# Abelian sandpile model

Abelian sandpile model
You are encouraged to solve this task according to the task description, using any language you may know.
 This page uses content from Wikipedia. The original article was at Abelian sandpile model. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance)

Implement the Abelian sandpile model also known as Bak–Tang–Wiesenfeld model. It's history, mathematical definition and properties can be found under it's wikipedia article.

The task requires the creation of a 2D grid of arbitrary size on which "piles of sand" can be placed. Any "pile" that has 4 or more sand particles on it collapses, resulting in four particles being subtracted from the pile and distributed among it's neighbors.

It is recommended to display the output in some kind of image format, as terminal emulators are usually too small to display images larger than a few dozen characters tall. As an example of how to accomplish this, see the Bitmap/Write a PPM file task.

Examples:

```0 0 0 0 0    0 0 0 0 0
0 0 0 0 0    0 0 1 0 0
0 0 4 0 0 -> 0 1 0 1 0
0 0 0 0 0    0 0 1 0 0
0 0 0 0 0    0 0 0 0 0

0 0 0 0 0    0 0 0 0 0
0 0 0 0 0    0 0 1 0 0
0 0 6 0 0 -> 0 1 2 1 0
0 0 0 0 0    0 0 1 0 0
0 0 0 0 0    0 0 0 0 0

0  0 0  0  0    0 0 1 0 0
0  0 0  0  0    0 2 1 2 0
0  0 16 0  0 -> 1 1 0 1 1
0  0 0  0  0    0 2 1 2 0
0  0 0  0  0    0 0 1 0 0
```

## Go

Translation of: Rust

Stack management in Go is automatic, starting very small (2KB) for each goroutine and expanding as necessary until the maximum allowed size is reached.

`package main import (    "fmt"    "log"    "os"    "strings") const dim = 16 // image size func check(err error) {    if err != nil {        log.Fatal(err)    }} // Outputs the result to the terminal using UTF-8 block characters.func drawPile(pile [][]uint) {    chars:= []rune(" ░▓█")    for _, row := range pile {        line := make([]rune, len(row))        for i, elem := range row {            if elem > 3 { // only possible when algorithm not yet completed.                elem = 3            }            line[i] = chars[elem]        }        fmt.Println(string(line))    }} // Creates a .ppm file in the current directory, which contains// a colored image of the pile.func writePile(pile [][]uint) {    file, err := os.Create("output.ppm")    check(err)    defer file.Close()    // Write the signature, image dimensions and maximum color value to the file.    fmt.Fprintf(file, "P3\n%d %d\n255\n", dim, dim)    bcolors := []string{"125 0 25 ", "125 80 0 ", "186 118 0 ", "224 142 0 "}    var line strings.Builder    for _, row := range pile {                for _, elem := range row {            line.WriteString(bcolors[elem])        }        file.WriteString(line.String() + "\n")        line.Reset()     }} // Main part of the algorithm, a simple, recursive implementation of the model.func handlePile(x, y uint, pile [][]uint) {    if pile[y][x] >= 4 {        pile[y][x] -= 4        // Check each neighbor, whether they have enough "sand" to collapse and if they do,        // recursively call handlePile on them.        if y > 0 {            pile[y-1][x]++            if pile[y-1][x] >= 4 {                handlePile(x, y-1, pile)            }        }        if x > 0 {            pile[y][x-1]++            if pile[y][x-1] >= 4 {                handlePile(x-1, y, pile)            }        }        if y < dim-1 {            pile[y+1][x]++            if pile[y+1][x] >= 4 {                handlePile(x, y+1, pile)            }        }        if x < dim-1 {            pile[y][x+1]++            if pile[y][x+1] >= 4 {                handlePile(x+1, y, pile)            }        }         // Uncomment this line to show every iteration of the program.        // Not recommended with large input values.        // drawPile(pile)         // Finally call the function on the current cell again,        // in case it had more than 4 particles.        handlePile(x, y, pile)    }} func main() {    // Create 2D grid and set size using the 'dim' constant.    pile := make([][]uint, dim)    for i := 0; i < dim; i++ {        pile[i] = make([]uint, dim)    }     // Place some sand particles in the center of the grid and start the algorithm.    hdim := uint(dim/2 - 1)    pile[hdim][hdim] = 16    handlePile(hdim, hdim, pile)    drawPile(pile)     // Uncomment this to save the final image to a file    // after the recursive algorithm has ended.    // writePile(pile)}`
Output:
```

░
▓░▓
░░ ░░
▓░▓
░

```

## Julia

Modified from code by Hayk Aleksanyan, viewable at github.com/hayk314/Sandpiles, license viewable there.

`module AbelSand # supports output functionality for the results of the sandpile simulations# outputs the final grid in CSV format, as well as an image file using CSV, DataFrames, Images function TrimZeros(A)    # given an array A trims any zero rows/columns from its borders    # returns a 4 tuple of integers, i1, i2, j1, j2, where the trimmed array corresponds to A[i1:i2, j1:j2]    # A can be either numeric or a boolean array     i1, j1 = 1, 1    i2, j2 = size(A)     zz = typeof(A[1, 1])(0)    # comparison of a value takes into account the type as well     # i1 is the first row which has non zero element    for i = 1:size(A, 1)        q = false        for k = 1:size(A, 2)            if A[i, k] != zz                q = true                i1 = i                break            end        end         if q == true            break        end    end     # i2 is the first from below row with non zero element    for i in size(A, 1):-1:1        q = false        for k = 1:size(A, 2)            if A[i, k] != zz                q = true                i2 = i                break            end        end         if q == true            break        end    end     # j1 is the first column with non zero element     for j = 1:size(A, 2)        q = false        for k = 1:size(A, 1)            if A[k, j] != zz                j1 = j                q = true                break            end        end         if q == true            break        end    end     # j2 is the last column with non zero element     for j in size(A, 2):-1:1        q=false        for k=1:size(A,1)            if A[k, j] != zz                j2 = j                q=true                break            end        end         if q==true            break        end    end     return i1, i2, j1, j2end function addLayerofZeros(A, extraLayer)    # adds layer of zeros from all corners to the given array A     if extraLayer <= 0        return A    end     N, M = size(A)      Z = zeros( typeof(A[1,1]), N + 2*extraLayer, M + 2*extraLayer)    Z[(extraLayer+1):(N + extraLayer ), (extraLayer+1):(M+extraLayer)] = A     return Z end function printIntoFile(A, extraLayer, strFileName, TrimSmallValues = false)    # exports a 2d matrix A into a csv file    # @extraLayer is an integers adding layer of 0-s sorrounding the output matrix     # trimming off very small values; tiny values affect the performance of CSV export    if TrimSmallValues == true        A = map(x -> if (abs(x - floor(x)) < 0.01) floor(x) else x end, A)     end     i1, i2, j1, j2  = TrimZeros( A )    A = A[i1:i2, j1:j2]     A = addLayerofZeros(A, extraLayer)     CSV.write(string(strFileName,".csv"), DataFrame(A), writeheader = false)     return A end function Array_magnifier(A, cell_mag, border_mag)    # A is the main array; @cell_mag is the magnifying size of the cell,    # @border_mag is the magnifying size of the border between lattice cells     # creates a new array where each cell of the original array A appears magnified by size = cell_mag      total_factor = cell_mag + border_mag     A1 = zeros(typeof(A[1, 1]), total_factor*size(A, 1), total_factor*size(A, 2))     for i = 1:size(A,1), j = 1:size(A,2), u = ((i-1)*total_factor+1):(i*total_factor),                                          v = ((j-1)*total_factor+1):(j*total_factor)        if(( u - (i - 1) * total_factor <= cell_mag) && (v - (j - 1) * total_factor <= cell_mag))            A1[u, v] = A[i, j]         end    end     return A1 end function saveAsGrayImage(A, fileName, cell_mag, border_mag, TrimSmallValues = false)    # given a 2d matrix A, we save it as a gray image after magnifying by the given factors    A1 = Array_magnifier(A, cell_mag, border_mag)    A1 = A1/maximum(maximum(A1))     # trimming very small values from A1 to improve performance    if TrimSmallValues == true        A1 = map(x -> if ( x < 0.01) 0.0 else round(x, digits = 2) end, A1)     end     save(string(fileName, ".png") , colorview(Gray, A1)) end function saveAsRGBImage(A, fileName, color_codes, cell_mag, border_mag)    # color_codes is a dictionary, where key is a value in A and value is an RGB triplet    # given a 2d array A, and color codes (mapping from values in A to RGB triples), save A    # into fileName as png image after applying the magnifying factors     A1 = Array_magnifier(A, cell_mag, border_mag)    color_mat = zeros(UInt8, (3, size(A1, 1), size(A1, 2)))     for i = 1:size(A1,1)        for j = 1:size(A1,2)            color_mat[:, i, j]  = get(color_codes, A1[i, j] , [0, 0, 0])        end    end     save(string(fileName, ".png") , colorview(RGB, color_mat/255)) end const N_size = 700       # the radius of the lattice Z^2, the actual size becomes (2*N+1)x(2*N+1)const dx = [1, 0, -1, 0] # for a given (x,y) in Z^2, (x + dx, y + dy) for all (dx,dy) covers the neighborhood of (x,y)const dy = [0, 1, 0, -1] struct L_coord    # represents a lattice coordinate    x::Int    y::Intend function FindCoordinate(Z::Array{L_coord,1}, a::Int, b::Int)    # in the given array Z of coordinates finds the (first) index of the tuple (a,b)    # if no match, returns -1     for i=1:length(Z)        if (Z[i].x == a) && (Z[i].y == b)            return i        end    end     return -1end function move(N)    # the main function moving the pile sand grains of size N at the origin of Z^2 until the sandpile becomes stable     Z_lat = zeros(UInt8, 2 * N_size + 1, 2 * N_size + 1)     # models the integer lattice Z^2, we will have at most 4 sands on each vertex    V_sites = falses(2 * N_size + 1, 2 * N_size + 1)         # all sites which are visited by the sandpile process, are being marked here    Odometer = zeros(UInt64, 2 * N_size + 1, 2 * N_size + 1) # stores the values of the odometer function      walking = L_coord[]    # the coordinates of sites which need to move     V_sites[N_size + 1, N_size + 1] = true     # i1, ... j2  -> show the boundaries of the box which is visited by the sandpile process    i1, i2, j1, j2 = N_size + 1, N_size + 1, N_size + 1, N_size + 1     n = N     t1 = time_ns()     while n > 0        n -= 1         Z_lat[N_size + 1, N_size + 1] += 1        if (Z_lat[N_size + 1, N_size + 1] >= 4)            push!(walking, L_coord(N_size + 1, N_size + 1))        end         while(length(walking) > 0)            w = pop!(walking)            x = w.x            y = w.y             Z_lat[x, y] -= 4            Odometer[x, y] += 4             for k = 1:4                Z_lat[x + dx[k], y + dy[k]] += 1                V_sites[x + dx[k], y + dy[k]] = true                if Z_lat[x + dx[k], y + dy[k]] >= 4                    if FindCoordinate(walking, x + dx[k] , y + dy[k]) == -1                        push!(walking, L_coord( x + dx[k], y + dy[k]))                    end                end            end             i1 = min(i1, x - 1)            i2 = max(i2, x + 1)            j1 = min(j1, y - 1)            j2 = max(j2, y + 1)        end      end #end of the main while    t2 = time_ns()     println("The final boundaries are:: ", (i2 - i1 + 1),"x",(j2 - j1 + 1), "\n")    print("time elapsed: " , (t2 - t1) / 1.0e9, "\n")     Z_lat = printIntoFile(Z_lat, 0, string("Abel_Z_", N))    Odometer = printIntoFile(Odometer, 1, string("Abel_OD_", N))     saveAsGrayImage(Z_lat, string("Abel_Z_", N), 20, 0)    color_code = Dict(1=>[255, 128, 255], 2=>[255, 0, 0],3 => [0, 128, 255])    saveAsRGBImage(Z_lat, string("Abel_Z_color_", N), color_code, 20, 0)     # for the total elapsed time, it's better to use the @time macros on the main call     return Z_lat, Odometer # these are trimmed in output module end # end of function move  end # module  using .AbelSand Z_lat, Odometer = AbelSand.move(100000) `
Output:

## Phix

Library: pGUI

Generates moving images similar to the julia output. The distributed version also has variable speed, additional display modes, and a random dropping toggle.

`-- demo\rosetta\Abelian_sandpile_model.exwinclude pGUI.e Ihandle dlg, canvascdCanvas cddbuffer sequence board = {{0,0,0},                  {0,0,0},                  {0,0,0}} procedure drop(integer y, x)    sequence moves = {}    while true do        board[y,x] += 1        if board[y,x]>=4 then            board[y,x] -= 4            moves &= {{y,x-1},{y,x+1},{y-1,x},{y+1,x}}        end if        -- extend board if rqd (maintain a border of zeroes)        if x=1 then                             -- extend left            for i=1 to length(board) do                board[i] = prepend(board[i],0)            end for            for i=1 to length(moves) do                moves[i] += 1            end for        elsif x=length(board) then           -- extend right            for i=1 to length(board) do                board[i] = append(board[i],0)            end for        end if        -- (copy the all-0 lines from the other end...)        if y=1 then                             -- extend up            board = prepend(board,board[\$])            for i=1 to length(moves) do                moves[i] += 1            end for        elsif y=length(board) then              -- extend down            board = append(board,board)        end if        if length(moves)=0 then exit end if        {y,x} = moves[\$]        moves = moves[1..\$-1]    end while       IupUpdate(canvas)end procedure function timer_cb(Ihandle /*ih*/)    integer y = floor(length(board)/2)+1,            x = floor(length(board)/2)+1    drop(y,x)    return IUP_DEFAULTend function function redraw_cb(Ihandle ih, integer /*posx*/, integer /*posy*/)    IupGLMakeCurrent(ih)    cdCanvasActivate(cddbuffer)    cdCanvasClear(cddbuffer)    for y=1 to length(board) do        for x=1 to length(board) do             integer c = board[y][x]            if c!=0 then                integer colour = {CD_VIOLET,CD_RED,CD_BLUE}[c]                cdCanvasPixel(cddbuffer, x, y, colour)            end if        end for    end for    cdCanvasFlush(cddbuffer)    return IUP_DEFAULTend function function map_cb(Ihandle ih)    IupGLMakeCurrent(ih)    atom res = IupGetDouble(NULL, "SCREENDPI")/25.4    cddbuffer = cdCreateCanvas(CD_GL, "300x100 %g", {res})    cdCanvasSetBackground(cddbuffer, CD_PARCHMENT)    return IUP_DEFAULTend function procedure main()    IupOpen()    canvas = IupGLCanvas("RASTERSIZE=300x100")    IupSetCallbacks({canvas}, {"ACTION", Icallback("redraw_cb"),                               "MAP_CB", Icallback("map_cb")})    dlg = IupDialog(canvas,"TITLE=\"Abelian sandpile model\"")    IupCloseOnEscape(dlg)    IupShow(dlg)    Ihandle timer = IupTimer(Icallback("timer_cb"), 10)    IupMainLoop()    IupClose()end procedure main()`

## Python

` import numpy as npimport matplotlib.pyplot as plt  def iterate(grid):    changed = False    for ii, arr in enumerate(grid):        for jj, val in enumerate(arr):            if val > 3:                grid[ii, jj] -= 4                if ii > 0:                    grid[ii - 1, jj] += 1                if ii < len(grid)-1:                    grid[ii + 1, jj] += 1                if jj > 0:                    grid[ii, jj - 1] += 1                if jj < len(grid)-1:                    grid[ii, jj + 1] += 1                changed = True    return grid, changed  def simulate(grid):    while True:        grid, changed = iterate(grid)        if not changed:            return grid  if __name__ == '__main__':    start_grid = np.zeros((10, 10))    start_grid[4:5, 4:5] = 64    final_grid = simulate(start_grid.copy())    plt.figure()    plt.gray()    plt.imshow(start_grid)    plt.figure()    plt.gray()    plt.imshow(final_grid) `

Output: </n> Before:

` [[0. 0. 0. 0. 0. 0. 0. 0. 0. 0.] [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.] [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.] [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.] [0. 0. 0. 0.64. 0. 0. 0. 0. 0.] [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.] [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.] [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.] [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.] [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]] `

After:

` [[0. 0. 0. 0. 0. 0. 0. 0. 0. 0.] [0. 0. 0. 1. 2. 1. 0. 0. 0. 0.] [0. 0. 2. 2. 2. 2. 2. 0. 0. 0.] [0. 1. 2. 2. 2. 2. 2. 1. 0. 0.] [0. 2. 2. 2. 0. 2. 2. 2. 0. 0.] [0. 1. 2. 2. 2. 2. 2. 1. 0. 0.] [0. 0. 2. 2. 2. 2. 2. 0. 0. 0.] [0. 0. 0. 1. 2. 1. 0. 0. 0. 0.] [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.] [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]] `

## Rust

`// Set image size.const DIM: usize = 16; // This function outputs the result to the console using UTF-8 block characters.fn draw_pile(pile: &Vec<Vec<usize>>) {    for row in pile {        let mut line = String::with_capacity(row.len());        for elem in row {            line.push(match elem {                0 => ' ',                1 => '░',                2 => '▒',                3 => '▓',                _ => '█'            });        }         println!("{}", line);    }} // This function creates a file called "output.ppm" in the directory the program was run, which contains// a colored image of the pile.fn write_pile(pile: &Vec<Vec<usize>>) {    use std::fs::File; // Used for opening the file.    use std::io::Write; // Used for writing to the file.     // Learn more about PPM here: http://netpbm.sourceforge.net/doc/ppm.html    let mut file = File::create("./output.ppm").unwrap();     // We write the signature, image dimensions and maximum color value to the file.    let _ = write!(file, "P3\n {} {}\n255\n", DIM, DIM).unwrap();     for row in pile {        let mut line = String::with_capacity(row.len()*6);        for elem in row {            line.push_str(match elem {                0 => "125 0 25 ", // Background color for cells that have no "sand" in them.                 // Depending on how many particles of sand is there in the cell we use a different shade of yellow.                1 => "125 80 0 ",                2 => "186 118 0 ",                3 => "224 142 0 ",                 // It is impossible to have more than 3 particles of sand in one cell after the program has run,                // however, Rust demands that all branches have to be considered in a match statement, so we                // explicitly tell the compiler, that this is an unreachable branch.                _ => unreachable!()             });        }         let _ = write!(file, "{}", line).unwrap();    }} // This is the main part of the algorithm, a simple, recursive implementation of the model.fn handle_pile(x: usize, y: usize, pile: &mut Vec<Vec<usize>>) {    if pile[y][x] >= 4 {        pile[y][x] -= 4;         // We check each neighbor, whether they have enough "sand" to collapse and if they do,         // we recursively call handle_pile on them.        if y > 0 {            pile[y-1][x] += 1;            if pile[y-1][x] >= 4 {handle_pile(x, y-1, pile)}}         if x > 0 {            pile[y][x-1] += 1;            if pile[y][x-1] >= 4 {handle_pile(x-1, y, pile)}}         if y < DIM-1 {            pile[y+1][x] += 1;            if pile[y+1][x] >= 4 {handle_pile(x, y+1, pile)}}         if x < DIM-1 {            pile[y][x+1] += 1;            if pile[y][x+1] >= 4 {handle_pile(x+1, y, pile)}}         // Uncomment this line to show every iteration of the program. Not recommended with large input values.        //draw_pile(&pile);         // Finally we call the function on the current cell again, in case it had more than 4 particles.        handle_pile(x,y,pile);    }}  fn main() {    use std::thread::Builder; // Used to spawn a new thread.     /* Rust by default uses a 2Mb stack, which gets quickly filled (resulting in a stack overflow) if we use any value larger than     * about 30,000 as our input value. To circumvent this, we spawn a thread with 32Mbs of stack memory, which can easily handle     * hundreds of thousands of sand particles. I tested the program using 256,000, but it should theoretically work with larger     * values too.      */     let _ = Builder::new().stack_size(33554432).spawn(|| {        // This is our 2D grid. It's size can be set using the DIM constant found at the top of the code.        let mut pile: Vec<Vec<usize>> = vec![vec![0;DIM]; DIM];         // We place this much sand in the center of the grid.        pile[DIM/2 - 1][DIM/2 - 1] = 16;         // We start the algorithm on the pile we just created.        handle_pile(DIM/2 - 1, DIM/2 - 1, &mut pile);          draw_pile(&pile)         // Uncomment this to save the image to a file after the recursive algorithm has ended.        //write_pile(&pile)    }).unwrap().join();}`

Output:

```

░
▒░▒
░░ ░░
▒░▒
░

```