"""Find a Hamiltonian path or cycle in a graph that is induced by a rectangular
board and a list of moves.


Copyright (c) 2016 <NAME> <<EMAIL>>
"""
from sys import stdout
import math


class Hamilton(object):
    """Find a Hamiltonian path or cycle in a graph that is induced by a
    rectangular board and a list of moves.
    """
    def __init__(
            self, moves, x_size, y_size, x_start, y_start, closed=False,
            max_retries=0):
        """
        :arg list moves: Definition of allowed moves.
        :arg int x_size: Height of the board.
        :arg int y_size: Width of the board.
        :arg int x: x-coordinate.
        :arg int y: y-coordinate.
        :arg bool closed: Find a closed path.
        :arg int max_reties: Stop after a given number of retries (0=disabled).
        """
        self._moves = moves
        self._x_size = x_size
        self._y_size = y_size
        self._max_retries = max_retries

        self._max_depth = self._x_size * self._y_size
        self._decimals = int(math.log(self._max_depth, 10) + 1)

        self.reset(x_start, y_start, closed)

    def __str__(self):
        return '\n'.join(
            map(lambda r: ''.join(
            map(lambda e: ' {{:{}d}}'.format(self._decimals).format(e), r)),
            self.board)) + '\n'

    def _push(self, moves):
        self._stack.append([len(moves), moves])

    def _pop(self):
        return self._stack.pop()[1]

    def _current(self):
        return self._stack[-1][1][self._stack[-1][0]]

    def _next(self):
        if not self._stack[-1][0]:
            return ()

        self._stack[-1][0] -= 1
        return self._current()

    def _valid_moves(self, x, y):
        """Determine all valid moves, given a position.

        :arg int x: x-coordinate.
        :arg int y: y-coordinate.

        :returns list: List of valid moves.
        """
        moves = []

        for move in self._moves:
            _x = x + move[0]
            _y = y + move[1]
            if (
                    _x >= 0 and _x < self._x_size and
                    _y >= 0 and _y < self._y_size and self.board[_x][_y] < 1):
                moves.append((_x, _y))

        return moves

    def _update(self, moves, amount):
        """Update accessibility of a list of moves.

        :arg list moves: List of moves.
        :arg int amount: Increase or decrease accessibility (1 or -1).
        """
        for move in moves:
            self.board[move[0]][move[1]] += amount

    def _prioritise(self, moves):
        """Prioritise a list of moves based on accessibility.

        :arg list moves: List of moves.

        :returns list: List of moves sorted by accessibility.
        """
        if not moves:
            return []

        weights = map(lambda x: self.board[x[0]][x[1]], moves)

        return list(zip(*sorted(zip(weights, moves))))[1]

    def _solve_recursive(self, x, y, depth=1):
        """
        :arg int x: x-coordinate.
        :arg int y: y-coordinate.
        :arg int depth: Move number.

        :returns bool: True for success, False for failure.
        """
        # Making any field inaccessible would be a better check.
        if self._closed and not self._valid_moves(
                self._x_start, self._y_start):
            return False

        self.board[x][y] = depth

        if depth == self._max_depth:
            return True

        moves = self._valid_moves(x, y)
        self._update(moves, 1)

        for move in self._prioritise(moves)[::-1]:
            if self._solve_recursive(move[0], move[1], depth + 1):
                return True

        self.retries += 1
        self.board[x][y] = -len(moves)
        self._update(moves, -1)

        return False

    def reset(self, x, y, closed=False):
        """Initialise the board and set the parameters for the path finding.

        :arg int x: x-coordinate.
        :arg int y: y-coordinate.
        :arg bool closed: Find a closed path.
        """
        self.board = [[0] * self._y_size for _ in range(self._x_size)]

        for i in range(self._x_size):
            for j in range(self._y_size):
                self.board[i][j] = -len(self._valid_moves(i, j))

        self._x_start = x
        self._y_start = y
        self._closed = closed
        self._stack = []
        self.retries = 0

    def solve_recursive(self):
        """Find a Hamiltonian path or cycle.

        :returns bool: True for success, False for failure.
        """
        return self._solve_recursive(self._x_start, self._y_start)

    def _stop_condition(self, move):
        if not move:
            return False

        if self._closed:
            if not self._valid_moves(self._x_start, self._y_start):
                return False

        priorities = list(map(
            lambda x: len(self._valid_moves(x[0], x[1])), self._stack[-1][1]))
        if len(priorities) > 1:
            # If one of the squares is unreachable, there is no point in
            # exploring other possibilities.
            if priorities[-1] == 0:
                return False
            if self._closed:
                # If we see two squares that are accessible from one
                # position, either one of those squares is an end point. 
                if priorities[-2] < 2:
                    #stdout.write('\n{}'.format(self))
                    return False
            else:
                # If we see three squares that are accessible from one
                # position, either two of those squares is an end point. Since
                # we can only have one end point, we can stop here.
                if len(priorities) > 2 and priorities[-3] < 2:
                    return False

        return True

    def solve(self):
        """Find a Hamiltonian path or cycle.

        :returns bool: True for success, False for failure.
        """
        depth = 1
        self.board[self._x_start][self._y_start] = depth

        moves = self._prioritise(
            self._valid_moves(self._x_start, self._y_start))
        self._update(moves, 1)
        self._push(moves)

        while True:
            #stdout.write('{}'.format(depth))
            move = self._next()
            if self._stop_condition(move):
                depth += 1
                self.board[move[0]][move[1]] = depth
                if depth == self._max_depth:
                    return True
                moves = self._prioritise(self._valid_moves(move[0], move[1]))
                self._update(moves, 1)
                self._push(moves)
            else:
                if self._max_retries and self.retries >= self._max_retries:
                    return False
                self.retries += 1
                moves = self._pop()
                if not self._stack:
                    return False
                undo = self._current()
                self.board[undo[0]][undo[1]] = -len(moves)
                self._update(moves, -1)
                depth -= 1

"""Find a Hamiltonian path or cycle in a graph that is induced by a rectangular
board and a list of moves.


Copyright (c) 2016 <NAME> <<EMAIL>>
"""
from sys import stdout
import math


class Hamilton(object):
    """Find a Hamiltonian path or cycle in a graph that is induced by a
    rectangular board and a list of moves.
    """
    def __init__(
            self, moves, x_size, y_size, x_start, y_start, closed=False,
            max_retries=0):
        """
        :arg list moves: Definition of allowed moves.
        :arg int x_size: Height of the board.
        :arg int y_size: Width of the board.
        :arg int x: x-coordinate.
        :arg int y: y-coordinate.
        :arg bool closed: Find a closed path.
        :arg int max_reties: Stop after a given number of retries (0=disabled).
        """
        self._moves = moves
        self._x_size = x_size
        self._y_size = y_size
        self._max_retries = max_retries

        self._max_depth = self._x_size * self._y_size
        self._decimals = int(math.log(self._max_depth, 10) + 1)

        self.reset(x_start, y_start, closed)

    def __str__(self):
        return '\n'.join(
            map(lambda r: ''.join(
            map(lambda e: ' {{:{}d}}'.format(self._decimals).format(e), r)),
            self.board)) + '\n'

    def _push(self, moves):
        self._stack.append([len(moves), moves])

    def _pop(self):
        return self._stack.pop()[1]

    def _current(self):
        return self._stack[-1][1][self._stack[-1][0]]

    def _next(self):
        if not self._stack[-1][0]:
            return ()

        self._stack[-1][0] -= 1
        return self._current()

    def _valid_moves(self, x, y):
        """Determine all valid moves, given a position.

        :arg int x: x-coordinate.
        :arg int y: y-coordinate.

        :returns list: List of valid moves.
        """
        moves = []

        for move in self._moves:
            _x = x + move[0]
            _y = y + move[1]
            if (
                    _x >= 0 and _x < self._x_size and
                    _y >= 0 and _y < self._y_size and self.board[_x][_y] < 1):
                moves.append((_x, _y))

        return moves

    def _update(self, moves, amount):
        """Update accessibility of a list of moves.

        :arg list moves: List of moves.
        :arg int amount: Increase or decrease accessibility (1 or -1).
        """
        for move in moves:
            self.board[move[0]][move[1]] += amount

    def _prioritise(self, moves):
        """Prioritise a list of moves based on accessibility.

        :arg list moves: List of moves.

        :returns list: List of moves sorted by accessibility.
        """
        if not moves:
            return []

        weights = map(lambda x: self.board[x[0]][x[1]], moves)

        return list(zip(*sorted(zip(weights, moves))))[1]

    def _solve_recursive(self, x, y, depth=1):
        """
        :arg int x: x-coordinate.
        :arg int y: y-coordinate.
        :arg int depth: Move number.

        :returns bool: True for success, False for failure.
        """
        # Making any field inaccessible would be a better check.
        if self._closed and not self._valid_moves(
                self._x_start, self._y_start):
            return False

        self.board[x][y] = depth

        if depth == self._max_depth:
            return True

        moves = self._valid_moves(x, y)
        self._update(moves, 1)

        for move in self._prioritise(moves)[::-1]:
            if self._solve_recursive(move[0], move[1], depth + 1):
                return True

        self.retries += 1
        self.board[x][y] = -len(moves)
        self._update(moves, -1)

        return False

    def reset(self, x, y, closed=False):
        """Initialise the board and set the parameters for the path finding.

        :arg int x: x-coordinate.
        :arg int y: y-coordinate.
        :arg bool closed: Find a closed path.
        """
        self.board = [[0] * self._y_size for _ in range(self._x_size)]

        for i in range(self._x_size):
            for j in range(self._y_size):
                self.board[i][j] = -len(self._valid_moves(i, j))

        self._x_start = x
        self._y_start = y
        self._closed = closed
        self._stack = []
        self.retries = 0

    def solve_recursive(self):
        """Find a Hamiltonian path or cycle.

        :returns bool: True for success, False for failure.
        """
        return self._solve_recursive(self._x_start, self._y_start)

    def _stop_condition(self, move):
        if not move:
            return False

        if self._closed:
            if not self._valid_moves(self._x_start, self._y_start):
                return False

        priorities = list(map(
            lambda x: len(self._valid_moves(x[0], x[1])), self._stack[-1][1]))
        if len(priorities) > 1:
            # If one of the squares is unreachable, there is no point in
            # exploring other possibilities.
            if priorities[-1] == 0:
                return False
            if self._closed:
                # If we see two squares that are accessible from one
                # position, either one of those squares is an end point. 
                if priorities[-2] < 2:
                    #stdout.write('\n{}'.format(self))
                    return False
            else:
                # If we see three squares that are accessible from one
                # position, either two of those squares is an end point. Since
                # we can only have one end point, we can stop here.
                if len(priorities) > 2 and priorities[-3] < 2:
                    return False

        return True

    def solve(self):
        """Find a Hamiltonian path or cycle.

        :returns bool: True for success, False for failure.
        """
        depth = 1
        self.board[self._x_start][self._y_start] = depth

        moves = self._prioritise(
            self._valid_moves(self._x_start, self._y_start))
        self._update(moves, 1)
        self._push(moves)

        while True:
            #stdout.write('{}'.format(depth))
            move = self._next()
            if self._stop_condition(move):
                depth += 1
                self.board[move[0]][move[1]] = depth
                if depth == self._max_depth:
                    return True
                moves = self._prioritise(self._valid_moves(move[0], move[1]))
                self._update(moves, 1)
                self._push(moves)
            else:
                if self._max_retries and self.retries >= self._max_retries:
                    return False
                self.retries += 1
                moves = self._pop()
                if not self._stack:
                    return False
                undo = self._current()
                self.board[undo[0]][undo[1]] = -len(moves)
                self._update(moves, -1)
                depth -= 1

Find a Hamiltonian path or cycle in a graph that is induced by a rectangular board and a list of moves.   Copyright (c) 2016 <NAME> <<EMAIL>> Find a Hamiltonian path or cycle in a graph that is induced by a     rectangular board and a list of moves. :arg list moves: Definition of allowed moves.         :arg int x_size: Height of the board.         :arg int y_size: Width of the board.         :arg int x: x-coordinate.         :arg int y: y-coordinate.         :arg bool closed: Find a closed path.         :arg int max_reties: Stop after a given number of retries (0=disabled). Determine all valid moves, given a position.          :arg int x: x-coordinate.         :arg int y: y-coordinate.          :returns list: List of valid moves. Update accessibility of a list of moves.          :arg list moves: List of moves.         :arg int amount: Increase or decrease accessibility (1 or -1). Prioritise a list of moves based on accessibility.          :arg list moves: List of moves.          :returns list: List of moves sorted by accessibility. :arg int x: x-coordinate.         :arg int y: y-coordinate.         :arg int depth: Move number.          :returns bool: True for success, False for failure. # Making any field inaccessible would be a better check. Initialise the board and set the parameters for the path finding.          :arg int x: x-coordinate.         :arg int y: y-coordinate.         :arg bool closed: Find a closed path. Find a Hamiltonian path or cycle.          :returns bool: True for success, False for failure. # If one of the squares is unreachable, there is no point in # exploring other possibilities. # If we see two squares that are accessible from one # position, either one of those squares is an end point. #stdout.write('\n{}'.format(self)) # If we see three squares that are accessible from one # position, either two of those squares is an end point. Since # we can only have one end point, we can stop here. Find a Hamiltonian path or cycle.          :returns bool: True for success, False for failure. #stdout.write('{}'.format(depth))