{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "
Peter Norvig
13 March 2018
\n", "\n", "# Maze Generation\n", "\n", "Let's make some mazes! I'm thinking of mazes like this one, which is a rectangular grid of squares, with walls on some of the sides of squares, and openings on other sides:\n", "\n", "![Wikipedia](https://upload.wikimedia.org/wikipedia/commons/thumb/8/88/Maze_simple.svg/475px-Maze_simple.svg.png)\n", "\n", "The main constraint is that there should be a path from entrance to exit, and it should be ***fun*** to solve the maze with pencil, paper, and brain power—not too easy, but also not impossible. \n", "\n", "As I think about how to model a maze on the computer, it seems like a **graph** is the right model: the nodes of\n", "the graph are the squares of the grid, and the edges of the graph are the openings between adjacent squares. So what properties of a graph make a good maze?\n", "- There must be a path from entrance to exit.\n", "- There must not be too such many paths; maybe it is best if there is only one. \n", "- Probably the graph should be *singly connected*—there shouldn't be islands of squares that are unreachable from the start. And maybe we want exactly one path between any two squares.\n", "- The path should have many twists; it would be too easy if it was mostly straight.\n", "\n", "I know that a **tree** has all these properties except the last one. So my goal has become: *Superimpose a tree over the grid, covering every square, and make sure the paths are twisty.* Here's how I'll do it:\n", "\n", "- Start with a grid with no edges (every square is surrounded by walls on all sides). \n", "- Add edges (that is, knock down walls) for the entrance at upper left and exit at lower right.\n", "- Place the root of the tree in some square.\n", "- Then repeat until the tree covers the whole grid:\n", " * Select some node already in the tree.\n", " * Randomly select a neighbor that hasn't been added to the tree yet.\n", " * Add an edge (knock down the wall) from the node to the neighbor.\n", " \n", "In the example below, the root, `A`, has been placed in the upper-left corner, and two branches,\n", "`A-B-C-D` and `A-b-c-d`, have been randomly chosen (well, not actually random; they are starting to create the same maze as in the diagram above):\n", "\n", " o o--o--o--o--o--o--o--o--o--o\n", " | A b c| | | | | | | |\n", " o o--o o--o--o--o--o--o--o--o\n", " | B| | d| | | | | | | |\n", " o o--o--o--o--o--o--o--o--o--o\n", " | C D| | | | | | | | |\n", " o--o--o--o--o--o--o--o--o--o--o\n", " | | | | | | | | | | |\n", " o--o--o--o--o--o--o--o--o--o--o\n", " | | | | | | | | | | |\n", " o--o--o--o--o--o--o--o--o--o o\n", " \n", "Next I select node `d` and extend it to `e` (at which point there are no available neighbors, so `e` will not be selected in the future), and then I select `D` and extend from there all the way to `N`, at each step selecting the node I just added:\n", "\n", " o o--o--o--o--o--o--o--o--o--o\n", " | A b c| | | | | | | |\n", " o o--o o--o--o--o--o--o--o--o\n", " | B| e d| | N| | | | | |\n", " o o--o--o--o o--o--o--o--o--o\n", " | C D| | | M| | | | | |\n", " o--o o--o--o o--o--o--o--o--o\n", " | F E| | K L| | | | | |\n", " o o--o--o o--o--o--o--o--o--o\n", " | G H I J| | | | | | |\n", " o--o--o--o--o--o--o--o--o--o o\n", " \n", "Continue like this until every square in the grid has been added to the tree. \n", "\n", "\n", "## Implementing Random Trees\n", "\n", "I'll make the following implementation choices:\n", "\n", "- A tree will be represented as a list of edges.\n", "- An `Edge` is a tuple of two nodes. I'll keep them sorted so that `Edge(A, B)` is the same as `Edge(B, A)`.\n", "- A node in a tree can be anything: a number, a letter, a square, ...\n", "- The algorithm for `random_tree(nodes, neighbors, pop)` is as follows:\n", " * We will keep track of three collections:\n", " - `tree`: a list of edges that constitutes the tree.\n", " - `nodes`: the set of nodes that have not yet been added to the tree, but will be.\n", " - `frontier`: a queue of nodes in the tree that are eligible to have an edge added.\n", " * On each iteration:\n", " - Use `pop` to pick a `node` from the frontier, and find the neighbors that are not already in the tree.\n", " - If there are any neighbors, randomly pick one (`nbr`), add `Edge(node, nbr)` to `tree`, remove the\n", " neighbor from `nodes`, and keep both the node and the neighbor on the frontier. If there are no neighbors,\n", " drop the node from the frontier.\n", " * When no `nodes` remain, return `tree`." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [], "source": [ "import random\n", "from collections import deque, namedtuple\n", "\n", "def Edge(node1, node2): return tuple(sorted([node1, node2]))\n", "\n", "def random_tree(nodes: set, neighbors: callable, pop: callable) -> [Edge]:\n", " \"Repeat: pop a node and add Edge(node, nbr) until all nodes have been added to tree.\"\n", " tree = []\n", " root = nodes.pop()\n", " frontier = deque([root])\n", " while nodes:\n", " node = pop(frontier)\n", " nbrs = neighbors(node) & nodes\n", " if nbrs:\n", " nbr = random.choice(list(nbrs))\n", " tree.append(Edge(node, nbr))\n", " nodes.remove(nbr)\n", " frontier.extend([node, nbr])\n", " return tree" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Implementing Random Mazes\n", "\n", "Now let's use `random_tree` to implement `random_maze`. Some more choices:\n", "\n", "* A `Maze` is a named tuple with three fields: the `width` and `height` of the grid, and a list of `edges` between squares. \n", "* A square is denoted by an `(x, y)` tuple of integer coordinates.\n", "* The function `neighbors4` gives the four surrounding squares." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "Maze = namedtuple('Maze', 'width, height, edges')\n", "\n", "def neighbors4(square):\n", " \"The 4 neighbors of an (x, y) square.\"\n", " (x, y) = square\n", " return {(x + 1, y), (x - 1, y), (x, y + 1), (x, y - 1)}\n", "\n", "def squares(width, height): \n", " \"All squares in a grid of these dimensions.\"\n", " return {(x, y) for x in range(width) for y in range(height)}\n", "\n", "def random_maze(width, height, pop=deque.pop):\n", " \"Use random_tree to generate a random maze.\"\n", " nodes = squares(width, height)\n", " tree = random_tree(nodes, neighbors4, pop)\n", " return Maze(width, height, tree)" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "Maze(width=10, height=5, edges=[((6, 3), (7, 3)), ((6, 3), (6, 4)), ((6, 4), (7, 4)), ((7, 4), (8, 4)), ((8, 3), (8, 4)), ((8, 2), (8, 3)), ((7, 2), (8, 2)), ((7, 1), (7, 2)), ((7, 0), (7, 1)), ((7, 0), (8, 0)), ((8, 0), (8, 1)), ((8, 1), (9, 1)), ((9, 0), (9, 1)), ((9, 1), (9, 2)), ((9, 2), (9, 3)), ((9, 3), (9, 4)), ((6, 0), (7, 0)), ((5, 0), (6, 0)), ((5, 0), (5, 1)), ((5, 1), (6, 1)), ((6, 1), (6, 2)), ((5, 2), (6, 2)), ((4, 2), (5, 2)), ((3, 2), (4, 2)), ((3, 2), (3, 3)), ((2, 3), (3, 3)), ((2, 2), (2, 3)), ((2, 1), (2, 2)), ((2, 0), (2, 1)), ((1, 0), (2, 0)), ((0, 0), (1, 0)), ((0, 0), (0, 1)), ((0, 1), (1, 1)), ((1, 1), (1, 2)), ((0, 2), (1, 2)), ((0, 2), (0, 3)), ((0, 3), (1, 3)), ((1, 3), (1, 4)), ((0, 4), (1, 4)), ((1, 4), (2, 4)), ((2, 4), (3, 4)), ((3, 4), (4, 4)), ((4, 3), (4, 4)), ((4, 3), (5, 3)), ((5, 3), (5, 4)), ((2, 0), (3, 0)), ((3, 0), (4, 0)), ((4, 0), (4, 1)), ((3, 1), (4, 1))])" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "random_maze(10,5)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "That's not very pretty to look at. I'm going to need a way to visualize a maze.\n", "\n", "# Printing a maze" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "o o--o--o--o--o--o--o--o--o--o\n", "| | | |\n", "o--o--o--o o--o o o o--o o\n", "| | | | |\n", "o o o--o--o o--o--o--o--o o\n", "| | | | | | |\n", "o o--o o o--o o o--o o--o\n", "| | | | | | |\n", "o o o--o--o--o--o o o--o o\n", "| | |\n", "o--o--o--o--o--o--o--o--o--o o\n" ] } ], "source": [ "def print_maze(maze, dot='o', lin='--', bar='|', sp=' '):\n", " \"Print maze in ASCII.\"\n", " exit = Edge((maze.width-1, maze.height-1), (maze.width-1, maze.height))\n", " edges = set(maze.edges) | {exit}\n", " print(dot + sp + lin.join(dot * maze.width)) # Top line, including entrance\n", " def vert_wall(x, y): return (' ' if Edge((x, y), (x+1, y)) in edges else bar)\n", " def horz_wall(x, y): return (sp if Edge((x, y), (x, y+1)) in edges else lin)\n", " for y in range(maze.height):\n", " print(bar + cat(sp + vert_wall(x, y) for x in range(maze.width)))\n", " print(dot + cat(horz_wall(x, y) + dot for x in range(maze.width)))\n", " \n", "cat = ''.join\n", " \n", "print_maze(random_maze(10, 5))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "*Much better!* But can I do better still?\n", "\n", "# Plotting a maze\n", "\n", "I'll use `matplotlib` to plot lines where the edges aren't:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [], "source": [ "%matplotlib inline\n", "import matplotlib.pyplot as plt\n", "\n", "def plot_maze(maze):\n", " \"Plot a maze by drawing lines between adjacent squares, except for pairs in maze.edges\"\n", " plt.figure(figsize=(8, 4))\n", " plt.axis('off')\n", " plt.gca().invert_yaxis()\n", " w, h = maze.width, maze.height\n", " exits = {Edge((0, 0), (0, -1)), Edge((w-1, h-1), (w-1, h))}\n", " edges = set(maze.edges) | exits\n", " for sq in squares(w, h):\n", " for nbr in neighbors4(sq):\n", " if Edge(sq, nbr) not in edges:\n", " plot_wall(sq, nbr)\n", " plt.show()\n", "\n", "def plot_wall(s1, s2):\n", " \"Plot a thick black line between squares s1 and s2.\"\n", " (x1, y1), (x2, y2) = s1, s2\n", " if x1 == x2: # horizontal wall\n", " y = max(y1, y2)\n", " X, Y = [x1, x1+1], [y, y]\n", " else: # vertical wall\n", " x = max(x1, x2)\n", " X, Y = [x, x], [y1, y1+1]\n", " plt.plot(X, Y, 'k-', linewidth=4.0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's compare the two visualization functions:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "o o--o--o--o--o--o--o--o--o--o\n", "| | |\n", "o o--o o--o o o--o--o--o o\n", "| | | | | | | |\n", "o o o--o o o o--o o--o o\n", "| | | | | | | |\n", "o o o o--o--o o o--o o--o\n", "| | | | | | |\n", "o--o o--o--o o o o--o--o o\n", "| | |\n", "o--o--o--o--o--o--o--o--o--o o\n" ] } ], "source": [ "M = random_maze(10, 5)\n", "plot_maze(M) \n", "print_maze(M)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# `pop` strategies\n", "\n", "Now I want to compare how the maze varies based on theree different choices for the `pop` parameter. \n", "\n", "(1) The default is `deque.pop`\n", "which means that the tree is created **depth-first**; we always select the `node` at the end of the `frontier`, so the tree follows a single branch along a randomly-twisted path until the path doubles back on itself and there are no more neighbors; at that point we select the most recent square for which there are neighbors:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plot_maze(random_maze(40, 20, deque.pop))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The maze with `deque.pop` looks pretty good. Reminds me of those [cyber brain](https://www.vectorstock.com/royalty-free-vector/cyber-brain-vector-3071965) images.\n", "\n", "(2) An alternative is `queue.popleft`, which creates the maze roughly **breadth-first**—we start at some root square , add an edge to it, and from then on we always select first a parent edge before we select a child edge. The net result is a design that appears to radiate out in concentric layers from the root (which is chosen by `random_tree` and is not necessarily the top-left square; below it looks like the root is in the upper-left quadrant). " ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plot_maze(random_maze(40, 20, deque.popleft))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The `deque.popleft` maze is interesting as a design, but to me it doesn't work well as a maze.\n", "\n", "(3) We can select a cell at random by shuffling the frontier before popping an element off of it:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "def poprandom(seq):\n", " \"Shuffle a mutable sequence (deque or list) and then pop an element.\"\n", " random.shuffle(seq)\n", " return seq.pop()\n", "\n", "plot_maze(random_maze(40, 20, poprandom))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is an interesting compromise: it has some structure, but still works nicely as a maze, in my opinion.\n", "\n", "What other variations can you come up with to generate interesting mazes?" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.0" } }, "nbformat": 4, "nbformat_minor": 2 }