In this project, you will implement Kruskal's algorithm and Dijkstra's algorithm to help you both generate and solve mazes.
This is a team assignment. This entire project is due on Sat, Mar 10 at 11:30pm.
When working on this assignment, we expect you meet these baseline expectations at all times:
Here are some baseline expectations we expect you to meet:
Follow the course collaboration policies
DO NOT use any classes from
java.util.*. There are only two exceptions to this rule:
You may import and use java.util.Iterator and
java.util.NoSuchElementException.
You may import and use anything from java.util.*
within your testing code.
DO NOT modify instructor-provided code (unless told otherwise)
Download the project from this link and open it in your IDE. See the instructions from project 1 if you need a reminder on how to do this.
Copy all of your data structures, algorithms, and tests from the previous project to this new one.
Your ArrayHeap will not immediately compile. This is ok. We will explain what changes you need to make in the next section.
Finally, make sure everything works.
Try running TestGraph.java and make sure the tests run.
Try running SanityCheck.java, and try running checkstyle.
Checkstyle should still report the same 5 errors with SanityCheck.java
as it did with the previous projects.
Task: make ArrayHeap compile.
Take a look at IPriorityQueue. You may notice that the
interface now allows all subclasses to implement an optional
method: remove(...).
You may optionally implement this method while implementing Dijkstra's algorithm later in the project.
For now, add the method to your ArrayHeap class and
throw an UnsupportedOperation exception so that your code
will compile.
Task: make sure you can run MainWindow
Navigate to the mazes.gui package and run MainWindow.java.
This will launch a program that will (eventually) generate and solve mazes.
The GUI consists of two main regions. The large region at the top, which will initially contain your maze, and a few options and buttons at the bottom.
Here is a brief explanation of the user interface:
The "Base maze shape" combobox controls what underlying "shape" your maze will have. The default is "Grid", which creates a maze where each room is a rectangle.
Try switching the option to "Voronoi" and click the "Generate new maze" button to the left. This will generate a maze where the rooms are more irregular.
The "Maze generator" combobox controls the actual maze generation process – it takes the initial set of room and removes edges to generate a plausible maze.
The default option is "Do not delete any edges", which, as you may have guessed, does not delete any edges.
Now, try picking one of the "Delete random edges" options and generate a new maze. You should now see some of the edges removed and see something that more closely resembles a maze.
Unfortunately, the "randomly remove edges" strategy is not that great. If we remove 30% of the edges, the maze is too easy. If we remove 50% of the edges, we often end up with an unsolvable maze. (The red dots in the upper-left and lower-right corners are our starting and ending points).
The final option labeled "Run (randomized) Kruskal" is what you will eventually need to implement. It turns out we can use the properties of minimum spanning trees to generate a much more interesting (and yet always solvable) maze!
The "Find shortest path" button will, once implemented, draw the shortest path between the two red dots.
Clicking this button should currently cause the program to crash – we haven't implemented it yet.
If you want to customize the different options (e.g. change the number of rows
and columns in the grid maze, change the percentage of edges removed), look at the
MainWindow.launch method. Feel free to change any of the constants
there -- we will not be looking at this file when grading, so we don't care
what changes you make.
Task: complete the ArrayDisjointSet class.
Notes:
Be sure to use implement your disjoint set using the array-based representation we discussed in lecture.
We have provided you a few tests for this class (TestArrayDisjointSet), but they're deliberately minimal. We strongly encourage you to add more tests to make sure your ArrayDisjointSet is working correctly.
Task: implement the Graph constructor and numVertices
and numEdges methods.
Notes:
You can find this class inside the misc.graphs package.
Your constructor is currently designed to accept a list of vertices and a list of edges.
However, you may find this somewhat inconvenient to work with, especially
once you start implementing the findShortestPathBetween(...) method.
So, you will probably want to add extra fields so that you can store the graph in a more useful representation (e.g. as an adjacency list or adjacency matrix).
This class uses generics in a slightly more complicated way then you've seen in previous projects. We've tried to insulate you from this as much as possible, but it's possible you may run into unexpected issues or have difficulty getting your code to compile depending on what exactly you're doing.
If this happens to you, please ask for help either on Piazza or during office hours ASAP and we'll help you get unstuck as best as we can.
This is a class about data structures and algorithms, not a class about Java, so we don't want you to waste your time struggling with Java-related issues.
You should make sure to look over the misc.graphs.Edge class.
Any objects of type E implement the Edge methods.
However, be sure not modify this class.
You can find (again fairly minimal) tests for this class in
TestGraph.
Task: implement the Graph.findMinimumSpanningTree(...)
Notes:
You should implement findMinimumSpanningTree(...) using
Kruskal's algorithm.
You can use top topKSort(...) to sort your edges. This will mean
your algorithm will not run in \(\mathcal{O}((|E| + |V|)\alpha(|V|))\) time,
but that's ok.
If you do want to implement Kruskal's algorithm as efficiently as possible, feel free to implement a linear sort such as Bucket sort and use that instead.
The only caveat is that your sorting algorithm needs to be a private
helper method within your Graph class. It doesn't really make
sense for the sorting algorithm to live in a class about graphs, but we need
to do this due to limitations in our grading scripts. (They copy specific files
from your projects, so if you add extra code in unexpected places, they may not
be copied over.)
Task: implement the KruskalMazeCarver.returnWallsToRemove(...) method
If you remember when we were running the maze program from before, the "remove random walls" algorithms ended up generating pretty poor mazes. They either removed too many walls and created trivial mazes, or removed too few and created impossible ones.
What we really want is an algorithm that (a) generates a random-looking maze, (b) makes sure the maze is actually solvable, and (c) removes as few walls as possible.
It turns out that we can use algorithms such as Prim and Kruskal to do exactly that!
Here's the trick: we take the maze, treat each room as a vertex and each wall as an edge, assign each wall a random weight, and run any MST-finding algorithm. We then remove any wall that was a part of that MST.
This will end up satisfying all three criteria! By randomizing the wall weights, we remove random walls which satisfies criteria (a). A MST-finding algorithm, by definition, will ensure there exists a path from every vertex (every room) to every other one, satisfying criteria (b). And finally, because MST-finding algorithms try and avoid cycles, we avoid removing unnecesary edges and end up with a maze where there really is only one solution, satisfying criteria (c).
Your task here is to implement this algorithm within the KruskalMazeCarver
class.
Other notes:
You can find this class inside the mazes.generators.maze package.
Make sure you understand how to use the Maze and
Wall objects. The Wall object is a subclass of
Edge, which means you should be able to pass a list or set
of Walls into your Graph constructor (along with the corresponding
list of Rooms).
You may import and use java.util.Random while implementing this
class.
If you're stuck, try taking a look at RandomMazeCarver, which is
located within the same package. Your algorithm will be a little more complicated,
but it may serve as a good source of inspiration.
To test your method, try running the program and generate a few mazes after selecting the "Run (randomized) Kruskal" option.
Task: implement the Graph.findShortestPathBetween(...) method
Finally, you will implement the code to actually solve the mazes.
Notes:
You should implement findShortestPathBetween(...) using
Dijkstra's algorithm.
To represent infinity, use the Double.POSITIVE_INFINITY
constant.
While we definitely do encourage you to look at the pseudocode we provided in lecture, please be sure to take them with a grain of salt.
While we tried to write the pseudocode in a way that stuck a good balance between giving you a high-level overview vs making details clear, it's possible that we accidentally omitted a few details or included a minor bug somewhere.
In addition, you may find it convenient to arrange your code in a slightly different way then what the pseudocode suggests, especially since you're trying to solve a slightly different problem then what the lecture slides are solving: The version of Dijkstra's algorithm we provided in lecture is trying to find the shortest paths from the start to every other node; you're trying to find the shortest path from the start to a specific node.
One challenge you may run into is figuring out how exactly to update the costs of the vertices.
Feel free to solve this problem however you want – however, if you're stuck, we recommend using one of two approaches:
ArrayHeap class and implement the
remove method. Then, to update a vertex, you can remove
and insert back in vertices.Rather then inserting your vertices directly into your heap, you may want to create and insert a custom inner class instead. (Why do you suppose that is?)
Once you're done and all your tests are passing, try re-running the program and click the "Find shortest path" button. If everything went well, the program should now draw a red line connecting the start and the end!
(Or, if there is no valid path, display an alert box stating so.)
Task: submit answers to the following questions.
Each group member should answer and submit these questions independently at this canvas page.
You must submit your answers in either .txt or
.pdf form.
To make sure we have enough time to grade everything, we do not plan on offering extra credit on this assignment.
However, if you're bored and have some free time, something you could try exploring is implementing other algorithms for generating mazes.
In particular, Kruskal's algorithm tends to generate mazes that have lots of short "cul-de-sacs", which lowers the difficulty of the overall maze.
A better algorithm might instead try and generate longer, more "windy" paths.
This website has a good overview of a few different maze generation algorithms if you want to investigate more.
The following deliverables are due on Sat, Mar 10 at 11:30pm.
A single person from your partnership should
submit a zip file conaining the entire contents of your src folder
on Canvas.
Before submitting, be sure to double-check and make sure that...
remove(...) method to ArrayHeap.
We do not plan on testing this method, so it's fine if all it does is
throw an UnsupportedOperation exception.ArrayDisjointSet class is fully implemented and
passes all tests.Graph class is fully implemented and
passes all tests.KruskalMazeCarver class is fully
implemented.Both partners should turn in a .txt or
.pdf file containing their answers to part 7
on Canvas
if you haven't already. This is due at the same time as
the rest of your project.