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CS2040S: Data Structures and Algorithms Problem Set 8 solved

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Overview
It’s easy to get lost in life. This week we’re going to develop a program to help us get unlost. Even
better, we’re going to give you some extra superpowers: the ability to break down the obstacles in
your way, helping you to get to your destination even faster.
To start off, you are given a map of the maze you are lost in. Like most mazes, this maze
consists of one or more rooms. Each of the rooms has doors to one or more other rooms. Your
first job is to write a program that will explore the maze, as is, and discover the shortest path from
your current location to a destination location. Simultaneously, your program will be expected to
determine some other geographic information about the maze.
Your second job involves determining how best to use your superpower. You have been given
the power to bypass some fixed number of walls. Please remember to specify your superpower:
• Able to walk through walls.
• Able to fly over walls.
• Strong enough to knock down walls.
• Possesses dynamite.
• Some other superpower.
For zero points of extra credit, explain how your superpowers work and how you acquired them
(e.g., on the forum). In any case, your job is to find the shortest path from the source to the
destination, while bypassing no more than the allowed number of walls.
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Problem Details
Preliminaries. We have provided you with some basic framework code for handling the maze.
A maze consists of an r × c grid of rooms, with adjacent rooms separated by either a wall or an
empty space. Furthermore, the entire maze is bordered by walls.
The rooms are numbered starting from the top left corner (which is (0, 0)). The first coordinate
specifies the row number, and the second coordinate specifies the column number. For example, in
a 5 × 5 maze, the bottom left corner is (4, 0) and the top right corner is (0, 4).
Here is a pictorial example of a 5 × 5 maze:
0 1 2 3 4
###########
0 #R R R#R#R#
### # # # #
1 #R#R#R R R#
# ### ### #
2 #R R R#R R#
# ### # # #
3 #R#R R#R#R#
# # # ### #
4 #R#R#R#R R#
###########
In this diagram, each wall is depicted using a hash symbol, i.e., #, while each room is depicted
using the letter R (this is for visualization purposes only – in the actual maze files, the rooms are
represented as empty spaces as well!). Notice that there are exactly c rooms and at most c + 1
walls (including the left and right borders) on each row.
You may move between adjacent rooms in any of the four cardinal directions (north, south, east
and west) if there is no wall in that direction. Diagonal movement is not allowed. In the above
example, one can move from (0,0) to (0,1), but NOT from (0,0) to (1,0), nor from (0,0) to (1,1).
We provide you with two classes Maze and Room that represent the maze that your program
will solve. The size of the maze is represented by the number of rows and columns in the maze
(in the above example, both rows and columns are 5). The maze itself is represented by a matrix
of rooms. You will be able to check if there exists a wall in the four directions of the room through
the public methods hasNorthWall(), hasSouthWall(), hasEastWall() and hasWestWall(). On
top of that, there is a public boolean attribute onPath that you can set for each room (for printing
of mazes).
The Maze class has a static method readMaze(String fileName) that reads in a maze from a
text file and returns the maze object. We have provided several sample mazes for you to experiment
with. We also provide a simplistic way of visualizing a maze through the static class MazePrinter.
The static method void printMaze(Maze maze) of the MazePrinter class prints out a maze to
the standard output1
.
We also provide you with the class ImprovedMazePrinter which prints out a much prettier
version of the maze. (It was contributed by a former student: Mai Anh Vu.)
1For an additional zero extra bonus points, implement a prettier graphical maze display and share it with the rest
of the class.
2
###########
#PPPPP# # #
### #P# # #
# # #PPPPP#
# ### ###P#
# # P#
# ### # #P#
# # # #P#
# # # ###P#
# # # # P#
###########
Figure 1: Example printMaze output of a solved maze (with no superpowers)
In this problem set, you will implement two classes MazeSolver and MazeSolverWithPower,
that will implement the provided interfaces IMazeSolver and IMazeSolverWithPower
respectively.
Problem 8.a. The Average Coder
Joe the Average Coder is eager to solve this problem and implemented the class MazeSolverNaive
(found in M azeSolverN aive.java). In particular, Joe implemented the pathSearch method that
will return the minimum number of steps to get from a given starting coordinate to a target
coordinate. He did so using the Depth-First Search traversal that he learned in his Algorithms
and Data Structures class.
Take a look at Joe’s code. Given an arbitrary maze, will his algorithm be able to find the
shortest path from the start to the target? Why or why not? (You do not need to submit any code
here. Just explain your answer.)
Problem 8.b. Exploring the Maze
Now implement the class MazeSolver that correctly implements the IMazeSolver interface. First,
implement the method:
Integer pathSearch(int startRow, int startCol, int endRow, int endCol)
which searches for the shortest path from the specified start room to the specified end room.
It should return an integer representing the minimum number of steps needed if a path is found, or
null if no such path is available. When your search is complete, we should have that R.onPath == true
for every room R on the path and R.onPath == false for every room R not on the path. If done
correctly, you will be able to execute printMaze(Maze maze) after your search is completed to
draw the path taken by the solver, as in Figure 1.
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Next, implement the method: Integer numReachable(int k) that will return an integer
indicating how many rooms there are such that the minimum number of steps required to reach it
is k, based on your most recent pathSearch starting location. Your numReachable method should
count the number of rooms reachable from the initial location for each possible number of steps.
For example, how many rooms are there such that the minimum path distance is 0? How many
rooms are reachable with minimum 1 step? How many rooms are reachable with minimum 2 steps?
etc. For the example maze shown in Figure 1, with the most recent pathSearch start location
being (0,0), the following holds:
0 Step: 1 Room
1 Step: 1 Room
2 Steps: 2 Rooms
3 Steps: 1 Room
4 Steps: 2 Rooms
5 Steps: 4 Rooms
6 Steps: 5 Rooms
7 Steps: 5 Rooms
8 Steps: 3 Rooms
9 Steps: 1 Room
Notice that for each k, it outputs the number of rooms for which the minimum path distance is k
exactly. If you calculate this for every initial starting point in the maze, then you can determine
the diameter of the maze.
Your numReachable method should be efficient, as it will likely be called several times after a
pathSearch.
Problem 8.c. Maze Exploration for Real SuperPeople
In this part, as in the previous part, your job is to determine the shortest path from a specified
source to a specified destination. However, you are also given the ability to bypass (demolish, jump
over, or magically traverse) up to a limited number of walls along the way. Of course, you will not
be allowed to use your power to demolish the outer walls that surround your maze. You begin with
a fixed amount of superpowers, and every time you demolish a wall, your power reduces by one.
Your goal is to find the shortest path from the start to the end.
Create a new class MazeSolverWithPower that implements IMazeSolverWithPower which contains
the overridden method:
Integer pathSearch(int startRow, int startCol, int endRow, int endCol, int superpowers).
It should return an integer representing the minimum number of steps needed if a path is found,
or null if no such path is available. As before, it should also update, for each room, the onPath
attribute. (Notice that the IMazeSolverWithPower interface inherits from IMazeSolver, and hence
must correctly implement all the methods from there as well. In this case, if numReachable is called
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after a pathSearch with superpowers, then its answer should also be based on having the same
number of superpowers (in addition to having the same starting location))
Hint: One strategy is to represent the state as a combination of the current room and the
remaining amount of superpower. If you explore this graph of all possible states, you will visit all
possible rooms, with all possible remaining amounts of superpower.
Problem 8.d. (Bonus) Maze Generation
Sometimes you need to escape a maze, sometimes you need to build a maze2
. In this problem, we
would like to implement an algorithm for generating a maze. To get started, you may wish read
through this interesting review of various maze generation algorithms:
http://weblog.jamisbuck.org/2011/2/7/maze-generation-algorithm-recap
This blog entry summarizes 11 different methods for generating a maze. All of these techniques
yield mazes that have only one route from start to finish (i.e., there are no cycles—they generate a
tree). You might think about how to modify them to generate interesting graphs/mazes that have
more than one possible solution.
Inside the MazeGenerator class, implement the following method:
Maze generateMaze(int rows, int columns).
It should return a randomly generated maze given the number of rows and columns we would
like the maze to contain. If you feel that it will make the generated mazes more interesting, feel
free to add or remove parameters from the generateMaze method.
Submit your MazeGenerator class, along with a discussion of the design decisions that you
made, and your favorite maze that was generated by the algorithm (with no manual intervention).
Please share any fun mazes you come up with (whether by hand or by algorithm) on
the forum!
2See, for example, Inception (2010).
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