Maze generation algorithm

Maze generation algorithms are automated methods for the creation of mazes.

This maze was generated by a modified version of Prim's algorithm, below.

Graph theory based methods

Animation of graph theory based method (randomized depth-first search)

A maze can be generated by starting with a predetermined arrangement of cells (most commonly a rectangular grid but other arrangements are possible) with wall sites between them. This predetermined arrangement can be considered as a connected graph with the edges representing possible wall sites and the nodes representing cells. The purpose of the maze generation algorithm can then be considered to be making a subgraph in which it is challenging to find a route between two particular nodes.

If the subgraph is not connected, then there are regions of the graph that are wasted because they do not contribute to the search space. If the graph contains loops, then there may be multiple paths between the chosen nodes. Because of this, maze generation is often approached as generating a random spanning tree. Loops, which can confound naive maze solvers, may be introduced by adding random edges to the result during the course of the algorithm.

The animation shows the maze generation steps for a graph that is not on a rectangular grid. First, the computer creates a random planar graph G shown in blue, and its dual F shown in yellow. Second, the computer traverses F using a chosen algorithm, such as a depth-first search, coloring the path red. During the traversal, whenever a red edge crosses over a blue edge, the blue edge is removed. Finally, when all vertices of F have been visited, F is erased and two edges from G, one for the entrance and one for the exit, are removed.

Randomized depth-first search

Animation of generator using depth-first search

A different animation of a generator using depth-first search

This algorithm, also known as the "recursive backtracker" algorithm, is a randomized version of the depth-first search algorithm.

Frequently implemented with a stack, this approach is one of the simplest ways to generate a maze using a computer. Consider the space for a maze being a large grid of cells (like a large chess board), each cell starting with four walls. Starting from a random cell, the computer then selects a random neighbouring cell that has not yet been visited. The computer removes the wall between the two cells and marks the new cell as visited, and adds it to the stack to facilitate backtracking. The computer continues this process, with a cell that has no unvisited neighbours being considered a dead-end. When at a dead-end it backtracks through the path until it reaches a cell with an unvisited neighbour, continuing the path generation by visiting this new, unvisited cell (creating a new junction). This process continues until every cell has been visited, causing the computer to backtrack all the way back to the beginning cell. We can be sure every cell is visited.

As given above this algorithm involves deep recursion which may cause stack overflow issues on some computer architectures. The algorithm can be rearranged into a loop by storing backtracking information in the maze itself. This also provides a quick way to display a solution, by starting at any given point and backtracking to the beginning.

Horizontal Passage Bias

Mazes generated with a depth-first search have a low branching factor and contain many long corridors, because the algorithm explores as far as possible along each branch before backtracking.

Recursive implementation

Randomized depth-first search on a hexagonal grid

The depth-first search algorithm of maze generation is frequently implemented using backtracking. This can be described with a following recursive routine:

1. Given a current cell as a parameter
2. Mark the current cell as visited
3. While the current cell has any unvisited neighbour cells
   1. Choose one of the unvisited neighbours
   2. Remove the wall between the current cell and the chosen cell
   3. Invoke the routine recursively for the chosen cell

which is invoked once for any initial cell in the area.

Iterative implementation (with stack)

A disadvantage of the first approach is a large depth of recursion – in the worst case, the routine may need to recur on every cell of the area being processed, which may exceed the maximum recursion stack depth in many environments. As a solution, the same backtracking method can be implemented with an explicit stack, which is usually allowed to grow much bigger with no harm.

1. Choose the initial cell, mark it as visited and push it to the stack
2. While the stack is not empty
   1. Pop a cell from the stack and make it a current cell
   2. If the current cell has any neighbours which have not been visited
      1. Push the current cell to the stack
      2. Choose one of the unvisited neighbours
      3. Remove the wall between the current cell and the chosen cell
      4. Mark the chosen cell as visited and push it to the stack

Iterative randomized Kruskal's algorithm (with sets)

An animation of generating a 30 by 20 maze using Kruskal's algorithm

This algorithm is a randomized version of Kruskal's algorithm.

1. Create a list of all walls, and create a set for each cell, each containing just that one cell.
2. For each wall, in some random order:
   1. If the cells divided by this wall belong to distinct sets:
      1. Remove the current wall.
      2. Join the sets of the formerly divided cells.

There are several data structures that can be used to model the sets of cells. An efficient implementation using a disjoint-set data structure can perform each union and find operation on two sets in nearly constant amortized time (specifically, ${\displaystyle O(\alpha (V))}$ time; ${\displaystyle \alpha (x)<5}$ for any plausible value of ${\displaystyle x}$), so the running time of this algorithm is essentially proportional to the number of walls available to the maze.

It matters little whether the list of walls is initially randomized or if a wall is randomly chosen from a nonrandom list, either way is just as easy to code.

Because the effect of this algorithm is to produce a minimal spanning tree from a graph with equally weighted edges, it tends to produce regular patterns which are fairly easy to solve.

Iterative randomized Prim's algorithm (without stack, without sets)

An animation of generating a 30 by 20 maze using Prim's algorithm

This algorithm is a randomized version of Prim's algorithm.

1. Start with a grid full of walls.
2. Pick a cell, mark it as part of the maze. Add the walls of the cell to the wall list.
3. While there are walls in the list:
   1. Pick a random wall from the list. If only one of the cells that the wall divides is visited, then:
      1. Make the wall a passage and mark the unvisited cell as part of the maze.
      2. Add the neighboring walls of the cell to the wall list.
   2. Remove the wall from the list.

Note that simply running classical Prim's on a graph with random edge weights would create mazes stylistically identical to Kruskal's, because they are both minimal spanning tree algorithms. Instead, this algorithm introduces stylistic variation because the edges closer to the starting point have a lower effective weight.

Modified version

Although the classical Prim's algorithm keeps a list of edges, for maze generation we could instead maintain a list of adjacent cells. If the randomly chosen cell has multiple edges that connect it to the existing maze, select one of these edges at random. This will tend to branch slightly more than the edge-based version above.

Simplified version

The algorithm can be simplified even further by randomly selecting cells that neighbour already-visited cells, rather than keeping track of the weights of all cells or edges.

It will usually be relatively easy to find the way to the starting cell, but hard to find the way anywhere else.

Wilson's algorithm

Main article: Loop-erased random walk

Maze generation animation using Wilson's algorithm (gray represents an ongoing random walk). Once built the maze is solved using depth first search.

All the above algorithms have biases of various sorts: depth-first search is biased toward long corridors, while Kruskal's/Prim's algorithms are biased toward many short dead ends. Wilson's algorithm, ^[1] on the other hand, generates an unbiased sample from the uniform distribution over all mazes, using loop-erased random walks.

We begin the algorithm by initializing the maze with one cell chosen arbitrarily. Then we start at a new cell chosen arbitrarily, and perform a random walk until we reach a cell already in the maze—however, if at any point the random walk reaches its own path, forming a loop, we erase the loop from the path before proceeding. When the path reaches the maze, we add it to the maze. Then we perform another loop-erased random walk from another arbitrary starting cell, repeating until all cells have been filled.

This procedure remains unbiased no matter which method we use to arbitrarily choose starting cells. So we could always choose the first unfilled cell in (say) left-to-right, top-to-bottom order for simplicity.

Aldous-Broder algorithm

The Aldous-Broder algorithm also produces uniform spanning trees. However, it is one of the least efficient maze algorithms. ^[2]

1. Pick a random cell as the current cell and mark it as visited.
2. While there are unvisited cells:
   1. Pick a random neighbour.
   2. If the chosen neighbour has not been visited:
      1. Remove the wall between the current cell and the chosen neighbour.
      2. Mark the chosen neighbour as visited.
   3. Make the chosen neighbour the current cell.

Recursive division method

Illustration of Recursive Division

original chamber	division by two walls	holes in walls	continue subdividing...	completed
step 1	step 2	step 3	step 4	step 5

Mazes can be created with recursive division, an algorithm which works as follows: Begin with the maze's space with no walls. Call this a chamber. Divide the chamber with a randomly positioned wall (or multiple walls) where each wall contains a randomly positioned passage opening within it. Then recursively repeat the process on the subchambers until all chambers are minimum sized. This method results in mazes with long straight walls crossing their space, making it easier to see which areas to avoid.

For example, in a rectangular maze, build at random points two walls that are perpendicular to each other. These two walls divide the large chamber into four smaller chambers separated by four walls. Choose three of the four walls at random, and open a one cell-wide hole at a random point in each of the three. Continue in this manner recursively, until every chamber has a width of one cell in either of the two directions.

Fractal Tessellation algorithm

Maze generation animation using a tessellation algorithm

This is a simple and fast way to generate a maze. ^[3]

On each iteration, this algorithm creates a maze twice the size by copying itself 3 times. At the end of each iteration, 3 paths are opened between the 4 smaller mazes.

The advantage of this method is that it is very fast. The downside is that it is not possible to get a maze of a chosen size - but various tricks can be used to get around this problem.

Simple algorithms

3D version of Prim's algorithm. Vertical layers are labeled 1 through 4 from bottom to top. Stairs up are indicated with "/"; stairs down with "\", and stairs up-and-down with "x". Source code is included with the image description.

Other algorithms exist that require only enough memory to store one line of a 2D maze or one plane of a 3D maze. Eller's algorithm prevents loops by storing which cells in the current line are connected through cells in the previous lines, and never removes walls between any two cells already connected. ^[4] The Sidewinder algorithm starts with an open passage along the entire top row, and subsequent rows consist of shorter horizontal passages with one connection to the passage above. The Sidewinder algorithm is trivial to solve from the bottom up because it has no upward dead ends. ^[5] Given a starting width, both algorithms create perfect mazes of unlimited height.

Most maze generation algorithms require maintaining relationships between cells within it, to ensure the result will be solvable. Valid simply connected mazes can however be generated by focusing on each cell independently. A binary tree maze is a standard orthogonal maze where each cell always has a passage leading up or leading left, but never both. To create a binary tree maze, for each cell flip a coin to decide whether to add a passage leading up or left. Always pick the same direction for cells on the boundary, and the result will be a valid simply connected maze that looks like a binary tree, with the upper left corner its root. As with Sidewinder, the binary tree maze has no dead ends in the directions of bias.

Maze generated in Commodore 64 BASIC, using the code 10 PRINT CHR$(205.5+RND(1)); : GOTO 10

A related form of flipping a coin for each cell is to create an image using a random mix of forward slash and backslash characters. This doesn't generate a valid simply connected maze, but rather a selection of closed loops and unicursal passages. The manual for the Commodore 64 presents a BASIC program using this algorithm, using PETSCII diagonal line graphic characters instead for a smoother graphic appearance.

Cellular automaton algorithms

Certain types of cellular automata can be used to generate mazes. ^[6] Two well-known such cellular automata, Maze and Mazectric, have rulestrings B3/S12345 and B3/S1234. ^[6] In the former, this means that cells survive from one generation to the next if they have at least one and at most five neighbours. In the latter, this means that cells survive if they have one to four neighbours. If a cell has exactly three neighbours, it is born. It is similar to Conway's Game of Life in that patterns that do not have a living cell adjacent to 1, 4, or 5 other living cells in any generation will behave identically to it. ^[6] However, for large patterns, it behaves very differently from Life. ^[6]

For a random starting pattern, these maze-generating cellular automata will evolve into complex mazes with well-defined walls outlining corridors. Mazecetric, which has the rule B3/S1234 has a tendency to generate longer and straighter corridors compared with Maze, with the rule B3/S12345. ^[6] Since these cellular automaton rules are deterministic, each maze generated is uniquely determined by its random starting pattern. This is a significant drawback since the mazes tend to be relatively predictable.

Like some of the graph-theory based methods described above, these cellular automata typically generate mazes from a single starting pattern; hence it will usually be relatively easy to find the way to the starting cell, but harder to find the way anywhere else.

See also

• Maze solving algorithm
• Self-avoiding walk
• Brute-force search

References

1. Wilson, David Bruce (May 22–24, 1996). "Generating random spanning trees more quickly than the cover time". Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing. Symposium on Theory of Computing. Philadelphia: ACM. pp. 296–303. CiteSeerX   10.1.1.47.8598 . doi:10.1145/237814.237880. ISBN   0-89791-785-5.
2. Jamis Buck (17 January 2011). "Maze Generation: Aldous-Broder algorithm".
3. Ze Oliveira (18 June 2023). "Maze Walls".
4. Jamis Buck (29 December 2010). "Maze Generation: Eller's Algorithm".
5. Jamis Buck (3 February 2011). "Maze Generation: Sidewinder Algorithm".
6. Nathaniel Johnston; et al. (21 August 2010). "Maze - LifeWiki". LifeWiki. Retrieved 1 March 2011.

External links

• Think Labyrinth: Maze algorithms (details on these and other maze generation algorithms)
• Jamis Buck: HTML 5 Presentation with Demos of Maze generation Algorithms
• Maze generation visualization
• Java implementation of Prim's algorithm
• Implementations of DFS maze creation algorithm in multiple languages at Rosetta Code
• Armin Reichert: 34 maze algorithms in Java 8, with demo application
• Coding Challenge #10.1: Maze Generator with p5.js - Part 1: Maze generation algorithm in JavaScript with p5
• Maze Generator by Charles Bond, COMPUTE! Magazine, December 1981