CMPSC 442: Homework 2

CMPSC 442: Homework 2 [100 points]
TO PREPARE AND SUBMIT HOMEWORK
Follow these steps exactly, so the Gradescope autgrader can grade your homework. Failure to do so
will result in a zero grade:
1. You *must* download the homework template file homework2_cmpsc442.py from Canvas. Each
template file is a python file that gives you a headstart in creating your homework python script
with the correct function names for autograding.
2. You *must* download the code for the GUI homework2_lights_out_gui.py from Canvas. You
will use this to interactively play the Lights Out puzzle.
3. You *must* rename the file by replacing cmpsc442 with your PSU id from your official PSU. For
example, if your PSU email id is abcd1234, you would rename your file as
homework2_abcd1234.py to submit to Gradescope.
4. Upload your *py file to Gradescope by its due date. It is your responsibility to upload on time. If
the submission to Gradescope closes before you upload, your homework will be counted late, and
you might get a zero grade.
5. Make sure your file can import before you submit; the autograder imports your file. If it won't
import, you will get a zero.
Instructions
In this assignment, you will explore three classic puzzles from the perspective of uninformed search.
A skeleton file homework2_cmpsc442.py containing empty definitions for each question has been
provided. Since portions of this assignment will be graded automatically, none of the names or
function signatures in this file should be modified. However, you are free to introduce additional
variables or functions if needed.
You may import definitions from any standard Python library, and are encouraged to do so in cases
where you find yourself reinventing the wheel. If you are unsure where to start, consider taking a look
at the data structures and functions defined in the collections, itertools, math, and random
modules.
You will find that in addition to a problem specification, most programming questions also include a
pair of examples from the Python interpreter. These are meant to illustrate typical use cases to clarify
the assignment, and are not comprehensive test suites. In addition to performing your own testing,
you are strongly encouraged to verify that your code gives the expected output for these examples
before submitting.
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You are strongly encouraged to follow the Python style guidelines set forth in PEP 8, which was written
in part by the creator of Python. However, your code will not be graded for style.
1. N-Queens [25 points]
In this section, you will develop a solver for the n-queens problem, wherein n queens are to be placed
on an n x n chessboard so that no pair of queens can attack each other. Recall that in chess, a queen
can attack any piece that lies in the same row, column, or diagonal as itself.
A brief treatment of this problem for the case where n = 8 is given in Section 3.2 of the 3rd edition of
the textbook, which you may wish to consult for additional information. See the Canvas Homework 2
page for a link to the AIMA 3rd edition discussion of uninformed search problems
1. [5 points] Rather than performing a search over all possible placements of queens on the board,
it is sufficient to consider only those configurations for which each row contains exactly one
queen. Without taking any of the chess-specific constraints between queens into account,
implement the pair of functions num_placements_all(n) and
num_placements_one_per_row(n) that return the number of possible placements of n queens
on an n x n board without or with this additional restriction. Think carefully about why this
restriction is valid, and note the extent to which it reduces the size of the search space. You
should assume that all queens are indistinguishable for the purposes of your calculations.
2. [5 points] With the answer to the previous question in mind, a sensible representation for a
board configuration is a list of numbers between 0 and n - 1, where the ith number designates the
column of the queen in row i for 0 ≤ i < n. A complete configuration is then specified by a list
containing n numbers, and a partial configuration is specified by a list containing fewer than n
numbers. Write a function n_queens_valid(board) that accepts such a list and returns True if
no queen can attack another, or False otherwise. Note that the board size need not be included
as an additional argument to decide whether a particular list is valid.
>>> n_queens_valid([0, 0])
False
>>> n_queens_valid([0, 2])
True
>>> n_queens_valid([0, 1])
False
>>> n_queens_valid([0, 3, 1])
True
3. [15 points] Write a function n_queens_solutions(n) that yields all valid placements of n
queens on an n x n board, using the representation discussed above. The output may be
generated in any order you see fit. Your solution should be implemented as a depth-first search,
where queens are successively placed in empty rows until all rows have been filled. Hint: You
may find it helpful to define a helper function n_queens_helper(n, board) that yields all
valid placements which extend the partial solution denoted by board.
Though our discussion of search in class has primarily covered algorithms that return just a
single solution, the extension to a generator which yields all solutions is relatively simple. Rather
than using a return statement when a solution is encountered, yield that solution instead, and
then continue the search.
>>> solutions = n_queens_solutions(4)
>>> next(solutions)
>>> list(n_queens_solutions(6))
[[1, 3, 5, 0, 2, 4], [2, 5, 1, 4, 0, 3],
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[1, 3, 0, 2]
>>> next(solutions)
[2, 0, 3, 1]
[3, 0, 4, 1, 5, 2], [4, 2, 0, 5, 3, 1]]
>>> len(list(n_queens_solutions(8)))
92
2. Lights Out [40 points]
The Lights Out puzzle consists of an m x n grid of lights, each of which has two states: on and off. The
goal of the puzzle is to turn all the lights off, with the caveat that whenever a light is toggled, its
neighbors above, below, to the left, and to the right will be toggled as well. If a light along the edge of
the board is toggled, then fewer than four other lights will be affected, as the missing neighbors will be
ignored.
In this section, you will investigate the behavior of Lights Out puzzles of various sizes by implementing
a LightsOutPuzzle class. Once you have completed the problems in this section, you can test your
code in an interactive setting using the provided GUI. See the end of the section for more details.
1. [2 points] A natural representation for this puzzle is a two-dimensional list of Boolean values,
where True corresponds to the on state and False corresponds to the off state. In the
LightsOutPuzzle class, write an initialization method __init__(self, board) that stores an
input board of this form for future use. Also write a method get_board(self) that returns this
internal representation. You additionally may wish to store the dimensions of the board as
separate internal variables, though this is not required.
>>> b = [[True, False], [False, True]]
>>> p = LightsOutPuzzle(b)
>>> p.get_board()
[[True, False], [False, True]]
>>> b = [[True, True], [True, True]]
>>> p = LightsOutPuzzle(b)
>>> p.get_board()
[[True, True], [True, True]]
2. [3 points] Write a top-level function create_puzzle(rows, cols) that returns a new
LightsOutPuzzle of the specified dimensions with all lights initialized to the off state.
>>> p = create_puzzle(2, 2)
>>> p.get_board()
[[False, False], [False, False]]
>>> p = create_puzzle(2, 3)
>>> p.get_board()
[[False, False, False],
[False, False, False]]
3. [5 points] In the LightsOutPuzzle class, write a method perform_move(self, row, col)
that toggles the light located at the given row and column, as well as the appropriate neighbors.
>>> p = create_puzzle(3, 3)
>>> p.perform_move(1, 1)
>>> p.get_board()
[[False, True, False],
[True, True, True ],
[False, True, False]]
>>> p = create_puzzle(3, 3)
>>> p.perform_move(0, 0)
>>> p.get_board()
[[True, True, False],
[True, False, False],
[False, False, False]]
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4. [5 points] In the LightsOutPuzzle class, write a method scramble(self) which scrambles
the puzzle by calling perform_move(self, row, col) with probability 1/2 on each location on
the board. This guarantees that the resulting configuration will be solvable, which may not be
true if lights are flipped individually. Hint: After importing the random module with the
statement import random, the expression random.random() < 0.5 generates the values True
and False with equal probability.
5. [2 points] In the LightsOutPuzzle class, write a method is_solved(self) that returns
whether all lights on the board are off.
>>> b = [[True, False], [False, True]]
>>> p = LightsOutPuzzle(b)
>>> p.is_solved()
False
>>> b = [[False, False], [False, False]]
>>> p = LightsOutPuzzle(b)
>>> p.is_solved()
True
6. [3 points] In the LightsOutPuzzle class, write a method copy(self) that returns a new
LightsOutPuzzle object initialized with a deep copy of the current board. Changes made to the
original puzzle should not be reflected in the copy, and vice versa.
>>> p = create_puzzle(3, 3)
>>> p2 = p.copy()
>>> p.get_board() == p2.get_board()
True
>>> p = create_puzzle(3, 3)
>>> p2 = p.copy()
>>> p.perform_move(1, 1)
>>> p.get_board() == p2.get_board()
False
7. [5 points] In the LightsOutPuzzle class, write a method successors(self) that yields all
successors of the puzzle as (move, new-puzzle) tuples, where moves themselves are (row,
column) tuples. The second element of each successor should be a new LightsOutPuzzle object
whose board is the result of applying the corresponding move to the current board. The
successors may be generated in whichever order is most convenient.
>>> p = create_puzzle(2, 2)
>>> for move, new_p in p.successors():
... print(move, new_p.get_board())
...
(0, 0) [[True, True], [True, False]]
(0, 1) [[True, True], [False, True]]
(1, 0) [[True, False], [True, True]]
(1, 1) [[False, True], [True, True]]
>>> for i in range(2, 6):
... p = create_puzzle(i, i + 1)
... print(len(list(p.successors())))
...
6
12
20
30
8. [15 points] In the LightsOutPuzzle class, write a method find_solution(self) that returns
an optimal solution to the current board as a list of moves, represented as (row, column) tuples.
If more than one optimal solution exists, any of them may be returned. Your solver should be
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implemented using a breadth-first graph search, which means that puzzle states should not be
added to the frontier if they have already been visited, or are currently in the frontier. If the
current board is not solvable, the value None should be returned instead. You are highly
encouraged to reuse the methods defined in the previous exercises while developing your
solution.
Hint: For efficient testing of duplicate states, consider using tuples representing the boards of
the LightsOutPuzzle objects being explored rather than their internal list-based
representations. You will then be able to use the built-in set data type to check for the presence
or absence of a particular state in near-constant time.
>>> p = create_puzzle(2, 3)
>>> for row in range(2):
... for col in range(3):
... p.perform_move(row, col)
...
>>> p.find_solution()
[(0, 0), (0, 2)]
>>> b = [[False, False, False],
... [False, False, False]]
>>> b[0][0] = True
>>> p = LightsOutPuzzle(b)
>>> p.find_solution() is None
True
Once you have implemented the functions and methods described in this section, you can play with an
interactive version of the Lights Out puzzle using the provided GUI by running the following
command:
python homework2_lights_out_gui.py rows cols
The arguments rows and cols are positive integers designating the size of the puzzle.
In the GUI, you can click on a light to perform a move at that location, and use the side menu to
scramble or solve the puzzle. The GUI is merely a wrapper around your implementations of the
relevant functions, and may therefore serve as a useful visual tool for debugging.
3. Linear Disk Movement [30 points]
In this section, you will investigate the movement of disks on a linear grid.
The starting configuration of this puzzle is a row of L cells, with disks located on cells 0 through n - 1.
The goal is to move the disks to the end of the row using a constrained set of actions. At each step, a
disk can only be moved to an adjacent empty cell, or to an empty cell two spaces away, provided
another disk is located on the intervening cell. Given these restrictions, it can be seen that in many
cases, no movements will be possible for the majority of the disks. For example, from the starting
position, the only two options are to move the last disk from cell n - 1 to cell n, or to move the secondto-last disk from cell n - 2 to cell n.
1. [15 points] Write a function solve_identical_disks(length, n) that returns an optimal
solution to the above problem as a list of moves, where length is the number of cells in the row
and n is the number of disks. Each move in the solution should be a two-element tuple of the
form (from, to) indicating a disk movement from the first cell to the second. As suggested by its
name, this function should treat all disks as being identical.
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Your solver for this problem should be implemented using a breadth-first graph search. The
exact solution produced is not important, as long as it is of minimal length.
Unlike in the previous two sections, no requirement is made with regards to the manner in which
puzzle configurations are represented. Before you begin, think carefully about which data
structures might be best suited for the problem, as this choice may affect the efficiency of your
search.
>>> solve_identical_disks(4, 2)
[(0, 2), (1, 3)]
>>> solve_identical_disks(5, 2)
[(0, 2), (1, 3), (2, 4)]
>>> solve_identical_disks(4, 3)
[(1, 3), (0, 1)]
>>> solve_identical_disks(5, 3)
[(1, 3), (0, 1), (2, 4), (1, 2)]
2. [15 points] Write a function solve_distinct_disks(length, n) that returns an optimal
solution to the same problem with a small modification: in addition to moving the disks to the
end of the row, their final order must be the reverse of their initial order. More concretely, if we
abbreviate length as L, then a desired solution moves the first disk from cell 0 to cell L - 1, the
second disk from cell 1 to cell L - 2, . . . , and the last disk from cell n - 1 to cell L - n.
Your solver for this problem should again be implemented using a breadth-first graph search. As
before, the exact solution produced is not important, as long as it is of minimal length.
>>> solve_distinct_disks(4, 2)
[(0, 2), (2, 3), (1, 2)]
>>> solve_distinct_disks(5, 2)
[(0, 2), (1, 3), (2, 4)]
>>> solve_distinct_disks(4, 3)
[(1, 3), (0, 1), (2, 0), (3, 2), (1, 3),
(0, 1)]
>>> solve_distinct_disks(5, 3)
[(1, 3), (2, 1), (0, 2), (2, 4), (1, 2)]
4. Feedback [5 points]
1. [1 point] Approximately how long did you spend on this assignment?
2. [2 points] Which aspects of this assignment did you find most challenging? Were there any
significant stumbling blocks?
3. [2 points] Which aspects of this assignment did you like? Is there anything you would have
changed?