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 *

import math
import random
import sys
import time
import bisect
import string

#______________________________________________________________________________


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, as specified in the constructor. Override this
        method if checking against a single self.goal is not enough."""
        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."
        update(self, state=state, parent=parent, action=action,
               path_cost=path_cost, depth=0)
        if parent:
            self.depth = parent.depth + 1

    def __repr__(self):
        return "<Node %s>" % (self.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):
        "Fig. 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. [Fig. 3.1]"""

    def __init__(self, initial_state=None):
        update(self, state=initial_state, seq=[])

    def __call__(self, percept):
        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. [Fig. 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. [Fig. 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):
    "[Fig. 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):
    "[Fig. 3.14]"
    return best_first_graph_search(problem, lambda node: node.path_cost)


def depth_limited_search(problem, limit=50):
    "[Fig. 3.17]"
    def recursive_dls(node, problem, limit):
        if problem.goal_test(node.state):
            return node
        elif node.depth == limit:
            return 'cutoff'
        else:
            cutoff_occurred = False
            for child in node.expand(problem):
                result = recursive_dls(child, problem, limit)
                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):
    "[Fig. 3.18]"
    for depth in range(sys.maxsize):
        result = depth_limited_search(problem, depth)
        if result != 'cutoff':
            return result

#______________________________________________________________________________
# 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):
    "[Fig. 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(lambda x, y: cmp(x.f, y.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. [Fig. 4.2]"""
    current = Node(problem.initial)
    while True:
        neighbors = current.expand(problem)
        if not neighbors:
            break
        neighbor = argmax_random_tie(neighbors,
                                     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()):
    "[Fig. 4.5]"
    current = Node(problem.initial)
    for t in range(sys.maxsize):
        T = schedule(t)
        if T == 0:
            return current
        neighbors = current.expand(problem)
        if not neighbors:
            return current
        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):
    """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"""
    "[Fig. 4.11]"

    #functions used by and_or_search
    def or_search(state, problem, path):
        if problem.goal_test(state):
            return {}
        if state in path:
            return None
        for action in problem.action(state):
            plan = and_search(problem.result(state, action), problem, path + [state,])
            if not plan == 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=dict()
        for s in states:
            plan[s] = or_search(s, problem, path)
            if plan[s] == None:
                return None
        return plan

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


def online_dfs_agent(s1):
    "[Fig. 4.21]"
    unimplemented()


def lrta_star_agent(s1):
    "[Fig. 4.24]"
    unimplemented()

#______________________________________________________________________________
# 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."""
    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, ngen=1000, pmut=0.1):
    "[Fig. 4.8]"
    for i in range(ngen):
        new_population = []
        for i in len(population):
            fitnesses = list(map(fitness_fn, population))
            p1, p2 = weighted_sample_with_replacement(population, fitnesses, 2)
            child = p1.mate(p2)
            if random.uniform(0, 1) < pmut:
                child.mutate()
            new_population.append(child)
        population = new_population
    return argmax(population, fitness_fn)


class GAState:

    "Abstract class for individuals in a genetic search."

    def __init__(self, genes):
        self.genes = genes

    def mate(self, other):
        "Return a new individual crossing self and other."
        c = random.randrange(len(self.genes))
        return self.__class__(self.genes[:c] + other.genes[c:])

    def mutate(self):
        "Change a few of my genes."
        raise NotImplementedError

#_____________________________________________________________________________
# 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, distance) in list(self.dict[a].items()):
                self.connect1(b, a, distance)

    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, distance_to_node)
                d = distance(g.locations[neighbor], here) * curvature()
                g.connect(node, neighbor, int(d))
    return g

romania = UndirectedGraph(dict(
    A=dict(Z=75, S=140, T=118),
    B=dict(U=85, P=101, G=90, F=211),
    C=dict(D=120, R=146, P=138),
    D=dict(M=75),
    E=dict(H=86),
    F=dict(S=99),
    H=dict(U=98),
    I=dict(V=92, N=87),
    L=dict(T=111, M=70),
    O=dict(Z=71, S=151),
    P=dict(R=97),
    R=dict(S=80),
    U=dict(V=142)))
romania.locations = dict(
    A=(91, 492),    B=(400, 327),    C=(253, 288),   D=(165, 299),
    E=(562, 293),    F=(305, 449),    G=(375, 270),   H=(534, 350),
    I=(473, 506),    L=(165, 379),    M=(168, 339),   N=(406, 537),
    O=(131, 571),    P=(320, 368),    R=(233, 410),   S=(207, 457),
    T=(94, 410),    U=(456, 350),    V=(509, 444),   Z=(108, 531))

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

#______________________________________________________________________________


class NQueensProblem(Problem):

    """The problem of placing N queens on an NxN board with none attacking
    each other.  A state is represented as an N-element array, where
    a value of r in the c-th entry means there is a queen at column c,
    row r, and a value of None means that the c-th column has not been
    filled in yet.  We fill in columns left to right.
    >>> depth_first_tree_search(NQueensProblem(8))
    <Node [7, 3, 0, 2, 5, 1, 6, 4]>
    """

    def __init__(self, N):
        self.N = N
        self.initial = [None] * N

    def actions(self, state):
        "In the leftmost empty column, try all non-conflicting rows."
        if state[-1] is not None:
            return []  # All columns filled; no successors
        else:
            col = state.index(None)
            return [row for row in range(self.N)
                    if not self.conflicted(state, row, col)]

    def result(self, state, row):
        "Place the next queen at the given row."
        col = state.index(None)
        new = state[:]
        new[col] = row
        return new

    def conflicted(self, state, row, col):
        "Would placing a queen at (row, col) conflict with anything?"
        return any(self.conflict(row, col, state[c], c)
                   for c in range(col))

    def conflict(self, row1, col1, row2, col2):
        "Would putting two queens in (row1, col1) and (row2, col2) conflict?"
        return (row1 == row2  # same row
                or col1 == col2  # same column
                or row1-col1 == row2-col2  # same \ diagonal
                or row1+col1 == row2+col2)  # same / diagonal

    def goal_test(self, state):
        "Check if all columns filled, no conflicts."
        if state[-1] is None:
            return False
        return not any(self.conflicted(state, state[col], col)
                       for col in range(len(state)))

#______________________________________________________________________________
# Inverse Boggle: Search for a high-scoring Boggle board. A good domain for
# iterative-repair and related search techniques, as suggested by Justin Boyan.

ALPHABET = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'

cubes16 = ['FORIXB', 'MOQABJ', 'GURILW', 'SETUPL',
           'CMPDAE', 'ACITAO', 'SLCRAE', 'ROMASH',
           'NODESW', 'HEFIYE', 'ONUDTK', 'TEVIGN',
           'ANEDVZ', 'PINESH', 'ABILYT', 'GKYLEU']


def random_boggle(n=4):
    """Return a random Boggle board of size n x n.
    We represent a board as a linear list of letters."""
    cubes = [cubes16[i % 16] for i in range(n*n)]
    random.shuffle(cubes)
    return list(map(random.choice, cubes))

# The best 5x5 board found by Boyan, with our word list this board scores
# 2274 words, for a score of 9837

boyan_best = list('RSTCSDEIAEGNLRPEATESMSSID')


def print_boggle(board):
    "Print the board in a 2-d array."
    n2 = len(board)
    n = exact_sqrt(n2)
    for i in range(n2):

        if i % n == 0 and i > 0:
            print()
        if board[i] == 'Q':
            print('Qu', end=' ')
        else:
            print(str(board[i]) + ' ', end=' ')
    print()


def boggle_neighbors(n2, cache={}):
    """Return a list of lists, where the i-th element is the list of indexes
    for the neighbors of square i."""
    if cache.get(n2):
        return cache.get(n2)
    n = exact_sqrt(n2)
    neighbors = [None] * n2
    for i in range(n2):
        neighbors[i] = []
        on_top = i < n
        on_bottom = i >= n2 - n
        on_left = i % n == 0
        on_right = (i+1) % n == 0
        if not on_top:
            neighbors[i].append(i - n)
            if not on_left:
                neighbors[i].append(i - n - 1)
            if not on_right:
                neighbors[i].append(i - n + 1)
        if not on_bottom:
            neighbors[i].append(i + n)
            if not on_left:
                neighbors[i].append(i + n - 1)
            if not on_right:
                neighbors[i].append(i + n + 1)
        if not on_left:
            neighbors[i].append(i - 1)
        if not on_right:
            neighbors[i].append(i + 1)
    cache[n2] = neighbors
    return neighbors


def exact_sqrt(n2):
    "If n2 is a perfect square, return its square root, else raise error."
    n = int(math.sqrt(n2))
    assert n * n == n2
    return n

#_____________________________________________________________________________


class Wordlist:

    """This class holds a list of words. You can use (word in wordlist)
    to check if a word is in the list, or wordlist.lookup(prefix)
    to see if prefix starts any of the words in the list."""

    def __init__(self, filename, min_len=3):
        lines = open(filename).read().upper().split()
        self.words = [word for word in lines if len(word) >= min_len]
        self.words.sort()
        self.bounds = {}
        for c in ALPHABET:
            c2 = chr(ord(c) + 1)
            self.bounds[c] = (bisect.bisect(self.words, c),
                              bisect.bisect(self.words, c2))

    def lookup(self, prefix, lo=0, hi=None):
        """See if prefix is in dictionary, as a full word or as a prefix.
        Return two values: the first is the lowest i such that
        words[i].startswith(prefix), or is None; the second is
        True iff prefix itself is in the Wordlist."""
        words = self.words
        if hi is None:
            hi = len(words)
        i = bisect.bisect_left(words, prefix, lo, hi)
        if i < len(words) and words[i].startswith(prefix):
            return i, (words[i] == prefix)
        else:
            return None, False

    def __contains__(self, word):
        return self.lookup(word)[1]

    def __len__(self):
        return len(self.words)

#_____________________________________________________________________________


class BoggleFinder:

    """A class that allows you to find all the words in a Boggle board. """

    wordlist = None  # A class variable, holding a wordlist

    def __init__(self, board=None):
        if BoggleFinder.wordlist is None:
            BoggleFinder.wordlist = Wordlist("../data/EN-text/wordlist")
        self.found = {}
        if board:
            self.set_board(board)

    def set_board(self, board=None):
        "Set the board, and find all the words in it."
        if board is None:
            board = random_boggle()
        self.board = board
        self.neighbors = boggle_neighbors(len(board))
        self.found = {}
        for i in range(len(board)):
            lo, hi = self.wordlist.bounds[board[i]]
            self.find(lo, hi, i, [], '')
        return self

    def find(self, lo, hi, i, visited, prefix):
        """Looking in square i, find the words that continue the prefix,
        considering the entries in self.wordlist.words[lo:hi], and not
        revisiting the squares in visited."""
        if i in visited:
            return
        wordpos, is_word = self.wordlist.lookup(prefix, lo, hi)
        if wordpos is not None:
            if is_word:
                self.found[prefix] = True
            visited.append(i)
            c = self.board[i]
            if c == 'Q':
                c = 'QU'
            prefix += c
            for j in self.neighbors[i]:
                self.find(wordpos, hi, j, visited, prefix)
            visited.pop()

    def words(self):
        "The words found."
        return list(self.found.keys())

    scores = [0, 0, 0, 0, 1, 2, 3, 5] + [11] * 100

    def score(self):
        "The total score for the words found, according to the rules."
        return sum([self.scores[len(w)] for w in self.words()])

    def __len__(self):
        "The number of words found."
        return len(self.found)

#_____________________________________________________________________________


def boggle_hill_climbing(board=None, ntimes=100, verbose=True):
    """Solve inverse Boggle by hill-climbing: find a high-scoring board by
    starting with a random one and changing it."""
    finder = BoggleFinder()
    if board is None:
        board = random_boggle()
    best = len(finder.set_board(board))
    for _ in range(ntimes):
        i, oldc = mutate_boggle(board)
        new = len(finder.set_board(board))
        if new > best:
            best = new
            if verbose:
                print(best, _, board)
        else:
            board[i] = oldc  # Change back
    if verbose:
        print_boggle(board)
    return board, best


def mutate_boggle(board):
    i = random.randrange(len(board))
    oldc = board[i]
    # random.choice(boyan_best)
    board[i] = random.choice(random.choice(cubes16))
    return i, oldc

#______________________________________________________________________________

# Code to compare searchers on various problems.


class InstrumentedProblem(Problem):

    """Delegates to a problem, and keeps statistics."""

    def __init__(self, problem):
        self.problem = problem
        self.succs = self.goal_tests = self.states = 0
        self.found = None

    def actions(self, state):
        self.succs += 1
        return self.problem.actions(state)

    def result(self, state, action):
        self.states += 1
        return self.problem.result(state, action)

    def goal_test(self, state):
        self.goal_tests += 1
        result = self.problem.goal_test(state)
        if result:
            self.found = state
        return result

    def path_cost(self, c, state1, action, state2):
        return self.problem.path_cost(c, state1, action, state2)

    def value(self, state):
        return self.problem.value(state)

    def __getattr__(self, attr):
        return getattr(self.problem, attr)

    def __repr__(self):
        return '<%4d/%4d/%4d/%s>' % (self.succs, self.goal_tests,
                                     self.states, str(self.found)[:4])


def compare_searchers(problems, header,
                      searchers=[breadth_first_tree_search,
                                 breadth_first_search, depth_first_graph_search,
                                 iterative_deepening_search,
                                 depth_limited_search,
                                 recursive_best_first_search]):
    def do(searcher, problem):
        p = InstrumentedProblem(problem)
        searcher(p)
        return p
    table = [[name(s)] + [do(s, p) for p in problems] for s in searchers]
    print_table(table, header)


def compare_graph_searchers():
    """Prints a table of results like this:
>>> compare_graph_searchers()
Searcher                      Romania(A, B)        Romania(O, N)         Australia          
breadth_first_tree_search     <  21/  22/  59/B>   <1158/1159/3288/N>    <   7/   8/  22/WA>
breadth_first_search          <   7/  11/  18/B>   <  19/  20/  45/N>    <   2/   6/   8/WA>
depth_first_graph_search      <   8/   9/  20/B>   <  16/  17/  38/N>    <   4/   5/  11/WA>
iterative_deepening_search    <  11/  33/  31/B>   < 656/1815/1812/N>    <   3/  11/  11/WA>
depth_limited_search          <  54/  65/ 185/B>   < 387/1012/1125/N>    <  50/  54/ 200/WA>
recursive_best_first_search   <   5/   6/  15/B>   <5887/5888/16532/N>   <  11/  12/  43/WA>"""
    compare_searchers(problems=[GraphProblem('A', 'B', romania),
                                GraphProblem('O', 'N', romania),
                                GraphProblem('Q', 'WA', australia)],
                      header=['Searcher', 'Romania(A, B)', 'Romania(O, N)', 'Australia'])

#______________________________________________________________________________

__doc__ += """
>>> ab = GraphProblem('A', 'B', romania)
>>> breadth_first_tree_search(ab).solution()
['S', 'F', 'B']
>>> breadth_first_search(ab).solution()
['S', 'F', 'B']
>>> uniform_cost_search(ab).solution()
['S', 'R', 'P', 'B']
>>> depth_first_graph_search(ab).solution()
['T', 'L', 'M', 'D', 'C', 'P', 'B']
>>> iterative_deepening_search(ab).solution()
['S', 'F', 'B']
>>> len(depth_limited_search(ab).solution())
50
>>> astar_search(ab).solution()
['S', 'R', 'P', 'B']
>>> recursive_best_first_search(ab).solution()
['S', 'R', 'P', 'B']

>>> board = list('SARTELNID')
>>> print_boggle(board)
S  A  R 
T  E  L 
N  I  D 
>>> f = BoggleFinder(board)
>>> len(f)
206
"""

__doc__ += """
Random tests
>>> ' '.join(f.words())
'LID LARES DEAL LIE DIETS LIN LINT TIL TIN RATED ERAS LATEN DEAR TIE LINE INTER STEAL LATED LAST TAR SAL DITES RALES SAE RETS TAE RAT RAS SAT IDLE TILDES LEAST IDEAS LITE SATED TINED LEST LIT RASE RENTS TINEA EDIT EDITS NITES ALES LATE LETS RELIT TINES LEI LAT ELINT LATI SENT TARED DINE STAR SEAR NEST LITAS TIED SEAT SERAL RATE DINT DEL DEN SEAL TIER TIES NET SALINE DILATE EAST TIDES LINTER NEAR LITS ELINTS DENI RASED SERA TILE NEAT DERAT IDLEST NIDE LIEN STARED LIER LIES SETA NITS TINE DITAS ALINE SATIN TAS ASTER LEAS TSAR LAR NITE RALE LAS REAL NITER ATE RES RATEL IDEA RET IDEAL REI RATS STALE DENT RED IDES ALIEN SET TEL SER TEN TEA TED SALE TALE STILE ARES SEA TILDE SEN SEL ALINES SEI LASE DINES ILEA LINES ELD TIDE RENT DIEL STELA TAEL STALED EARL LEA TILES TILER LED ETA TALI ALE LASED TELA LET IDLER REIN ALIT ITS NIDES DIN DIE DENTS STIED LINER LASTED RATINE ERA IDLES DIT RENTAL DINER SENTI TINEAL DEIL TEAR LITER LINTS TEAL DIES EAR EAT ARLES SATE STARE DITS DELI DENTAL REST DITE DENTIL DINTS DITA DIET LENT NETS NIL NIT SETAL LATS TARE ARE SATI'

>>> boggle_hill_climbing(list('ABCDEFGHI'), verbose=False)
(['E', 'P', 'R', 'D', 'O', 'A', 'G', 'S', 'T'], 123)
"""