# Project III – N-Puzzle

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Cmpe 160 – Introduction to Object Oriented
Programming
Project III – N-Puzzle
1 Introduction
N-puzzle is a popular puzzle game that consists of N tiles where N can have
different values such as 8, 15, 24 and so on. The puzzle is divided into √
N + 1
rows and √
N + 1 columns. It consists of one empty space where the tiles can be
moved and thus the puzzle is solved when a particular goal pattern is formed.
There are several algorithms developed for N-puzzle problems which can be
categorized as Uninformed Search Algorithms and Informed Search Algorithms.
Uninformed Search Algorithms include
• Breadth-first Search
• Depth-first Search
• Iterative Deepening Search
some of which, we are familiar from tree algorithms. Informed Search Algorihms
include
• A* Search
• Greedy Search
To solve this problem, we will be using A* Search algorithm, which is a general
artificial intelligence methodology to solve similar problems. Informed Search
Algorithms, as the name suggest, must be informed about the current state by
some parameter, which is called the heuristic. A heuristic, also called heuristic function, is a function that ranks alternatives in search algorithms at each
branching step.
In the following sections, first we’ll explain A* Search Algorithm and our Heuristic, Taxicab Distance. Then we’ll be giving some details about the implementation.
2 A* Search Algorithm
A* algorithm is a searching algorithm that searches for the shortest path between the initial and the final state. A* algorithm has 3 parameters:
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• g: the cost of moving from initial cell to the current cell, which can be
expressed as the cost so far.
• h: heuristic value, that is, the estimated cost of moving from current cell
to the final cell.
• f: sum of g and h.
Ok, but what do they correspond to in our N-puzzle problem? Well, g, obviously, corresponds to the number of state transitions done so far for any state.
h corresponds to the Taxicab Distance [2] between the current state and the
final state (goal). f, again corresponds to h + g.
2.1 Taxicab Distance
Taxicab Distance between a state and the goal state is the sum of the vertical
and horizontal distances from the tiles to their goal positions. For an example,
examine the current state and goal state given below [1]:
Let’s calculate the Taxicab distance between each tile and their goal positions:
for tile 4 → 1 (1 row)
for tile 5 → 1 (1 row)
for tile 7 → 4 (2 rows + 2 columns)
for tile 8 → 2 (1 row + 1 column)
for tile 1 → 2 (1 row + 1 column)
for tile 2 → 2 (1 row + 1 column)
for tile 3 → 4 (2 rows + 2 columns)
for tile 6 → 2 (1 row + 1 column)
So, in total, Taxicab distance between our current state and the goal state is 18.
Let’s assume we reached to the current state from the initial state by 5 state
transitions. Therefore, for this state equation becomes: f = g+h = 5+18 = 23.
Taxicab distance is an easy-to-use distance metric that is used in Artificial
Intelligence algorithms along with other distance metrics such as Euclidean
distance and Hamming distance.
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3 Implementation
The project consists of two classes, namely Puzzle.java and Solver.java. In Puzzle.java we will be representing the current state of the game, and in Solver.java
we will be solving the puzzle using A* search algorithm.
3.1 Puzzle.java
This class represents a particular state of the board. The following field should
be defined for this class:
1. int[][] tile: In this field we store the state of the board in a 2D array.
Initialize this field as private so that other classes cannot reach this field.
You can create additional private fields as you want. But make sure that
they are private, not reachable from other classes.
Following public methods should be implemented. You are allowed to create
additional helper private methods if you need.
1. Puzzle(int[][] tiles): Constructor of the Puzzle class. Fill up the
int[][] tile field according to the given 2D array. Empty tile will ve
represented with 0. Do not assign the given parameter to the field directly.
Doing such is a bad practice because in general, we can’t guarantee that
given parameter won’t be modified by some other classes in our project.
Copy the content of the given array to int[][] tile. This practice makes
our class Immutable. The immutable objects are objects whose value can
not be changed after initialization.
2. int h(): This function is used to calculate Heuristic value, that is, Taxicab distance between this particular Puzzle object (current state) and the
goal state. Within this function, calculate the vertical and horizontal distances between each tile and goal positions. Return the summation of
distances. We will be using this information for our algorithm. In our goal
state, numbers should be in ascending order from left to right and from up
to bottom. Empty tile should be at the right-bottom of the puzzle.
3. boolean isCompleted(): Return true if current state is the same as the
goal state, otherwise false.
4. Iterable<Puzzle> getAdjacents(): This function should return all adjacent states to the current state of the puzzle. An adjacent state is another
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puzzle object we reach when the position of the empty tile is changed by
one row or column. Examine the current int[][] tile, and create new
Puzzle objects that are adjacent to the current state. Do not modify the
int[][] tile of the current Puzzle. This function will be very useful when
we explore each branching step in the search algorithm.
5. String toString(): Override the toString() method. It will be used to
print out the Puzzle objects to the output file. You should append ” ”
between the tiles and ”\n” after each row including the last row.
3.2 Solver.java
In this class we will be solving the puzzle using the algorithm described above.
We’ll also have a main method to test our program. You can create any number
of private fields and methods.
As we discussed earlier, we have to store f = g + h information for each
Puzzle objects. During the search algorithm, you can either use a PriorityQueue or a Stack to store each adjacent state in each step. However, storing the adjacents in a stack may become very inefficient for large input sizes
and PriorityQueue implementation may reduce the execution time drastically.
However, we have to somewhat ”order” Puzzle objects we store in our PriorityQueue so that, pq.peek() returns ”minimum” Puzzle according to this
”order”. In order to do so we’ll create a private subclass within the Solver
class named as PriorityObject, and this class will have another subclass that
implements Comparator<PriorityObject>. This class will have the following
fields. Please keep in mind that following implementation is provided
to guide you (everything is declared as private) – it’s not compulsory
to do so, you can come up with a different implementation to store
the Puzzle objects in PriorityQueue with an order.
1. private Puzzle board: Each PriorityObject actually corresponds to a
state (Puzzle).
2. private int f: f will be used to order the puzzles. In our A* algorithm,
we always want to prefer states with minimum f value.
3. private PriorityObject prev: We will store the previous PriorityObject
in this field. Whenever we create a PriorityObject, we have to store its
previous PriorityObject. This will be very useful to print out the states in
the path from inital state to the goal state.
4. private int g: g denotes the cost from inital state to this state, that is,
number of transitions between the states.
Our PriorityObject class will have the following methods:
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1. PriorityObject(Puzzle board, int g, PriorityObject prev):In the
constructor, assign the fields with the given parameters. Calculate f as
f = g + Taxicab Distance. Did we implement a function to calculate taxicab distance for a particular state before?
2. Comparator<PriorityObject> comparator(): Return a new
CustomComparator(), which is a subclass described below.
Implement a CustomComparator subclass (subclass of PriorityObject) that
implements Comparator<PriorityObject>. It will have one method:
int compare(PriorityObject o1, PriorityObject o2): Override the
compare method of Comparator<PriorityObject>. As usual, you should
return 1 if o1.f > o2.f. Return -1 if o1.f < o2.f and return 0 for equality.
Having talked about PriorityObject inner class, now let’s proceed to the
methods that will be implemented for Solver :
1. public static void main(String[] args): Within the main function,
open a file named as input.txt. First, you have to read an integer n that
represents the size of the board. For a 8-puzzle game, n will be 3. Then
you’ll read the tiles one by one to create 2D tile array. As stated above,
0 represents the empty tile. Create a Puzzle object and solve it using the
function below. Then open another file named output.txt and print out
number of minimum moves and each state that in the path to the final
state. Functions described below will be helpful during the solution. You
may use PrintStream to write into a file.
2. Solver(Puzzle root): This is the constructor of the Solver class. Throw
an IllegalArgumentException if root is null, otherwise call the Solve(root)
function to solve the puzzle.
3. void solve(Puzzle root): This function is where we solve the problem.
As stated earlier, you can use a stack or a priority queue to store the states.
However, priority queue is strongly recommended in order to reduce the
execution time of the algorithm. Store each state in a priority queue and
take states from priority queue with minimum f = g + h heuristic values.
If the recently popped state is final state, finish the execution. You may
want to store final state (final PriorityObject) in a private field to use it in
other functions.
There is an important optimization to observe here. Let’s say we are at
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state X and we pushed all adjacent states to the Priority Queue. After
popping one of the adjacents, we will again push X to the priority queue,
since obviously X is an adjacent of that node. In order to avoid such
duplicates, you can check if previous state was equal to the adjacent. If so,
do not push it to the priority queue. This is one of the points where storing
previous states as a field in PriorityObject becomes helpful.
4. Iterable<Puzzle> getSolution(): This function will be called from the
main after Solve(root) function is called. Iterable<Puzzle> getSolution()
must return an Iterable that consists of all Puzzle objects from initial state
to final state. We have stored previous states in our PriorityObject objects, now our approach will be very useful to us. We can collect all
states easily by back traversing from final state to the initial state using
private PriorityObject prev field. First element of the Iterable must
be the the Puzzle object that corresponds to the initial state.
5. int getMoves(): This function returns minimum number of moves for the
solution. You may want to store minimum number of moves in a private
field after solving the puzzle so that this function takes constant time.
4 Input & Output Format
At the very beginning of the input file, you will be provided with the value N.
Then following lines will consist of he initial state of the board. An example
input format can be found below:
file
3
1 2 3
0 4 6
7 5 8
In the first line, 3 is provided which corresponds to the N for our NxN puzzle.
Then in the following lines, initial state of the board is provided. In each line
there exists three tiles since our puzzle is 3×3. After reading the input, we have
to solve the puzzle using our algorithm explained above. For the output file,
at the very beginning of the file we have to print out the minimum number of
moves to solve the puzzle, which is in this case equal to 3. Then you have to
list all the states of the board that goes to the solution one by one. You have to
print out the number of rows/columns in the board (which is N) following with
the content of the board. States are separated by a newline. Corresponding
output file can be found below:
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file
Minimum number of moves = 3
3
1 2 3
0 4 6
7 5 8
3
1 2 3
4 0 6
7 5 8
3
1 2 3
4 5 6
7 0 8
3
1 2 3
4 5 6
7 8 0
In the first line, 3 corresponds to the minimum number of moves. Then in
the following lines, 3 corresponds to the N value of our NxN board.
5 Important Remarks
• You can use appropriate access modifiers, keywords (super, final, static
etc.) whenever you want. However, do not declare anything as public
except those listed in the Implementation section.
• Code documentation is important. Please provide comments for your fields,
classes and methods. Brief descriptions are enough.
• You can implement additional private helper methods of your choice.
• 1. What is a Comparator? What’s the point of creating a custom comparator that implements a Comparator<T>?
A Comparator helps us to impose a total ordering on some collection of
objects. Let’s say you want to sort a Collection consisting of Integers.
Integers have natural total order among them, therefore you can call the
function Collections.sort(myList) directly, and Java knows how to
sort Integers. But what about a data structure you created? How does
Java know how to sort them? At this point, we can create our own total
ordering and give it as a parameter. For example, let’s say we have
Book objects and we want to sort them by number of pages. We can
implement a CustomComparator for that Book class and give it as a
parameter to the sort function:
Collections.sort(myList, myList.get(0).pageOrder()). I have
prepared a small pdf explaining Comparable and Comparators. You
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can check it from here. To implement PriorityObject subclass, you
will do something similar to what is shown in last page of the pdf. This
pdf is just one of my personal notes, and the content is not related to
this project. Of course, you are strongly encouraged to use detailed
sources online to become familiar with the concept.
2. What is an immutable object?
An immutable object is the one that cannot be changed after creation.
For example, Strings are immutable in Java, meaning that you cannot
modify a string after creation. You can ask this point: But how can I
do something like this?:
String str = “hello”;
str = “modified”;
Actually, what you do in this scenario is not modifying the string but
creating a new one in the memory and changing where str points to.
You can think of like this: str used to point to a memory location in
which ”hello” was stored. Now, it points to a different position in which
”modified” is stored. So what happened to ”hello”? Well, since there is
no reference pointing to the memory location in which ”hello” is stored,
it will be deleted by the Garbage Collector [3] of Java. Remember that
strings were immutable in Python as well. In this project, we want to
create Puzzle class as immutable.
References
[1] https://tristanpenman.com/demos/n-puzzle/.
[2] https://en.wikipedia.org/wiki/Taxicab_geometry
[3] https://en.wikipedia.org/wiki/Garbage_collection_(computer_science)
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