text: """
  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.
  """
  
  import bisect
  import math
  import random
  import sys
  from collections import deque
  
  from utils import (is_in, argmax_random_tie, probability, weighted_sampler, memoize, print_table, open_data,
                     PriorityQueue, name, distance, vector_add, inf)
  
  
  class Problem:
      """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_state = problem.result(self.state, action)
          next_node = Node(next_state, self, action, problem.path_cost(self.path_cost, self.state, action, next_state))
          return next_node
  
      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_graph_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:
      """
      [Figure 3.1]
      Abstract framework for a problem-solving agent.
      """
  
      def __init__(self, initial_state=None):
          """State is an abstract 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, state, 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 breadth_first_tree_search(problem):
      """
      [Figure 3.7]
      Search the shallowest nodes in the search tree first.
      Search through the successors of a problem to find a goal.
      The argument frontier should be an empty queue.
      Repeats infinitely in case of loops.
      """
  
      frontier = deque([Node(problem.initial)])  # FIFO queue
  
      while frontier:
          node = frontier.popleft()
          if problem.goal_test(node.state):
              return node
          frontier.extend(node.expand(problem))
      return None
  
  
  def depth_first_tree_search(problem):
      """
      [Figure 3.7]
      Search the deepest nodes in the search tree first.
      Search through the successors of a problem to find a goal.
      The argument frontier should be an empty queue.
      Repeats infinitely in case of loops.
      """
  
      frontier = [Node(problem.initial)]  # Stack
  
      while frontier:
          node = frontier.pop()
          if problem.goal_test(node.state):
              return node
          frontier.extend(node.expand(problem))
      return None
  
  
  def depth_first_graph_search(problem):
      """
      [Figure 3.7]
      Search the deepest nodes in the search tree first.
      Search through the successors of a problem to find a goal.
      The argument frontier should be an empty queue.
      Does not get trapped by loops.
      If two paths reach a state, only use the first one.
      """
      frontier = [(Node(problem.initial))]  # Stack
  
      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_graph_search(problem):
      """[Figure 3.11]
      Note that this function can be implemented in a
      single line as below:
      return graph_search(problem, FIFOQueue())
      """
      node = Node(problem.initial)
      if problem.goal_test(node.state):
          return node
      frontier = deque([node])
      explored = set()
      while frontier:
          node = frontier.popleft()
          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, display=False):
      """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)
      frontier = PriorityQueue('min', f)
      frontier.append(node)
      explored = set()
      while frontier:
          node = frontier.pop()
          if problem.goal_test(node.state):
              if display:
                  print(len(explored), "paths have been expanded and", len(frontier), "paths remain in the frontier")
              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:
                  if f(child) < frontier[child]:
                      del frontier[child]
                      frontier.append(child)
      return None
  
  
  def uniform_cost_search(problem, display=False):
      """[Figure 3.14]"""
      return best_first_graph_search(problem, lambda node: node.path_cost, display)
  
  
  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 = inf
  
      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 = inf, inf
          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 = inf
          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 inf
  
  
  # ______________________________________________________________________________
  # 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, display=False):
      """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), display)
  
  
  # ______________________________________________________________________________
  # A* heuristics 
  
  class EightPuzzle(Problem):
      """ The problem of sliding tiles numbered from 1 to 8 on a 3x3 board, where one of the
      squares is a blank. A state is represented as a tuple of length 9, where  element at
      index i represents the tile number  at index i (0 if it's an empty square) """
  
      def __init__(self, initial, goal=(1, 2, 3, 4, 5, 6, 7, 8, 0)):
          """ Define goal state and initialize a problem """
          super().__init__(initial, goal)
  
      def find_blank_square(self, state):
          """Return the index of the blank square in a given state"""
  
          return state.index(0)
  
      def actions(self, state):
          """ Return the actions that can be executed in the given state.
          The result would be a list, since there are only four possible actions
          in any given state of the environment """
  
          possible_actions = ['UP', 'DOWN', 'LEFT', 'RIGHT']
          index_blank_square = self.find_blank_square(state)
  
          if index_blank_square % 3 == 0:
              possible_actions.remove('LEFT')
          if index_blank_square < 3:
              possible_actions.remove('UP')
          if index_blank_square % 3 == 2:
              possible_actions.remove('RIGHT')
          if index_blank_square > 5:
              possible_actions.remove('DOWN')
  
          return possible_actions
  
      def result(self, state, action):
          """ Given state and action, return a new state that is the result of the action.
          Action is assumed to be a valid action in the state """
  
          # blank is the index of the blank square
          blank = self.find_blank_square(state)
          new_state = list(state)
  
          delta = {'UP': -3, 'DOWN': 3, 'LEFT': -1, 'RIGHT': 1}
          neighbor = blank + delta[action]
          new_state[blank], new_state[neighbor] = new_state[neighbor], new_state[blank]
  
          return tuple(new_state)
  
      def goal_test(self, state):
          """ Given a state, return True if state is a goal state or False, otherwise """
  
          return state == self.goal
  
      def check_solvability(self, state):
          """ Checks if the given state is solvable """
  
          inversion = 0
          for i in range(len(state)):
              for j in range(i + 1, len(state)):
                  if (state[i] > state[j]) and state[i] != 0 and state[j] != 0:
                      inversion += 1
  
          return inversion % 2 == 0
  
      def h(self, node):
          """ Return the heuristic value for a given state. Default heuristic function used is 
          h(n) = number of misplaced tiles """
  
          return sum(s != g for (s, g) in zip(node.state, self.goal))
  
  
  # ______________________________________________________________________________
  
  
  class PlanRoute(Problem):
      """ The problem of moving the Hybrid Wumpus Agent from one place to other """
  
      def __init__(self, initial, goal, allowed, dimrow):
          """ Define goal state and initialize a problem """
          super().__init__(initial, goal)
          self.dimrow = dimrow
          self.goal = goal
          self.allowed = allowed
  
      def actions(self, state):
          """ Return the actions that can be executed in the given state.
          The result would be a list, since there are only three possible actions
          in any given state of the environment """
  
          possible_actions = ['Forward', 'TurnLeft', 'TurnRight']
          x, y = state.get_location()
          orientation = state.get_orientation()
  
          # Prevent Bumps
          if x == 1 and orientation == 'LEFT':
              if 'Forward' in possible_actions:
                  possible_actions.remove('Forward')
          if y == 1 and orientation == 'DOWN':
              if 'Forward' in possible_actions:
                  possible_actions.remove('Forward')
          if x == self.dimrow and orientation == 'RIGHT':
              if 'Forward' in possible_actions:
                  possible_actions.remove('Forward')
          if y == self.dimrow and orientation == 'UP':
              if 'Forward' in possible_actions:
                  possible_actions.remove('Forward')
  
          return possible_actions
  
      def result(self, state, action):
          """ Given state and action, return a new state that is the result of the action.
          Action is assumed to be a valid action in the state """
          x, y = state.get_location()
          proposed_loc = list()
  
          # Move Forward
          if action == 'Forward':
              if state.get_orientation() == 'UP':
                  proposed_loc = [x, y + 1]
              elif state.get_orientation() == 'DOWN':
                  proposed_loc = [x, y - 1]
              elif state.get_orientation() == 'LEFT':
                  proposed_loc = [x - 1, y]
              elif state.get_orientation() == 'RIGHT':
                  proposed_loc = [x + 1, y]
              else:
                  raise Exception('InvalidOrientation')
  
          # Rotate counter-clockwise
          elif action == 'TurnLeft':
              if state.get_orientation() == 'UP':
                  state.set_orientation('LEFT')
              elif state.get_orientation() == 'DOWN':
                  state.set_orientation('RIGHT')
              elif state.get_orientation() == 'LEFT':
                  state.set_orientation('DOWN')
              elif state.get_orientation() == 'RIGHT':
                  state.set_orientation('UP')
              else:
                  raise Exception('InvalidOrientation')
  
          # Rotate clockwise
          elif action == 'TurnRight':
              if state.get_orientation() == 'UP':
                  state.set_orientation('RIGHT')
              elif state.get_orientation() == 'DOWN':
                  state.set_orientation('LEFT')
              elif state.get_orientation() == 'LEFT':
                  state.set_orientation('UP')
              elif state.get_orientation() == 'RIGHT':
                  state.set_orientation('DOWN')
              else:
                  raise Exception('InvalidOrientation')
  
          if proposed_loc in self.allowed:
              state.set_location(proposed_loc[0], [proposed_loc[1]])
  
          return state
  
      def goal_test(self, state):
          """ Given a state, return True if state is a goal state or False, otherwise """
  
          return state.get_location() == tuple(self.goal)
  
      def h(self, node):
          """ Return the heuristic value for a given state."""
  
          # Manhattan Heuristic Function
          x1, y1 = node.state.get_location()
          x2, y2 = self.goal
  
          return abs(x2 - x1) + abs(y2 - y1)
  
  
  # ______________________________________________________________________________
  # 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, inf
          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 = inf
              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, inf)
      return result
  
  
  def hill_climbing(problem):
      """
      [Figure 4.2]
      From the initial node, keep choosing the neighbor with highest value,
      stopping when no neighbor is better.
      """
      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_choice = random.choice(neighbors)
          delta_e = problem.value(next_choice.state) - problem.value(current.state)
          if delta_e > 0 or probability(math.exp(delta_e / T)):
              current = next_choice
  
  
  def simulated_annealing_full(problem, schedule=exp_schedule()):
      """ This version returns all the states encountered in reaching 
      the goal state."""
      states = []
      current = Node(problem.initial)
      for t in range(sys.maxsize):
          states.append(current.state)
          T = schedule(t)
          if T == 0:
              return states
          neighbors = current.expand(problem)
          if not neighbors:
              return current.state
          next_choice = random.choice(neighbors)
          delta_e = problem.value(next_choice.state) - problem.value(current.state)
          if delta_e > 0 or probability(math.exp(delta_e / T)):
              current = next_choice
  
  
  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, [])
  
  
  # Pre-defined actions for PeakFindingProblem
  directions4 = {'W': (-1, 0), 'N': (0, 1), 'E': (1, 0), 'S': (0, -1)}
  directions8 = dict(directions4)
  directions8.update({'NW': (-1, 1), 'NE': (1, 1), 'SE': (1, -1), 'SW': (-1, -1)})
  
  
  class PeakFindingProblem(Problem):
      """Problem of finding the highest peak in a limited grid"""
  
      def __init__(self, initial, grid, defined_actions=directions4):
          """The grid is a 2 dimensional array/list whose state is specified by tuple of indices"""
          super().__init__(initial)
          self.grid = grid
          self.defined_actions = defined_actions
          self.n = len(grid)
          assert self.n > 0
          self.m = len(grid[0])
          assert self.m > 0
  
      def actions(self, state):
          """Returns the list of actions which are allowed to be taken from the given state"""
          allowed_actions = []
          for action in self.defined_actions:
              next_state = vector_add(state, self.defined_actions[action])
              if 0 <= next_state[0] <= self.n - 1 and 0 <= next_state[1] <= self.m - 1:
                  allowed_actions.append(action)
  
          return allowed_actions
  
      def result(self, state, action):
          """Moves in the direction specified by action"""
          return vector_add(state, self.defined_actions[action])
  
      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 = dict()
          self.unbacktracked = dict()
          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.pop(s1)
                      for (s, b) in self.result.keys():
                          if self.result[(s, b)] == unbacktracked_pop:
                              self.a = b
                              break
              else:
                  self.a = self.untried.pop(s1)
          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):
          super().__init__(initial, goal)
          self.graph = graph
  
      def actions(self, state):
          return self.graph.graph_dict[state].keys()
  
      def output(self, state, action):
          return self.graph.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 = min(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, 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 max(population, key=fitness_fn)
  
  
  def fitness_threshold(fitness_fn, f_thres, population):
      if not f_thres:
          return None
  
      fittest_individual = max(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 recombine_uniform(x, y):
      n = len(x)
      result = [0] * n
      indexes = random.sample(range(n), n)
      for i in range(n):
          ix = indexes[i]
          result[ix] = x[ix] if i < n / 2 else y[ix]
  
      return ''.join(str(r) for r in result)
  
  
  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 (vertices) 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, graph_dict=None, directed=True):
          self.graph_dict = graph_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.graph_dict.keys()):
              for (b, dist) in self.graph_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.graph_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.graph_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."""
          s1 = set([k for k in self.graph_dict.keys()])
          s2 = set([k2 for v in self.graph_dict.values() for k2, v2 in v.items()])
          nodes = s1.union(s2)
          return list(nodes)
  
  
  def UndirectedGraph(graph_dict=None):
      """Build a Graph where every edge (including future ones) goes both ways."""
      return Graph(graph_dict=graph_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 inf
                      return distance(g.locations[n], here)
  
                  neighbor = min(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
  """
  vacuum_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):
          super().__init__(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 inf)
  
      def find_min_edge(self):
          """Find minimum value of edges."""
          m = inf
          for d in self.graph.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 inf
  
  
  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
  
  
  # ______________________________________________________________________________
  
  
  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 -1 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):
          super().__init__(tuple([-1] * N))
          self.N = N
  
      def actions(self, state):
          """In the leftmost empty column, try all non-conflicting rows."""
          if state[-1] is not -1:
              return []  # All columns filled; no successors
          else:
              col = state.index(-1)
              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(-1)
          new = list(state[:])
          new[col] = row
          return tuple(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 or  # same row
                  col1 == col2 or  # same column
                  row1 - col1 == row2 - col2 or  # same \ diagonal
                  row1 + col1 == row2 + col2)  # same / diagonal
  
      def goal_test(self, state):
          """Check if all columns filled, no conflicts."""
          if state[-1] is -1:
              return False
          return not any(self.conflicted(state, state[col], col)
                         for col in range(len(state)))
  
      def h(self, node):
          """Return number of conflicting queens for a given node"""
          num_conflicts = 0
          for (r1, c1) in enumerate(node.state):
              for (r2, c2) in enumerate(node.state):
                  if (r1, c1) != (r2, c2):
                      num_conflicts += self.conflict(r1, c1, r2, c2)
  
          return num_conflicts
  
  
  # ______________________________________________________________________________
  # 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, file, min_len=3):
          lines = file.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(open_data("EN-text/wordlist.txt"))
          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}/{}>'.format(self.succs, self.goal_tests,
                                                 self.states, str(self.found)[:4])
  
  
  def compare_searchers(problems, header,
                        searchers=[breadth_first_tree_search,
                                   breadth_first_graph_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 search results."""
      compare_searchers(problems=[GraphProblem('Arad', 'Bucharest', romania_map),
                                  GraphProblem('Oradea', 'Neamt', romania_map),
                                  GraphProblem('Q', 'WA', australia_map)],
                        header=['Searcher', 'romania_map(Arad, Bucharest)',
                                'romania_map(Oradea, Neamt)', 'australia_map'])
  