu=140, Timisoara=118),
    Bucharest=dict(Urziceni=85, Pitesti=101, Giurgiu=90, Fagaras=211),
    Craiova=dict(Drobeta=120, Rimnicu=146, Pitesti=138),
    Drobeta=dict(Mehadia=75),
    Eforie=dict(Hirsova=86),
    Fagaras=dict(Sibiu=99),
    Hirsova=dict(Urziceni=98),
    Iasi=dict(Vaslui=92, Neamt=87),
    Lugoj=dict(Timisoara=111, Mehadia=70),
    Oradea=dict(Zerind=71, Sibiu=151),
    Pitesti=dict(Rimnicu=97),
    Rimnicu=dict(Sibiu=80),
    Urziceni=dict(Vaslui=142)))
romania_map.locations = dict(
    Arad=(91, 492), Bucharest=(400, 327), Craiova=(253, 288),
    Drobeta=(165, 299), Eforie=(562, 293), Fagaras=(305, 449),
    Giurgiu=(375, 270), Hirsova=(534, 350), Iasi=(473, 506),
    Lugoj=(165, 379), Mehadia=(168, 339), Neamt=(406, 537),
    Oradea=(131, 571), Pitesti=(320, 368), Rimnicu=(233, 410),
    Sibiu=(207, 457), Timisoara=(94, 410), Urziceni=(456, 350),
    Vaslui=(509, 444), Zerind=(108, 531))

""" [Figure 4.9]
Eight possible states of the vacumm world
Each state is represented as
   *       "State of the left room"      "State of the right room"   "Room in which the agent
                                                                      is present"
1 - DDL     Dirty                         Dirty                       Left
2 - DDR     Dirty                         Dirty                       Right
3 - DCL     Dirty                         Clean                       Left
4 - DCR     Dirty                         Clean                       Right
5 - CDL     Clean                         Dirty                       Left
6 - CDR     Clean                         Dirty                       Right
7 - CCL     Clean                         Clean                       Left
8 - CCR     Clean                         Clean                       Right
"""
vacumm_world = Graph(dict(
    State_1=dict(Suck=['State_7', 'State_5'], Right=['State_2']),
    State_2=dict(Suck=['State_8', 'State_4'], Left=['State_2']),
    State_3=dict(Suck=['State_7'], Right=['State_4']),
    State_4=dict(Suck=['State_4', 'State_2'], Left=['State_3']),
    State_5=dict(Suck=['State_5', 'State_1'], Right=['State_6']),
    State_6=dict(Suck=['State_8'], Left=['State_5']),
    State_7=dict(Suck=['State_7', 'State_3'], Right=['State_8']),
    State_8=dict(Suck=['State_8', 'State_6'], Left=['State_7'])
    ))

""" [Figure 4.23]
One-dimensional state space Graph
"""
one_dim_state_space = Graph(dict(
    State_1=dict(Right='State_2'),
    State_2=dict(Right='State_3', Left='State_1'),
    State_3=dict(Right='State_4', Left='State_2'),
    State_4=dict(Right='State_5', Left='State_3'),
    State_5=dict(Right='State_6', Left='State_4'),
    State_6=dict(Left='State_5')
    ))
one_dim_state_space.least_costs = dict(
    State_1=8,
    State_2=9,
    State_3=2,
    State_4=2,
    State_5=4,
    State_6=3)

""" [Figure 6.1]
Principal states and territories of Australia
"""
australia_map = UndirectedGraph(dict(
    T=dict(),
    SA=dict(WA=1, NT=1, Q=1, NSW=1, V=1),
    NT=dict(WA=1, Q=1),
    NSW=dict(Q=1, V=1)))
australia_map.locations = dict(WA=(120, 24), NT=(135, 20), SA=(135, 30),
                               Q=(145, 20), NSW=(145, 32), T=(145, 42),
                               V=(145, 37))


class GraphProblem(Problem):

    """The problem of searching a graph from one node to another."""

    def __init__(self, initial, goal, graph):
        Problem.__init__(self, initial, goal)
        self.graph = graph

    def actions(self, A):
        """The actions at a graph node are just its neighbors."""
        return list(self.graph.get(A).keys())

    def result(self, state, action):
        """The result of going to a neighbor is just that neighbor."""
        return action

    def path_cost(self, cost_so_far, A, action, B):
        return cost_so_far + (self.graph.get(A, B) or infinity)

    def find_min_edge(self):
        """Find minimum value of edges."""
        m = infinity
        for d in self.graph.dict.values():
            local_min = min(d.values())
            m = min(m, local_min)

        return m

    def h(self, node):
        """h function is straight-line distance from a node's state to goal."""
        locs = getattr(self.graph, 'locations', None)
        if locs:
            if type(node) is str:
                return int(distance(locs[node], locs[self.goal]))

            return int(distance(locs[node.state], locs[self.goal]))
        else:
            return infinity


class GraphProblemStochastic(GraphProblem):
    """
    A version of GraphProblem where an action can lead to
    nondeterministic output i.e. multiple possible states.

    Define the graph as dict(A = dict(Action = [[<Result 1>, <Result 2>, ...], <cost>], ...), ...)
    A the dictionary format is different, make sure the graph is created as a directed graph.
    """

    def result(self, state, action):
        return self.graph.get(state, action)

    def path_cost(self):
        raise NotImplementedError