Dijkstra's Algorithm

Introduction¶

Since this assignment includes code for a UI application, lets first take a moment to go over the distinct components of the code to frame the parts you will be implementing.

Application GUI¶

The mazes package contains the main application code for this assignment; find and run Main.java inside the package. 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 that displays the current maze, and a few options and buttons at the bottom.

• The “Maze base shape” drop-down 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” drop-down controls the actual maze generation process—it takes the initial set of room and removes walls to generate a plausible maze.

  The default option is “Do not delete any walls”, which, as you may have guessed, does not delete any walls.

  Now, try picking one of the “Delete random walls” options and generate a new maze. You should now see some of the walls removed and see something that more closely resembles a maze.

  Unfortunately, the “randomly remove walls” strategy is not that great. If we remove 30% of the walls, the maze is too easy. If we remove 50% of the walls, 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 be implementing later. 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, once implemented, will draw the shortest path between the two red dots.

  Clicking this button should currently throw an exception—you’ll be implementing this later as well.

• 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 walls removed), you can modify the main method to do so—we will not be looking at this file when grading, so feel free to change anything there.

Navigating the Codebase¶

You don’t need to understand everything immediately, but it will be difficult to complete the assignment if you don’t know where anything is.

Mazes¶

Of course, there’s more in the mazes package than just the Main class.

• The mazes.entities package contains basic data structures representing individual components of the maze.
  • The Wall and Room classes should be the only ones you need to interact with directly to complete the assignment.
• The mazes.gui package includes all the GUI code; you shouldn’t need to reference these files, but you’re welcome to take a look if you’re curious.
• The mazes.logic package contains all the main application logic—the code used to generate and solve mazes.
  • mazes.logic.generators contains the generators for grid-based and Voronoi-based mazes.
  • mazes.logic.carvers contains the maze “carvers”—the random removal carver and the Kruskal-based on that you will implement later.
  • mazes.logic also includes a MazeGraph class that serves as the primary graph representation used throughout the application, but the class itself does not contain much code and merely extends a graph from the graphs package instead.

Graphs¶

Meanwhile, the graphs package is a generic library of graph data structures and algorithms.

• graphs.Graph: a basic directed graph, with generic type parameters for vertex and edge types.
  • graphs.KruskalGraph: extends Graph to be undirected, and adds a few more methods required by Kruskal’s algorithm.
    • graphs.AdjacencyListUndirectedGraph: an adjacency list implementation of KruskalGraph. MazeGraph extends this class.
  • There are additional Graph implementations in the test module.
• graphs.BaseEdge: the base edge class; always directed and weighted.
  • graphs.Edge: the primary implementation of BaseEdge.
  • graphs.EdgeWithData: a BaseEdge that additionally contains some extra, generic data; used by MazeGraph.
• graphs.minspantrees: contains algorithms for finding minimum spanning trees
• graphs.shortestpaths: contains algorithms for finding shortest paths

Priority Queues¶

Since Dijkstra’s algorithm relies heavily on a priority queue, we’ve provided an alternate priority queue implementation in the skeleton code. If you wish to try using your own heap from the previous assignment anyway, you may change DijkstraShortestPathFinder to do so, as described in the class itself.

Disjoint Sets¶

Lastly, the disjointsets package contains the disjoint sets data structures required by Kruskal’s algorithm. A default, inefficient implementation is included in the skeleton. Later, you will implement an optimized version.

Solving Mazes¶

The primary maze-solving logic is in mazes.logic.MazeSolver, but the class basically just constructs a MazeGraph and uses a ShortestPathsFinder. This means that you won’t need to interact directly with MazeSolver; instead, in order to get maze solving working, you just need to implement a ShortestPathsFinder: namely, the DijkstraShortestPathsFinder.

We won’t go over all the interfaces here; only the most important ones. Make sure to look around in the code to find documentation for other classes.

Shortest-Path Finder¶

In the Dijkstra-based implementation that you will be implementing, the method above is broken into two sub-methods:

The ShortestPath object returned is essentially a container for edges, but also includes some other convenience methods.

Since all information needed is provided as method parameters, normal implementations shouldn’t require any fields or other persistent state. (This functionality could be defined as a static method, but having an associated object allows our grader code to be more flexible.)

Graph¶

Sample Lazy Implementation¶

It will probably be useful to take a look at this class before you begin implementing Dijkstra’s algorithm. (The code doesn’t actually compute correct shortest paths most of the time, but the syntax for using Graph, BasicEdge, and ShortestPath is correct.)

Dijkstra Implementation¶

With all the interfaces out of the way, you can finally start implementing Dijkstra’s algorithm.

Some implementation notes:

• You can use the Double.POSITIVE_INFINITY constant to represent infinity.
• Recall that Dijkstra’s algorithm requires that we start by initializing the distances of all possible vertices to infinity. This isn’t actually possible with our graph interface, and also may not be feasible in practice for graphs with many vertices—more than a computer could store in memory, or potentially even infinitely many vertices. To save memory, we will implement a slightly-different version of Dijkstra’s algorithm with a couple fundamental differences:
  • Instead of initializing values for all vertices at the beginning of the algorithm, we’ll initialize values for only the starting vertex. This necessitates that we also change our relaxation operation to match, to initialize values for new vertices as they are encountered.
  • Additionally, since the maze application has a particular destination in mind, our algorithm does not need to compute the full SPT; we can stop as soon as we have found the shortest path from the starting vertex to the destination.

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.)

Testing¶

Unfortunately, if you find yourself passing our unit tests, but failing stress tests on the grader, your best course of action will be to carefully review your code until you figure out where your code deviates from Dijkstra’s algorithm.

Some of our stress tests will also check your code’s runtime; one will time your code on random graphs with many vertices and edges (so the shortest paths end up being only several edges in length), whereas another will test your runtime with very long paths. Again, if you fail these tests, make sure that you’re following the pseudocode correctly.

As always, if you find yourself debugging for over an hour, definitely post on Piazza or come to office hours for assistance—we’re here to help you learn, and struggling on your own through multiple hours of debugging is not an efficient way to learn. (Additionally, by bringing our attention to bugs that our unit tests don’t catch, you help us write better tests for future quarters.)

Finally, for your convenience, here are some diagrams of graphs from provided tests.

Submission¶

Code from the second portion of the assignment does not interact with the code from this portion, so now is a good time to try submitting.

When you’re ready, continue on to the next section: Kruskal’s algorithm.