Maze solving algorithm

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There are a number of different maze solving algorithms, that is, automated methods for the solving of mazes. The random mouse, wall follower, Pledge, and Trémaux algorithms are designed to be used inside the maze by a traveler with no prior knowledge of the maze, whereas the dead-end filling and shortest path algorithms are designed to be used by a person or computer program that can see the whole maze at once.

Mazes containing no loops are known as "standard", or "perfect" mazes, and are equivalent to a tree in graph theory. Thus many maze solving algorithms are closely related to graph theory. Intuitively, if one pulled and stretched out the paths in the maze in the proper way, the result could be made to resemble a tree.[1]

Random mouse algorithm[edit]

This is a trivial method that can be implemented by a very unintelligent robot or perhaps a mouse. It is simply to proceed in a straight line until a junction is reached, and then to make a random decision about the next direction to follow. Although such a method would always eventually find the right solution, this algorithm can be extremely slow.

Wall follower[edit]

Traversal using Right-hand rule
Maze with two solutions.
Solution to above maze. Notice the solution is the boundary between the connected components of the wall of the maze, each represented by a different colour.

The wall follower, the best-known rule for traversing mazes, is also known as either the left-hand rule or the right-hand rule. If the maze is simply connected, that is, all its walls are connected together or to the maze's outer boundary, then by keeping one hand in contact with one wall of the maze the player is guaranteed not to get lost and will reach a different exit if there is one; otherwise, he or she will return to the entrance having traversed every corridor in the maze at least once.

Another perspective into why wall following works is topological. If the walls are connected, then they may be deformed into a loop or circle.[2] Then wall following reduces to walking around a circle from start to finish. To further this idea, notice that by grouping together connected components of the maze walls, the boundaries between these are precisely the solutions, even if there is more than one solution (see figures on the right).

If the maze is not simply connected (i.e. if the start or endpoints are in the center of the structure or the pathways cross over and under each other), this method will not be guaranteed to help the goal to be reached.

Wall-following can be done in 3D or higher-dimensional mazes if its higher-dimensional passages can be projected onto the 2D plane in a deterministic manner. For example, if in a 3D maze "up" passages can be assumed to lead northwest, and "down" passages can be assumed to lead southeast, then standard wall following rules can then be applied. However, unlike in 2D, this requires that the current orientation be known, to determine which direction is the first on the left or right.

Pledge algorithm[edit]

Robot in a wooden maze

Disjoint mazes can still be solved with the wall follower method, if the entrance and exit to the maze are on the outer walls of the maze. If however, the solver starts inside the maze, it might be on a section disjoint from the exit, and wall followers will continually go around their ring. The Pledge algorithm (named after Jon Pledge of Exeter) can solve this problem.[3]

The Pledge algorithm, designed to circumvent obstacles, requires an arbitrarily chosen direction to go toward. When an obstacle is met, one hand (say the right hand) is kept along the obstacle while the angles turned are counted. When the solver is facing the original direction again, and the angular sum of the turns made is 0, the solver leaves the obstacle and continues moving in its original direction.

The hand is removed from the wall only when both "sum of turns made" and "current heading" are at zero. This allows the algorithm to avoid traps shaped like an upper case letter "G". Assuming the algorithm turns left at the first wall, one gets turned around a full 360 degrees by the walls. An algorithm that only keeps track of "current heading" leads into an infinite loop as it leaves the lower rightmost wall heading left and runs into the curved section on the left hand side again. The Pledge algorithm does not leave the rightmost wall due to the "sum of turns made" not being zero at that point. It follows the wall all the way around, finally leaving it heading left outside and just underneath the letter shape.

This algorithm allows a person with a compass to find her way from any point inside to an outer exit of any finite and fair two-dimensional maze, regardless of the initial position of the solver. However, this algorithm will not work in doing the reverse, namely finding the way from an entrance on the outside of a maze to some end goal within it.

Trémaux's algorithm[edit]

Trémaux's algorithm, invented by Charles Pierre Trémaux,[4] is an efficient method to find the way out of a maze that requires drawing lines on the floor to mark a path, and is guaranteed to work for all mazes that have well-defined passages.[5] A path is either unvisited, marked once or marked twice. Every time a direction is chosen it is marked by drawing a line on the floor (from junction to junction). In the beginning a random direction is chosen (if there is more than one). On arriving at a junction that has not been visited before (no other marks), pick a random direction (and mark the path). When arriving at a marked junction and if your current path is marked only once then turn around and walk back (and mark the path a second time). If this is not the case, pick the direction with the fewest marks (and mark it, as always).. When you finally reach the solution, paths marked exactly once will indicate a direct way back to the start. If there is no exit, this method will take you back to the start where all paths are marked twice. In this case each path is walked down exactly twice, once in each direction. The resulting walk is called a bidirectional double-tracing.[6]

Essentially, this algorithm, which was discovered in the 19th century, has been used about hundred years later as depth-first search.[7][8]

Dead-end filling[edit]

Dead-end filling is an algorithm for solving mazes that fills all dead ends, leaving only the correct way unfilled. It can be used for solving mazes on paper or with a computer program, but it is not useful to a person inside an unknown maze since this method looks at the entire maze at once. The method is to 1) find all of the dead-ends in the maze, and then 2) "fill in" the path from each dead-end until the first junction met. A video of dead-end filling in action can be seen here: [1][2].

Dead-end filling cannot accidentally "cut off" the start from the finish since each step of the process preserves the topology of the maze. Furthermore, the process won't stop "too soon" since the end result cannot contain any dead-ends. Thus if dead-end filling is done on a perfect maze (maze with no loops), then only the solution will remain. If it is done on a partially braid maze (maze with some loops), then every possible solution will remain but nothing more. [3]

Recursive algorithm[edit]

If given an omniscient view of the maze, you are able to create a very simple recursive algorithm that is able to tell you how to get to the end. The algorithm will be given a starting X and Y value. If the X and Y values are not on a wall, the method will call itself with all adjacent X and Y values, making sure that it did not already use those X and Y values before. If the X and Y values are those of the end location, it will save all the previous instances of the method as the correct path. Here is a sample code in Java:

int[][] maze = new int[width][height]; // The maze
boolean[][] wasHere = new boolean[width][height];
boolean[][] correctPath = new boolean[width][height]; // The solution to the maze
int startX, startY; // Starting X and Y values of maze
int endX, endY;     // Ending X and Y values of maze
 
public void solveMaze() {
    maze = generateMaze(); // Create Maze (1 = path, 2 = wall)
    for (int row = 0; row < maze.length; row++)  
        // Sets boolean Arrays to default values
        for (int col = 0; col < maze[row].length; col++){
            wasHere[row][col] = false;
            correctPath[row][col] = false;
        }
    boolean b = recursiveSolve(startX, startY);
    // Will leave you with a boolean array (correctPath) 
    // with the path indicated by true values.
    // If b is false, there is no solution to the maze
}
public boolean recursiveSolve(int x, int y) {
    if (x == endX && y == endY) return true; // If you reached the end
    if (maze[x][y] == 2 || wasHere[x][y]) return false;  
    // If you are on a wall or already were here
    wasHere[x][y] = true;
    if (x != 0) // Checks if not on left edge
        if (recursiveSolve(x-1, y)) { // Recalls method one to the left
            correctPath[x][y] = true; // Sets that path value to true;
            return true;
        }
    if (x != width - 1) // Checks if not on right edge
        if (recursiveSolve(x+1, y)) { // Recalls method one to the right
            correctPath[x][y] = true;
            return true;
        }
    if (y != 0)  // Checks if not on top edge
        if (recursiveSolve(x, y-1)) { // Recalls method one up
            correctPath[x][y] = true;
            return true;
        }
    if (y != height- 1) // Checks if not on bottom edge
        if (recursiveSolve(x, y+1)) { // Recalls method one down
            correctPath[x][y] = true;
            return true;
        }
    return false;
}

Shortest path algorithm[edit]

A maze with many solutions and no dead-ends, where it may be useful to find the shortest path

When a maze has multiple solutions, the solver may want to find the shortest path from start to finish. There are several algorithms to find shortest paths, most of them coming from graph theory. One possible algorithm finds the shortest path by implementing a breadth-first search, while another, the A* algorithm, uses a heuristic technique. The breadth-first search algorithm uses a queue to visit cells in increasing distance order from the start until the finish is reached. Each visited cell needs to keep track of its distance from the start or which adjacent cell nearer to the start caused it to be added to the queue. When the finish location is found, follow the path of cells backwards to the start, which is the shortest path. The breadth-first search in its simplest form has its limitations, like finding the shortest path in weighted graphs.

See also[edit]

References[edit]

  1. ^ Maze to Tree on YouTube
  2. ^ Maze Transformed on YouTube
  3. ^ Abelson; diSessa (1980), Turtle Geometry: the computer as a medium for exploring mathematics 
  4. ^ Public conference, December 2, 2010 - by professor Jean Pelletier-Thibert in Academie de Macon (Burgundy - France) - (Abstract published in the Annals academic, March 2011 - ISSN: 0980-6032)
    Charles Tremaux (° 1859 - † 1882) Ecole Polytechnique of Paris (X:1876), French engineer of the telegraph
  5. ^ Édouard Lucas: Récréations Mathématiques Volume I, 1882.
  6. ^ Even, Shimon (2011), Graph Algorithms (2nd ed.), Cambridge University Press, pp. 46–48, ISBN 978-0-521-73653-4 .
  7. ^ Sedgewick, Robert (2002), Algorithms in C++: Graph Algorithms (3rd ed.), Pearson Education, ISBN 978-0-201-36118-6 .

External links[edit]