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+---
+layout: post
+title: Random Terrain Generation, A Clojure Walkthrough
+---
+
+![terrain][terrain]
+
+I recently started looking into the diamond-square algorithm (you can find a
+great article on it [here][diamondsquare]). The following is a short-ish
+walkthrough of how I tackled the problem in clojure and the results. You can
+find the [leiningen][lein] repo [here][repo] and follow along within that, or
+simply read the code below to get an idea.
+
+```clojure
+(ns diamond-square.core)
+
+; == The Goal ==
+; Create a fractal terrain generator using clojure
+
+; == The Algorithm ==
+; Diamond-Square. We start with a grid of points, each with a height of 0.
+;
+; 1. Take each corner point of the square, average the heights, and assign that
+; to be the height of the midpoint of the square. Apply some random error to
+; the midpoint.
+;
+; 2. Creating a line from the midpoint to each corner we get four half-diamonds.
+; Average the heights of the points (with some random error) and assign the
+; heights to the midpoints of the diamonds.
+;
+; 3. We now have four square sections, start at 1 for each of them (with
+; decreasing amount of error for each iteration).
+;
+; This picture explains it better than I can:
+; https://raw2.github.com/mediocregopher/diamond-square/master/resources/dsalg.png
+; (http://nbickford.wordpress.com/2012/12/21/creating-fake-landscapes/dsalg/)
+;
+; == The Strategy ==
+; We begin with a vector of vectors of numbers, and iterate over it, filling in
+; spots as they become available. Our grid will have the top-left being (0,0),
+; y being pointing down and x going to the right. The outermost vector
+; indicating row number (y) and the inner vectors indicate the column number (x)
+;
+; = Utility =
+; First we create some utility functions for dealing with vectors of vectors.
+
+(defn print-m
+ "Prints a grid in a nice way"
+ [m]
+ (doseq [n m]
+ (println n)))
+
+(defn get-m
+ "Gets a value at the given x,y coordinate of the grid, with [0,0] being in the
+ top left"
+ [m x y]
+ ((m y) x))
+
+(defn set-m
+ "Sets a value at the given x,y coordinat of the grid, with [0,0] being in the
+ top left"
+ [m x y v]
+ (assoc m y
+ (assoc (m y) x v)))
+
+(defn add-m
+ "Like set-m, but adds the given value to the current on instead of overwriting
+ it"
+ [m x y v]
+ (set-m m x y
+ (+ (get-m m x y) v)))
+
+(defn avg
+ "Returns the truncated average of all the given arguments"
+ [& l]
+ (int (/ (reduce + l) (count l))))
+
+; = Grid size =
+; Since we're starting with a blank grid we need to find out what sizes the
+; grids can be. For convenience the size (height and width) should be odd, so we
+; easily get a midpoint. And on each iteration we'll be halfing the grid, so
+; whenever we do that the two resultrant grids should be odd and halfable as
+; well, and so on.
+;
+; The algorithm that fits this is size = 2^n + 1, where 1 <= n. For the rest of
+; this guide I'll be referring to n as the "degree" of the grid.
+
+
+(def exp2-pre-compute
+ (vec (map #(int (Math/pow 2 %)) (range 31))))
+
+(defn exp2
+ "Returns 2^n as an integer. Uses pre-computed values since we end up doing
+ this so much"
+ [n]
+ (exp2-pre-compute n))
+
+(def grid-sizes
+ (vec (map #(inc (exp2 %)) (range 1 31))))
+
+(defn grid-size [degree]
+ (inc (exp2 degree)))
+
+; Available grid heights/widths are as follows:
+;[3 5 9 17 33 65 129 257 513 1025 2049 4097 8193 16385 32769 65537 131073
+;262145 524289 1048577 2097153 4194305 8388609 16777217 33554433 67108865
+;134217729 268435457 536870913 1073741825])
+
+(defn blank-grid
+ "Generates a grid of the given degree, filled in with zeros"
+ [degree]
+ (let [gsize (grid-size degree)]
+ (vec (repeat gsize
+ (vec (repeat gsize 0))))))
+
+(comment
+ (print-m (blank-grid 3))
+)
+
+; = Coordinate Pattern (The Tricky Part) =
+; We now have to figure out which coordinates need to be filled in on each pass.
+; A pass is defined as a square step followed by a diamond step. The next pass
+; will be the square/dimaond steps on all the smaller squares generated in the
+; pass. It works out that the number of passes required to fill in the grid is
+; the same as the degree of the grid, where the first pass is 1.
+;
+; So we can easily find patterns in the coordinates for a given degree/pass,
+; I've laid out below all the coordinates for each pass for a 3rd degree grid
+; (which is 9x9).
+
+; Degree 3 Pass 1 Square
+; [. . . . . . . . .]
+; [. . . . . . . . .]
+; [. . . . . . . . .]
+; [. . . . . . . . .]
+; [. . . . 1 . . . .] (4,4)
+; [. . . . . . . . .]
+; [. . . . . . . . .]
+; [. . . . . . . . .]
+; [. . . . . . . . .]
+
+; Degree 3 Pass 1 Diamond
+; [. . . . 2 . . . .] (4,0)
+; [. . . . . . . . .]
+; [. . . . . . . . .]
+; [. . . . . . . . .]
+; [2 . . . . . . . 2] (0,4) (8,4)
+; [. . . . . . . . .]
+; [. . . . . . . . .]
+; [. . . . . . . . .]
+; [. . . . 2 . . . .] (4,8)
+
+; Degree 3 Pass 2 Square
+; [. . . . . . . . .]
+; [. . . . . . . . .]
+; [. . 3 . . . 3 . .] (2,2) (6,2)
+; [. . . . . . . . .]
+; [. . . . . . . . .]
+; [. . . . . . . . .]
+; [. . 3 . . . 3 . .] (2,6) (6,6)
+; [. . . . . . . . .]
+; [. . . . . . . . .]
+
+; Degree 3 Pass 2 Diamond
+; [. . 4 . . . 4 . .] (2,0) (6,0)
+; [. . . . . . . . .]
+; [4 . . . 4 . . . 4] (0,2) (4,2) (8,2)
+; [. . . . . . . . .]
+; [. . 4 . . . 4 . .] (2,4) (6,4)
+; [. . . . . . . . .]
+; [4 . . . 4 . . . 4] (0,6) (4,6) (8,6)
+; [. . . . . . . . .]
+; [. . 4 . . . 4 . .] (2,8) (6,8)
+
+; Degree 3 Pass 3 Square
+; [. . . . . . . . .]
+; [. 5 . 5 . 5 . 5 .] (1,1) (3,1) (5,1) (7,1)
+; [. . . . . . . . .]
+; [. 5 . 5 . 5 . 5 .] (1,3) (3,3) (5,3) (7,3)
+; [. . . . . . . . .]
+; [. 5 . 5 . 5 . 5 .] (1,5) (3,5) (5,5) (7,5)
+; [. . . . . . . . .]
+; [. 5 . 5 . 5 . 5 .] (1,7) (3,7) (5,7) (7,7)
+; [. . . . . . . . .]
+
+; Degree 3 Pass 3 Square
+; [. 6 . 6 . 6 . 6 .] (1,0) (3,0) (5,0) (7,0)
+; [6 . 6 . 6 . 6 . 6] (0,1) (2,1) (4,1) (6,1) (8,1)
+; [. 6 . 6 . 6 . 6 .] (1,2) (3,2) (5,2) (7,2)
+; [6 . 6 . 6 . 6 . 6] (0,3) (2,3) (4,3) (6,3) (8,3)
+; [. 6 . 6 . 6 . 6 .] (1,4) (3,4) (5,4) (7,4)
+; [6 . 6 . 6 . 6 . 6] (0,5) (2,5) (4,5) (6,5) (8,5)
+; [. 6 . 6 . 6 . 6 .] (1,6) (3,6) (5,6) (7,6)
+; [6 . 6 . 6 . 6 . 6] (0,7) (2,7) (4,7) (6,7) (8,7)
+; [. 6 . 6 . 6 . 6 .] (1,8) (3,8) (5,8) (7,8)
+;
+; I make two different functions, one to give the coordinates for the square
+; portion of each pass and one for the diamond portion of each pass. To find the
+; actual patterns it was useful to first look only at the pattern in the
+; y-coordinates, and figure out how that translated into the pattern for the
+; x-coordinates.
+
+(defn grid-square-coords
+ "Given a grid degree and pass number, returns all the coordinates which need
+ to be computed for the square step of that pass"
+ [degree pass]
+ (let [gsize (grid-size degree)
+ start (exp2 (- degree pass))
+ interval (* 2 start)
+ coords (map #(+ start (* interval %))
+ (range (exp2 (dec pass))))]
+ (mapcat (fn [y]
+ (map #(vector % y) coords))
+ coords)))
+;
+; (grid-square-coords 3 2)
+; => ([2 2] [6 2] [2 6] [6 6])
+
+(defn grid-diamond-coords
+ "Given a grid degree and a pass number, returns all the coordinates which need
+ to be computed for the diamond step of that pass"
+ [degree pass]
+ (let [gsize (grid-size degree)
+ interval (exp2 (- degree pass))
+ num-coords (grid-size pass)
+ coords (map #(* interval %) (range 0 num-coords))]
+ (mapcat (fn [y]
+ (if (even? (/ y interval))
+ (map #(vector % y) (take-nth 2 (drop 1 coords)))
+ (map #(vector % y) (take-nth 2 coords))))
+ coords)))
+
+; (grid-diamond-coords 3 2)
+; => ([2 0] [6 0] [0 2] [4 2] [8 2] [2 4] [6 4] [0 6] [4 6] [8 6] [2 8] [6 8])
+
+; = Height Generation =
+; We now work on functions which, given a coordinate, will return what value
+; coordinate will have.
+
+(defn avg-points
+ "Given a grid and an arbitrary number of points (of the form [x y]) returns
+ the average of all the given points that are on the map. Any points which are
+ off the map are ignored"
+ [m & coords]
+ (let [grid-size (count m)]
+ (apply avg
+ (map #(apply get-m m %)
+ (filter
+ (fn [[x y]]
+ (and (< -1 x) (> grid-size x)
+ (< -1 y) (> grid-size y)))
+ coords)))))
+
+(defn error
+ "Returns a number between -e and e, inclusive"
+ [e]
+ (- (rand-int (inc (* 2 e))) e))
+
+; The next function is a little weird. It primarily takes in a point, then
+; figures out the distance from that point to the points we'll take the average
+; of. The locf (locator function) is used to return back the actual points to
+; use. For the square portion it'll be the points diagonal from the given one,
+; for the diamond portion it'll be the points to the top/bottom/left/right from
+; the given one.
+;
+; Once it has those points, it finds the average and applies the error. The
+; error function is nothing more than a number between -interval and +interval,
+; where interval is the distance between the given point and one of the averaged
+; points. It is important that the error decreases the more passes you do, which
+; is why the interval is used.
+;
+; The error function is what should be messed with primarily if you want to
+; change what kind of terrain you generate (a giant mountain instead of
+; hills/valleys, for example). The one we use is uniform for all intervals, so
+; it generates a uniform terrain.
+
+(defn- grid-fill-point
+ [locf m degree pass x y]
+ (let [interval (exp2 (- degree pass))
+ leftx (- x interval)
+ rightx (+ x interval)
+ upy (- y interval)
+ downy (+ y interval)
+ v (apply avg-points m
+ (locf x y leftx rightx upy downy))]
+ (add-m m x y (+ v (error interval)))))
+
+(def grid-fill-point-square
+ "Given a grid, the grid's degree, the current pass number, and a point on the
+ grid, fills in that point with the average (plus some error) of the
+ appropriate corner points, and returns the resultant grid"
+ (partial grid-fill-point
+ (fn [_ _ leftx rightx upy downy]
+ [[leftx upy]
+ [rightx upy]
+ [leftx downy]
+ [rightx downy]])))
+
+(def grid-fill-point-diamond
+ "Given a grid, the grid's degree, the current pass number, and a point on the
+ grid, fills in that point with the average (plus some error) of the
+ appropriate edge points, and returns the resultant grid"
+ (partial grid-fill-point
+ (fn [x y leftx rightx upy downy]
+ [[leftx y]
+ [rightx y]
+ [x upy]
+ [x downy]])))
+
+; = Filling in the Grid =
+; We finally compose the functions we've been creating to fill in the entire
+; grid
+
+(defn- grid-fill-point-passes
+ "Given a grid, a function to fill in coordinates, and a function to generate
+ those coordinates, fills in all coordinates for a given pass, returning the
+ resultant grid"
+ [m fill-f coord-f degree pass]
+ (reduce
+ (fn [macc [x y]] (fill-f macc degree pass x y))
+ m
+ (coord-f degree pass)))
+
+(defn grid-pass
+ "Given a grid and a pass number, does the square then the diamond portion of
+ the pass"
+ [m degree pass]
+ (-> m
+ (grid-fill-point-passes
+ grid-fill-point-square grid-square-coords degree pass)
+ (grid-fill-point-passes
+ grid-fill-point-diamond grid-diamond-coords degree pass)))
+
+; The most important function in this guide, does all the work
+(defn terrain
+ "Given a grid degree, generates a uniformly random terrain on a grid of that
+ degree"
+ ([degree]
+ (terrain (blank-grid degree) degree))
+ ([m degree]
+ (reduce
+ #(grid-pass %1 degree %2)
+ m
+ (range 1 (inc degree)))))
+
+(comment
+ (print-m
+ (terrain 5))
+)
+
+; == The Results ==
+; We now have a generated terrain, probably. We should check it. First we'll
+; create an ASCII representation. But to do that we'll need some utility
+; functions.
+
+(defn max-terrain-height
+ "Returns the maximum height found in the given terrain grid"
+ [m]
+ (reduce max
+ (map #(reduce max %) m)))
+
+(defn min-terrain-height
+ "Returns the minimum height found in the given terrain grid"
+ [m]
+ (reduce min
+ (map #(reduce min %) m)))
+
+(defn norm
+ "Given x in the range (A,B), normalizes it into the range (0,new-height)"
+ [A B new-height x]
+ (int (/ (* (- x A) new-height) (- B A))))
+
+(defn normalize-terrain
+ "Given a terrain map and a number of \"steps\", normalizes the terrain so all
+ heights in it are in the range (0,steps)"
+ [m steps]
+ (let [max-height (max-terrain-height m)
+ min-height (min-terrain-height m)
+ norm-f (partial norm min-height max-height steps)]
+ (vec (map #(vec (map norm-f %)) m))))
+
+; We now define which ASCII characters we want to use for which heights. The
+; vector starts with the character for the lowest height and ends with the
+; character for the heighest height.
+
+(def tiles
+ [\~ \~ \" \" \x \x \X \$ \% \# \@])
+
+(defn tile-terrain
+ "Given a terrain map, converts it into an ASCII tile map"
+ [m]
+ (vec (map #(vec (map tiles %))
+ (normalize-terrain m (dec (count tiles))))))
+
+(comment
+ (print-m
+ (tile-terrain
+ (terrain 5)))
+
+; [~ ~ " " x x x X % $ $ $ X X X X X X $ x x x X X X x x x x " " " ~]
+; [" ~ " " x x X X $ $ $ X X X X X X X X X X X X X X x x x x " " " "]
+; [" " " x x x X X % $ % $ % $ $ X X X X $ $ $ X X X X x x x x " " "]
+; [" " " x x X $ % % % % % $ % $ $ X X $ $ $ $ X X x x x x x x " " x]
+; [" x x x x X $ $ # % % % % % % $ X $ X X % $ % X X x x x x x x x x]
+; [x x x X $ $ $ % % % % % $ % $ $ $ % % $ $ $ $ X X x x x x x x x x]
+; [X X X $ % $ % % # % % $ $ % % % % $ % $ $ X $ X $ X X x x x X x x]
+; [$ $ X $ $ % $ % % % % $ $ $ % # % % % X X X $ $ $ X X X x x x x x]
+; [% X X % % $ % % % $ % $ % % % # @ % $ $ X $ X X $ X x X X x x x x]
+; [$ $ % % $ $ % % $ $ X $ $ % % % % $ $ X $ $ X X X X X X x x x x x]
+; [% % % X $ $ % $ $ X X $ $ $ $ % % $ $ X X X $ X X X x x X x x X X]
+; [$ $ $ X $ $ X $ X X X $ $ $ $ % $ $ $ $ $ X $ X x X X X X X x X X]
+; [$ $ $ $ X X $ X X X X X $ % % % % % $ X $ $ $ X x X X X $ X X $ $]
+; [X $ $ $ $ $ X X X X X X X % $ % $ $ $ X X X X X x x X X x X X $ $]
+; [$ $ X X $ X X x X $ $ X X $ % X X X X X X X X X x X X x x X X X X]
+; [$ $ X X X X X X X $ $ $ $ $ X $ X X X X X X X x x x x x x x X X X]
+; [% % % $ $ X $ X % X X X % $ $ X X X X X X x x x x x x x x x X X $]
+; [$ % % $ $ $ X X $ $ $ $ $ $ X X X X x X x x x x " x x x " x x x x]
+; [$ X % $ $ $ $ $ X X X X X $ $ X X X X X X x x " " " " " " " " x x]
+; [$ X $ $ % % $ X X X $ X X X x x X X x x x x x " " " " " ~ " " " "]
+; [$ $ X X % $ % X X X X X X X X x x X X X x x x " " " " " " ~ " " "]
+; [$ $ X $ % $ $ X X X X X X x x x x x x x x x " " " " " " " " " ~ ~]
+; [$ $ $ $ $ X X $ X X X X X x x x x x x x x " " " " " " " ~ " " " ~]
+; [$ % X X $ $ $ $ X X X X x x x x x x x x x x " " " " ~ " " ~ " " ~]
+; [% $ $ X $ X $ X $ X $ X x x x x x x x x x x " " " " ~ ~ ~ " ~ " ~]
+; [$ X X X X $ $ $ $ $ X x x x x x x x x x x " " " " ~ ~ ~ ~ ~ ~ ~ ~]
+; [X x X X x X X X X X X X X x x x x x x x x x " " " ~ ~ " " ~ ~ ~ ~]
+; [x x x x x x X x X X x X X X x x x x x x x " x " " " " " ~ ~ ~ ~ ~]
+; [x x x x x x x x X X X X $ X X x X x x x x x x x x " ~ ~ ~ ~ ~ ~ ~]
+; [" x x x x x X x X X X X X X X X X x x x x x x " " " " ~ ~ ~ ~ ~ ~]
+; [" " " x x x X X X X $ $ $ X X X X X X x x x x x x x x " " ~ ~ ~ ~]
+; [" " " " x x x X X X X X $ $ X X x X X x x x x x x x " " " " " ~ ~]
+; [~ " " x x x x X $ X $ X $ $ X x X x x x x x x x x x x x x " " " ~]
+)
+
+; = Pictures! =
+; ASCII is cool, but pictures are better. First we import some java libraries
+; that we'll need, then define the colors for each level just like we did tiles
+; for the ascii representation.
+
+(import
+ 'java.awt.image.BufferedImage
+ 'javax.imageio.ImageIO
+ 'java.io.File)
+
+(def colors
+ [0x1437AD 0x04859D 0x007D1C 0x007D1C 0x24913C
+ 0x00C12B 0x38E05D 0xA3A3A4 0x757575 0xFFFFFF])
+
+; Finally we reduce over a BufferedImage instance to output every tile as a
+; single pixel on it.
+
+(defn img-terrain
+ "Given a terrain map and a file name, outputs a png representation of the
+ terrain map to that file"
+ [m file]
+ (let [img (BufferedImage. (count m) (count m) BufferedImage/TYPE_INT_RGB)]
+ (reduce
+ (fn [rown row]
+ (reduce
+ (fn [coln tile]
+ (.setRGB img coln rown (colors tile))
+ (inc coln))
+ 0 row)
+ (inc rown))
+ 0 (normalize-terrain m (dec (count colors))))
+ (ImageIO/write img "png" (File. file))))
+
+(comment
+ (img-terrain
+ (terrain 10)
+ "resources/terrain.png")
+
+ ; https://raw2.github.com/mediocregopher/diamond-square/master/resources/terrain.png
+)
+
+; == Conclusion ==
+; There's still a lot of work to be done. The algorithm starts taking a
+; non-trivial amount of time around the 10th degree, which is only a 1025x1025px
+; image. I need to profile the code and find out where the bottlenecks are. It's
+; possible re-organizing the code to use pmaps instead of reduces in some places
+; could help.
+```
+
+[terrain]: /img/dsqr-terrain.png
+[diamondsquare]: http://www.gameprogrammer.com/fractal.html
+[lein]: https://github.com/technomancy/leiningen
+[repo]: https://github.com/mediocregopher/diamond-square