Maze Generation

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.

Algorithm Visualization

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.

References

Wikipedia

Implementations

1
# TODO: Implement in Python

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const n = 6; // rows (change these!)
const m = 6; // columns (change these!)

const hEnd = m * 4 - (m - 1);
const vEnd = n * 3 - (n - 1);

const G = [];

for (let i = 0; i < vEnd; i++) { // by row
  G[i] = new Array(hEnd);
  for (let j = 0; j < hEnd; j++) { // by column
    G[i][j] = ' ';

    if (i === 0 && j === 0) { // top-left corner
      G[i][j] = '┌';
    } else if (i === 0 && j === hEnd - 1) { // top-right corner
      G[i][j] = '┐';
    } else if (i === vEnd - 1 && j === 0) { // bottom-left corner
      G[i][j] = '└';
    } else if (i === vEnd - 1 && j === hEnd - 1) { // bottom-right corner
      G[i][j] = '┘';
    } else if ((j % 3 === 0) && (i % vEnd !== 0 && i !== vEnd - 1 && i % 2 === 1)) {
      G[i][j] = '│';
    } else if (i % 2 === 0) {
      G[i][j] = '─';
    }

    if (m > 1) { // More than one column
      if (j % 3 === 0 && j !== 0 && j !== hEnd - 1 && i === 0) {
        G[i][j] = '┬';
      }
      if (j % 3 === 0 && j !== 0 && j !== hEnd - 1 && i === vEnd - 1) {
        G[i][j] = '┴';
      }
    }

    if (n > 1) { // More than one row
      if (i % 2 === 0 && i !== 0 && i !== vEnd - 1 && j === 0) {
        G[i][j] = '├';
      }
      if (i % 2 === 0 && i !== 0 && i !== vEnd - 1 && j === hEnd - 1) {
        G[i][j] = '┤';
      }
    }

    if (n > 1 && m > 1) { // More than one row and column
      if (i % 2 === 0 && j % 3 === 0 && i !== 0 && j !== 0 && i !== vEnd - 1 && j !== hEnd - 1) {
        G[i][j] = '┼';
      }
    }
  }
}

function buildMaze() {
  const mySet = new disjointSet();
  const width = m;
  const height = n;
  let setSize = 0;
  const graph = [];
  const visitedMap = {};
  const walls = {};
  const rightWalls = [];
  const downWalls = [];
  let location = 0;

  mySet.addElements(width * height);

  console.log('initializing grid (all walls are up)');
  
  // init 'graph'
  // each room has two walls, a down and right wall.
  for (let i = 0; i < width; i++) {
    graph[i] = new Array(height);
    for (let j = 0; j < height; j++) {
      graph[i][j] = location;

      walls[location] = { down: true, right: true };
      visitedMap[location] = false;

      // If you can label the rooms with just 2 digits
      if (width * height < 100) {
        const locationString = location.toString();

        G[j * 2 + 1][i * 3 + 1] = locationString[0];
        G[j * 2 + 1][i * 3 + 2] = locationString[1];

      }

      rightWalls.push({ x: i, y: j });
      downWalls.push({ x: i, y: j });
      location++;
    }
  }

  console.log('shuffled the walls for random selection');
  
  // Randomly shuffle the walls
  shuffle(rightWalls);
  shuffle(downWalls);

  // Picking random walls to remove
  while (setSize !== mySet.elements - 1) {
    const randomWall = Math.floor((Math.random() * 2) + 1);
    if (randomWall === 1 && downWalls.length > 0) {
      // Down wall
      const currentRoom = downWalls.pop();
      const iX = currentRoom.x;
      const iY = currentRoom.y;
      const iYdown = iY + 1;
      if (iYdown < height) {
        const u = graph[iX][iY];
        const v = graph[iX][iYdown];
        
        if (mySet.find(u) !== mySet.find(v)) {
          
          console.log(`Rooms: ${u} & ${v} now belong to the same set, delete wall between them`);

          Tracer.delay();
          
          mySet.setUnion(u, v);
          setSize++;
          // delete wall
          walls[u].down = false;
        } else {
          
          console.log(`Rooms: ${u} & ${v} would create a cycle! This is not good!`);
          Tracer.delay();
          
        }
        
      }
    } else if (randomWall === 2 && rightWalls.length > 0) {
      // Right Wall
      const currentRoom = rightWalls.pop();
      const iX = currentRoom.x;
      const iY = currentRoom.y;
      const iXright = iX + 1;
      if (iXright < width) {
        const u = graph[iX][iY];
        const v = graph[iXright][iY];
        
        if (mySet.find(u) !== mySet.find(v)) {
          
          console.log(`Rooms: ${u} & ${v} now belong to the same set, delete wall between them`);

          Tracer.delay();
          
          mySet.setUnion(u, v);
          setSize++;
          // delete wall
          walls[u].right = false;
        } else {
          
          console.log(`Rooms: ${u} & ${v} would create a cycle! This is not good!`);
          Tracer.delay();
          
        }
        
      }
    }
  }

  console.log('deleting the walls');
  
  // update deleted walls
  for (let i = 0; i < width; i++) {
    for (let j = 0; j < height; j++) {
      const currentWall = walls[graph[i][j]];

      if (currentWall.down === false) {
        G[j * 2 + 2][i * 3 + 1] = ' ';
        G[j * 2 + 2][i * 3 + 2] = ' ';
        
      }

      if (currentWall.right === false) {
        G[j * 2 + 1][i * 3 + 3] = ' ';
        
      }
      
    }
  }
  
  console.log('cleaning up the grid!');
  
  cleanUpGrid(width, height);

  // Clear out walls for the start and end locations.
  const randomStart = Math.floor(Math.random() * width);
  const randomEnd = Math.floor(Math.random() * width);

  console.log('setting the Start (S) & End (E) locations');

  // Start Location
  G[0][randomStart * 3 + 1] = ' ';
  G[0][randomStart * 3 + 2] = ' ';
  G[1][randomStart * 3 + 1] = 'S';

  // End Location
  G[vEnd - 1][randomEnd * 3 + 1] = ' ';
  G[vEnd - 1][randomEnd * 3 + 2] = ' ';
  G[vEnd - 2][randomEnd * 3 + 1] = 'E';

  cleanUpStartLocation(randomStart);
  cleanUpEndLocation(randomEnd);

  console.log('maze is completed!');

  // set the data
  
}

function cleanUpStartLocation(start) {
  if (G[0][start * 3] === '┬' && G[1][start * 3] === '│') {
    G[0][start * 3] = '┐';
  }
  if (G[0][start * 3 + 3] === '┬' && G[1][start * 3 + 3] === '│') {
    G[0][start * 3 + 3] = '┌';
  }
  if (G[0][start * 3] === '┌') {
    G[0][start * 3] = '│';
  }
  if (G[0][start * 3 + 3] === '┐') {
    G[0][start * 3 + 3] = '│';
  }
}

function cleanUpEndLocation(end) {
  if (G[vEnd - 1][end * 3] === '┴' && G[vEnd - 2][end * 3] === '│') {
    G[vEnd - 1][end * 3] = '┘';
  }
  if (G[vEnd - 1][end * 3 + 3] === '┴' && G[vEnd - 2][end * 3 + 3] === '│') {
    G[vEnd - 1][end * 3 + 3] = '└';
  }
  if (G[vEnd - 1][end * 3] === '└') {
    G[vEnd - 1][end * 3] = '│';
  }
  if (G[vEnd - 1][end * 3 + 3] === '┘') {
    G[vEnd - 1][end * 3 + 3] = '│';
  }
}

function cleanUpGrid(width, height) {
  // Remove room numbers
  for (let i = 0; i < width; i++) {
    for (let j = 0; j < height; j++) {
      G[j * 2 + 1][i * 3 + 1] = ' ';
      G[j * 2 + 1][i * 3 + 2] = ' ';
    }
  }

  // clean up grid for looks
  for (let i = 0; i < vEnd; i++) {
    for (let j = 0; j < hEnd; j++) {
      if (G[i][j] === '├') {
        if (G[i][j + 1] === ' ') {
          G[i][j] = '│';
        }
      }

      if (G[i][j] === '┤') {
        if (G[i][j - 1] === ' ') {
          G[i][j] = '│';
        }
      }

      if (G[i][j] === '┬') {
        if (G[i + 1][j] === ' ') {
          G[i][j] = '─';
        }
      }

      if (G[i][j] === '┴') {
        if (G[i - 1][j] === ' ') {
          G[i][j] = '─';
        }
      }

      if (G[i][j] === '┼') {
        if (G[i][j + 1] === ' ' && G[i - 1][j] === ' ' && G[i][j - 1] !== ' ' && G[i + 1][j] !== ' ') {
          G[i][j] = '┐';
        } else if (G[i][j - 1] === ' ' && G[i - 1][j] === ' ' && G[i + 1][j] !== ' ' && G[i][j + 1] !== ' ') {
          G[i][j] = '┌';
        } else if (G[i][j - 1] === ' ' && G[i + 1][j] === ' ' && G[i - 1][j] !== ' ' && G[i][j + 1] !== ' ') {
          G[i][j] = '└';
        } else if (G[i][j + 1] === ' ' && G[i + 1][j] === ' ' && G[i - 1][j] !== ' ' && G[i][j - 1] !== ' ') {
          G[i][j] = '┘';
        } else if (G[i][j + 1] === ' ' && G[i][j - 1] === ' ' && (G[i + 1][j] === ' ' || G[i - 1][j] === ' ')) {
          G[i][j] = '│';
        } else if (G[i + 1][j] === ' ' && G[i - 1][j] === ' ' && (G[i][j - 1] === ' ' || G[i][j + 1] === ' ')) {
          G[i][j] = '─';
        } else if (G[i][j + 1] === ' ' && G[i][j - 1] === ' ') {
          G[i][j] = '│';
        } else if (G[i + 1][j] === ' ' && G[i - 1][j] === ' ') {
          G[i][j] = '─';
        } else if (G[i + 1][j] === ' ' && G[i - 1][j] !== ' ' && G[i][j - 1] !== ' ' && G[i][j + 1] !== ' ') {
          G[i][j] = '┴';
        } else if (G[i - 1][j] === ' ' && G[i + 1][j] !== ' ' && G[i][j + 1] !== ' ' && G[i][j - 1] !== ' ') {
          G[i][j] = '┬';
        } else if (G[i][j + 1] === ' ' && G[i - 1][j] !== ' ' && G[i + 1][j] !== ' ' && G[i][j - 1] !== ' ') {
          G[i][j] = '┤';
        } else if (G[i][j - 1] === ' ' && G[i - 1][j] !== ' ' && G[i + 1][j] !== ' ' && G[i][j + 1] !== ' ') {
          G[i][j] = '├';
        }
      }
    }
  }
}

class disjointSet {
  constructor() {
    this.set = [];
    this.elements = 0;
  }

  addElements(numberOfElements) {
    for (let i = 0; i < numberOfElements; i++) {
      this.elements++;
      this.set.push(-1);
    }
  }

  find(element) {
    if (this.set[element] < 0) {
      return element;
    }
    return this.set[element] = this.find(this.set[element]);
  }

  setUnion(_a, _b) {
    const a = this.find(_a);
    const b = this.find(_b);

    if (a !== b) {
      const newSize = (this.set[a] + this.set[b]);
      if (this.compareSize(a, b)) {
        this.set[b] = a;
        this.set[a] = newSize;
      } else {
        this.set[a] = b;
        this.set[b] = newSize;
      }
    }
  }

  compareSize(a, b) {
    if (this.set[a] === this.set[b]) {
      return true;
    } else if (this.set[a] < this.set[b]) {
      return true;
    }
    return false;
  }
}

// http://bost.ocks.org/mike/shuffle/
function shuffle(array) {
  let m = array.length;
  let t;
  let i;
  // While there remain elements to shuffle…
  while (m) {
    // Pick a remaining element…
    i = Math.floor(Math.random() * m--);
    // And swap it with the current element.
    t = array[m];
    array[m] = array[i];
    array[i] = t;
  }
  return array;
}

buildMaze();