Mazes

A maze is a twisty and convoluted arrangement of corridors that challenges the solver to find a path from the entry to the exit. This part of the assignment is about using ADTs to represent, process, and solve mazes.

An introduction to mazes

Labyrinths and mazes have fascinated humans since ancient times (remember Theseus and the Minotaur?), but mazes can be more than just recreation. The mathematician Leonhard Euler was one of the first to analyze mazes mathematically, and in doing so, he founded the branch of mathematics known as topology. Many algorithms that operate on mazes are closely related to graph theory and have applications to diverse tasks such as designing circuit boards, routing network traffic, motion planning, and social networking.

Your goal for the maze portion of the assignment is to implement neat algorithms to solve a maze, while gaining practice with ADTs.

Before jumping into the code, please carefully read this background information on how we will represent mazes, which ADTs to use, and the format of the provided maze data files.

A Grid represents a maze

A maze can be modeled as a two-dimensional array where each element is either a wall or a corridor.

• The Grid from the Stanford library is an ideal data structure for this. A maze is represented as a Grid<bool>.
• Each grid element is one "cell" of the maze. The boolean value for each element indicates whether that cell is a corridor.
• The element grid[row][col] is true when there is a open corridor at location (row, col) and false if it is a wall.

A maze is read from a text file. Each line of the file is one row of the maze. Within a row, the character @ is a wall, and - is a corridor. Here is a sample 5x7 maze file (5 rows by 7 columns):

-------
-@@@@@-
-----@-
-@@@-@-
-@---@-

Our starter project provides the function readMazeFile to read a text maze into a Grid<bool>. The folder "Other files/res" contains our provided maze files.

A GridLocation is a row/col pair

The GridLocation struct is a companion type to represent a row/col location in a Grid. A GridLocation has two fields, row and col, that are packaged together into one aggregate type. The sample code below demonstrates using a GridLocation variable and assigning and accessing its row and col fields.

 // Declare a new GridLocation
 GridLocation chosen; // default initialized to 0,0
 chosen.row = 3;      // assign row of chosen
 chosen.col = 4;      // assign col of chosen
 
 // Initialize row,col of exit as part of its declaration
 GridLocation exit = { maze.numRows()-1, maze.numCols()-1 }; // last row, last col 

 // You can use a GridLocation to index into a Grid (maze is a Grid)
 // This checks if maze[chosen] is an open corridor
 if (maze[chosen]) // chosen was set to {3, 4} so this accesses maze[3][4]
 ...

 // You can directly compare two GridLocations
 if (chosen == exit)
 ...

 // You can also individually access the GridLocation's fields for row and col
 if (chosen.row == 0 && chosen.col == 0) 
 ...

A Vector of GridLocations is a path

A path through the maze is an ordered sequence of connecting GridLocations. A Vector<GridLocation> is just the tool for the job! A valid maze solution path starts at the maze entrance, steps through connecting GridLocations, and ends at the maze exit.

In the "Other files/res" folder, there are a few .soln text files that contain solutions for some of our maze files. Here is the solution file for the 5x7 maze shown earlier. The solution path is a sequence of 11 GridLocations, starting at r0c0 (row 0, col 0), and ending at r4c6.

{r0c0, r0c1, r0c2, r0c3, r0c4, r0c5, r0c6, r1c6, r2c6, r3c6, r4c6}

Our starter code provides the function readSolutionFile that reads a solution path into a Vector<GridLocation>. Not every maze will have a corresponding .soln file, and part of your job will be to write code to generate solutions!

Resources for the Stanford ADTs

• Documentation for Stanford collections: Grid, GridLocation, Stack, Queue, Vector
• Section 6.1 of the textbook introduces struct types
• In the lower-left corner of your Qt Creator window, there is a magnifying glass by a search field. If you type the name of a header file such as grid.h or gridlocation.h , Qt will display the corresponding header file.
• A few reminders for using our ADT/templates:
  • The assignment operator for our ADTs makes a deep copy. Assigning from one Vector/Set/Stack to another creates a copy of the ADT and a copy of all its elements.
  • When using C++ template, you have to indicate the particular element type you will store. Using Vector without specifying the element type just won't fly, and a Vector<int> is not the same thing as a Vector<string>. The error messages you receive when you have mismatches in template types can be cryptic and hard to interpret. Bring your template woes to LaIR or the Ed forum, and we can help untangle them with you.

On to the code!

In maze.cpp, you will write two helper functions that process Grids and paths of GridLocations, along with two maze-solving functions. You will also be writing comprehensive test cases.

1) Write helper function generateValidMoves()

Your first task is to implement the helper function to generate the neighbors for a given location:

Set<GridLocation> generateValidMoves(Grid<bool>& maze, GridLocation cur)

Given a maze represented as a Grid of bools and a current GridLocation cur, this function returns a Set of all valid moves from cur. Valid moves are those GridLocations that are:

• Exactly one "step" away from cur in one of the four cardinal directions (N, S, E, W)
• Within bounds for the Grid
• An open corridor, not a wall

There are a few provided tests for generateValidMoves, but these tests are not fully comprehensive. Write at least 3 additional tests to make sure your helper function works correctly. Remember to label your tests as STUDENT_TEST.

2) Write helper function validatePath()

Next you'll write a function to confirm that a path is a valid maze solution:

void validatePath(Grid<bool>& maze, Vector<GridLocation>& path)

The image above displays a valid solution; each colored dot marks a GridLocation along the path. A path is a valid solution to a maze if it meets the following criteria:

• The path is not empty
• The path starts at the entry (upper left corner) of the maze
• The path ends at the exit (lower right corner) of the maze
• Each location in the path is a valid move from the previous path location
  • Hint: rather than re-implement the same logic you already did for generateValidMoves, simply call that function and check whether a move is contained in the set of valid moves.
• The path must not contain a loop, i.e. the same location cannot appear more than once in the path
  • Hint: a Set<GridLocation> is a good data structure for tracking seen locations and avoiding a repeat.

The function validatePath completes successfully if all of the criteria for a valid solution are met. If it instead detects that the path violates one of the above constraints, validatePath should call the error function from error.h to report what is wrong. The error function stops execution and prints your explanatory message:

error("Here is my message about what has gone wrong");

The function validatePath will require careful testing to confirm that it correctly detects all forms of valid and invalid paths. We've provided a few tests to get you started, but you will need additional tests of your own to complete the job. Write at least 3 student test cases for validatePath.

Note the use of the new test types EXPECT_ERROR and EXPECT_NO_ERROR in our provided tests. An EXPECT_ERROR test case evaluates an expression, expecting that the operation will raise an error. While executing the test, the SimpleTest framework will "catch" the error, record that it happened, and resume. Because the error was expected, the test case is marked Correct. If an error was expected but didn't materialize, the test case is marked Incorrect. EXPECT_NO_ERROR operates in the reverse: it expects the code to run without errors and if so, it is Correct. If an error is raised, it is marked Incorrect. More information on the different test macros can be found in the CS106B Testing Guide.

After writing validatePath, not only will you be familiar with using the ADTs to represent mazes, now you have a function to help when testing the functions you will write next. Having thoroughly tested your validatePath on a variety of invalid paths means you can be confident that it is the oracle of truth when it comes to confirming a solution. Your future self thanks you!

3) Write solveMazeBFS()

Now you're ready to tackle a function that searches the maze for a solution path:

Vector<GridLocation> solveMazeBFS(Grid<bool>& maze)

Solving a maze can be seen as a specific instance of a path-finding problem, where the challenge is to find a route from the entrance to the exit. Path-finding comes up in a variety of situations such as packet routing on the internet, robot motion planning, analyzing gene mutations, spelling correction, and more.

Breadth-first search (BFS) is a classic and elegant algorithm for finding a path. A breadth-first search considers paths outward from the entry location in a radial fashion until it finds the exit. The first paths examined take one hop from the entry. If any of these reach the exit location, success! If not, the search expands to those paths that are two hops long. At each subsequent step, the search expands radially, examining all paths of length three, then of length four, and so on, stopping at the first path that reaches the exit.

Breadth-first search is typically implemented using a queue. The queue stores partial paths that represent possibilities to explore. The first paths enqueued are all length one, followed by enqueuing the length two paths, and so on. Because a queue processes elements in FIFO order, all shorter paths are dequeued and processed before the longer paths make their way to the front of queue. This means paths are tried in order of increasing length and thus a solution path found via breadth-first search will be the shortest possible such solution.

At each step, the algorithm considers the current path at the front of the queue. If the current path ends at the exit, it must be a completed solution path. If not, the algorithm considers the current path, extends it to reach locations that are one hop further away in the maze, and enqueues those extended paths to be examined in later rounds.

Here are the steps followed by a breadth-first search:

1. Create an empty Queue of paths. Each path is a Vector<GridLocation> and the Queue of paths is of type Queue<Vector<GridLocation>>.
   • A nested ADT type like this looks a little scary at first, but it is just the right tool for this job!
2. Create a length-one path containing just the entry location. Enqueue that path.
   • In our mazes, the entry is always the location in the upper-left corner, and the exit is the lower-right.
3. While there are still more paths to explore:
   • Dequeue the current path from queue.
   • If the current path path ends at exit:
     • You're done. The current path is the solution!
   • Otherwise:
     • Determine the viable neighbors from the end location of the current path. A viable neighbor is a valid move that has not yet been visited during the search
     • For each viable neighbor, make copy of current path, extend by adding neighbor and enqueue it.

A couple of notes to keep in mind as you're implementing BFS:

• You may make the following assumptions:
  • The maze is well-formed, rectangular, contains at least two rows and two columns, and the grid locations at upper-left (enter) and lower-right (exit) are both open corridors.
  • The maze has a solution.
• The search should not revisit previously visited locations or create a path with a cycle. For example, if the current path leads from location r0c0 to r1c0, you should not extend the path by moving back to location r0c0.
• You should not call validatePath within your solveMazeBFS function, but you can call it in your test cases to confirm the validity of paths found by solveMazeBFS.
• For a helpful visualization and debugging aid, try adding a graphical animation using our provided mazegraphics.

We have provided some tests to get your started. Add 2 or more student tests that further verify the correct functionality of the solveMazeBFS function. There are additional maze files in the res folder and you can also create your own files.

4) Write solveMazeDFS()

Your final task is to implement depth-first search:

Vector<GridLocation> solveMazeDFS(Grid<bool>& maze)

Depth-first search is an alternate path-finding algorithm that works similarly to breadth-first search, but tries the possible paths in a different order. Once a depth-first search starts on a path, it goes deep, continuing until it reaches its ultimate end and only if unsuccessful, moves on to try other paths. Both algorithms consider all possible paths in search of a solution, the difference being in which order they consider the paths.

The power of ADTs makes it magically easy to convert between the two algorithms. Depth-first search uses the same steps for a breadth-first search but everywhere you see Queue with enqueue/dequeue operations, substitute a Stack with push/pop operations. Whereas BFS uses a Queue of paths, enqueuing paths to the back and dequeuing the next path to explore from the front, a DFS instead uses a Stack and pushes paths to the top and pops the next path to explore from the top. Using a Queue considers paths in FIFO (first-in-first-out) order, but a Stack will process the paths in LIFO (last-in-first-out) order.

This means that implementing solveMazeDFS is going to be a snap - just a switcheroo in which ADTs are being used. For this function:

• copy/paste your working implementation from solveMazeBFS
• change the data type of allPaths from Queue<Vector<GridLocation>> to Stack<Vector<GridLocation>>
• change Queue operations (enqueue/dequeue) into Stack operations (push/pop)

Voila, you've done it! Add 2 tests to confirm the solutions found by solveMazeDFS are valid according to validatePath. You now have two powerful algorithms to solve a maze and have gotten lots of practice with ADTs – way to go!

Answer the following questions in about 3 sentences each. This isn't a hard requirement, just a ballpark :)

Use graphics functions to animate solving the maze (optional)

The Stanford C++ library includes components for drawing and user interaction, but we won't ask you to dig into those features. Instead, we will supply any needed graphics routines pre-written in a simple, easy-to-use form. For this project, the provided mazegraphics module has function to display a maze and highlight a path. If you're interested, the mazegraphics.h header file for more details about:

• drawMaze(Grid<bool>& grid)
• highlightPath(Vector<GridLocation>& path, string color, int mSecsToPause = 0)

To visualize the maze solving process in either of your solve functions, call drawMaze once to draw the maze and then repeatedly call highlightPath to mark the locations on the path currently being explored. If desired, supply the optional third argument of highlightPath to pause the animation after each step to slow things down. This gives you time to watch the animation and see how the path is updated and follow along with the algorithm and debug its operation.

Now sit back and watch the show as your clever algorithms find the path to escape any maze you throw at it! 👏 Congratulations! 👏

References

There are many interesting facets to mazes and much fascinating mathematics underlying them. Mazes will come up again several times this quarter. Chapter 9 of the textbook uses a recursive depth-first search as path-finding algorithm. At the end of the quarter when we talk about graphs, we'll explore the equivalence between mazes and graphs and note how many interesting results in mazes came from breakthroughs in graph algorithms.

• Walter Pullen, Maze Classification. http://www.astrolog.org/labyrnth/algrithm.htm Website with lots of great info on mazes and maze algorithms
• Jamis Buck. Maze Algorithms. https://www.jamisbuck.org/mazes/ Fun animations of maze algorithms. He also wrote the excellent book about Mazes for Programmers: Code Your Own Twisty Little Passages.

Extensions

If you have completed the assignment and want to explore further, here are some ideas for extensions.

• Instead of reading pre-written mazes from a file, you could instead generate a new random maze on demand. There are an amazeing (I could not resist…) variety of algorithms for maze construction, ranging from the simple to the sublime. Here are a few names to get you started: backtracking, depth-first, growing tree, sidewinder, along with algorithms named for their inventor: Aldous-Broder, Eller, Prim, Kruskal, Wilson, and many others.
• Try out other maze solving algorithms. How does BFS and DFS measure up against the approaches in terms of solving power or runtime efficiency? Take a look at other solvers such as random mouse, wall following, recursive depth-first, Bellman-Ford, or others.
• There are many other neat maze-based games out there that make fun extensions. You might gather ideas from Robert Abbott's http://www.logicmazes.com or design a maze game board after inspiration from these musings on mazes from Chris Smith.

The last word from XKCD https://xkcd.com/2407/