Maze generation algorithms are automated methods for the creation of mazes.
Graph theory based methods
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, 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
Play mediaThis 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.
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
Play mediaThe depth-first search algorithm of maze generation is frequently implemented using backtracking. This can be described with a following recursive routine:
- Given a current cell as a parameter,
- Mark the current cell as visited
- While the current cell has any unvisited neighbour cells
- Choose one of the unvisited neighbours
- Remove the wall between the current cell and the chosen cell
- Invoke the routine recursively for a chosen cell
which is invoked once for any initial cell in the area.
Iterative implementation
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.
- Choose the initial cell, mark it as visited and push it to the stack
- While the stack is not empty
- Pop a cell from the stack and make it a current cell
- If the current cell has any neighbours which have not been visited
- Push the current cell to the stack
- Choose one of the unvisited neighbours
- Remove the wall between the current cell and the chosen cell
- Mark the chosen cell as visited and push it to the stack
Randomized Kruskal's algorithm
Play mediaThis algorithm is a randomized version of Kruskal's algorithm.
- Create a list of all walls, and create a set for each cell, each containing just that one cell.
- For each wall, in some random order:
- If the cells divided by this wall belong to distinct sets:
- Remove the current wall.
- Join the sets of the formerly divided cells.
- If the cells divided by this wall belong to distinct sets:
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, time; for any plausible value of ), 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.
Randomized Prim's algorithm
Play media