Abelian sandpile model

From Rosetta Code
Task
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 up to 2^30, wow!
javascript running on web
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

C[edit]

Writes out the initial and final sand piles to the console and the final sand pile to a PPM file.

 
#include<stdlib.h>
#include<string.h>
#include<stdio.h>
 
int main(int argc, char** argv)
{
int i,j,sandPileEdge, centerPileHeight, processAgain = 1,top,down,left,right;
int** sandPile;
char* fileName;
static unsigned char colour[3];
 
if(argc!=3){
printf("Usage: %s <Sand pile side> <Center pile height>",argv[0]);
return 0;
}
 
sandPileEdge = atoi(argv[1]);
centerPileHeight = atoi(argv[2]);
 
if(sandPileEdge<=0 || centerPileHeight<=0){
printf("Sand pile and center pile dimensions must be positive integers.");
return 0;
}
 
sandPile = (int**)malloc(sandPileEdge * sizeof(int*));
 
for(i=0;i<sandPileEdge;i++){
sandPile[i] = (int*)calloc(sandPileEdge,sizeof(int));
}
 
sandPile[sandPileEdge/2][sandPileEdge/2] = centerPileHeight;
 
printf("Initial sand pile :\n\n");
 
for(i=0;i<sandPileEdge;i++){
for(j=0;j<sandPileEdge;j++){
printf("%3d",sandPile[i][j]);
}
printf("\n");
}
 
while(processAgain == 1){
 
processAgain = 0;
top = 0;
down = 0;
left = 0;
right = 0;
 
for(i=0;i<sandPileEdge;i++){
for(j=0;j<sandPileEdge;j++){
if(sandPile[i][j]>=4){
if(i-1>=0){
top = 1;
sandPile[i-1][j]+=1;
if(sandPile[i-1][j]>=4)
processAgain = 1;
}
if(i+1<sandPileEdge){
down = 1;
sandPile[i+1][j]+=1;
if(sandPile[i+1][j]>=4)
processAgain = 1;
}
if(j-1>=0){
left = 1;
sandPile[i][j-1]+=1;
if(sandPile[i][j-1]>=4)
processAgain = 1;
}
if(j+1<sandPileEdge){
right = 1;
sandPile[i][j+1]+=1;
if(sandPile[i][j+1]>=4)
processAgain = 1;
}
sandPile[i][j] -= (top + down + left + right);
if(sandPile[i][j]>=4)
processAgain = 1;
}
}
}
}
 
printf("Final sand pile : \n\n");
 
for(i=0;i<sandPileEdge;i++){
for(j=0;j<sandPileEdge;j++){
printf("%3d",sandPile[i][j]);
}
printf("\n");
}
 
fileName = (char*)malloc((strlen(argv[1]) + strlen(argv[2]) + 23)*sizeof(char));
 
strcpy(fileName,"Final_Sand_Pile_");
strcat(fileName,argv[1]);
strcat(fileName,"_");
strcat(fileName,argv[2]);
strcat(fileName,".ppm");
 
FILE *fp = fopen(fileName,"wb");
 
fprintf(fp,"P6\n%d %d\n255\n",sandPileEdge,sandPileEdge);
 
for(i=0;i<sandPileEdge;i++){
for(j=0;j<sandPileEdge;j++){
colour[0] = (sandPile[i][j] + i)%256;
colour[1] = (sandPile[i][j] + j)%256;
colour[2] = (sandPile[i][j] + i*j)%256;
fwrite(colour,1,3,fp);
}
}
 
fclose(fp);
 
printf("\nImage file written to %s\n",fileName);
 
return 0;
}
 

Console output :

[email protected]:~/doodles$ ./a.out 10 64
Initial sand pile :

  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 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
Final sand pile :

  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  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

Image file written to Final_Sand_Pile_10_64.ppm

C++[edit]

Works with: g++ version 9.2.0 20061115
Library: xtensor
Library: xtensor-io


 
#include <iostream>
#include "xtensor/xarray.hpp"
#include "xtensor/xio.hpp"
#include "xtensor-io/ximage.hpp"
 
xt::xarray<int> init_grid (unsigned long x_dim, unsigned long y_dim)
{
xt::xarray<int>::shape_type shape = { x_dim, y_dim };
xt::xarray<int> grid(shape);
 
grid(x_dim/2, y_dim/2) = 64000;
 
return grid;
}
 
int print_grid(xt::xarray<int>& grid)
{
// for output to the terminal uncomment next line
// only makes sense for small grid < 32x32;
// std::cout << grid << std::endl << std::endl;
 
// output result to an image
xt::dump_image("grid.jpg", grid);
 
return 0;
}
 
bool iterate_grid(xt::xarray<int>& grid, const unsigned long& x_dim, const unsigned long& y_dim)
{
bool changed = false;
 
for (unsigned long i=0; i < x_dim; ++i)
{
for (unsigned long j=0; j < y_dim; ++j)
{
if ( grid(i, j) >= 4 )
{
grid(i, j) -= 4;
changed = true;
try
{
grid.at(i-1, j) += 1;
grid.at(i+1, j) += 1;
grid.at(i, j-1) += 1;
grid.at(i, j+1) += 1;
}
catch (const std::out_of_range& oor)
{
}
}
}
}
 
return changed;
}
 
int main(int argc, char* argv[])
{
const unsigned long x_dim { 200 };
const unsigned long y_dim { 200 };
 
xt::xarray<int> grid = init_grid(x_dim, y_dim);
bool changed { true };
 
iterate_grid(grid, x_dim, y_dim);
 
while (changed == true)
{
changed = iterate_grid(grid, x_dim, y_dim);
}
print_grid(grid);
 
return 0;
}
 
 
Compile with following CMakeList.txt
cmake_minimum_required(VERSION 3.1)
project(abelian_sandpile)

find_package(xtl REQUIRED)
find_package(xtensor REQUIRED)
# if xtensor was built with xsimd support:
# find_package(xsimd REQUIRED)
set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} -fopenmp")
include_directories(/usr/include/OpenImageIO)
find_library(OIIO "OpenImageIO")

add_executable(abelian_sandpile src/abelian_sandpile.cpp)

target_compile_options(abelian_sandpile PRIVATE -march=native -std=c++14)


target_link_libraries(abelian_sandpile xtensor ${OIIO})

Fōrmulæ[edit]

In this page you can see the solution of this task.

Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text (more info). Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for transportation effects more than visualization and edition.

The option to show Fōrmulæ programs and their results is showing images. Unfortunately images cannot be uploaded in Rosetta Code.

Forth[edit]

Works with: gforth version 0.7.3


#! /usr/bin/gforth -d 20M
\ Abelian Sandpile Model
 
0 assert-level !
 
\ command-line
 
: parse-number s>number? invert throw drop ;
: parse-size ." size  : " next-arg parse-number dup . cr ;
: parse-height ." height: " next-arg parse-number dup . cr ;
: parse-args cr parse-size parse-height ;
 
parse-args constant HEIGHT constant SIZE
 
: allot-erase create here >r dup allot r> swap erase ;
: size^2 SIZE dup * cells ;
: 2cells [ 2 cells ] literal ;
: -2cells [ 2cells negate ] literal ;
 
size^2 allot-erase arr
 
\ array processing
: ix swap SIZE * + cells arr + ;
: center SIZE 2/ dup ;
: write-cell ix @ u. ;
: write-row SIZE 0 ?do dup i write-cell loop drop cr ;
: arr. SIZE 0 ?do i write-row loop ;
 
\ stack processing
 
: stack-empty? dup -1 = ;
: stack-full? stack-empty? invert ;
 
\ pgm-handling
 
: concat { a1 l1 a2 l2 } l1 l2 + allocate throw dup dup a1 swap l1 cmove a2 swap l1 + l2 cmove l1 l2 + ;
: write-pgm ." P2" cr SIZE u. SIZE u. cr ." 3" cr arr. ;
: u>s 0 <# #s #> ;
: filename s" sandpile-" SIZE u>s concat s" -" concat HEIGHT u>s concat s" .pgm" concat ;
: to-pgm filename w/o create-file throw ['] write-pgm over outfile-execute close-file throw ;
 
\ sandpile
 
: prep-arr HEIGHT center ix ! ;
: prep-stack -1 HEIGHT 4 u>= if center then ;
: prepare prep-arr prep-stack ;
: ensure if else 2drop 0 2rdrop exit then ;
: col>=0 dup 0>= ensure ;
: col<SIZE dup SIZE < ensure ;
: row>=0 over 0>= ensure ;
: row<SIZE over SIZE < ensure ;
: legal? col>=0 col<SIZE row>=0 row<SIZE 2drop true ;
: north 1. d- ;
: east 1+ ;
: south 1. d+ ;
: west 1- ;
: reduce 2dup ix dup -4 swap +! @ 4 < if 2drop then ;
: increase 2dup legal? if 2dup ix dup 1 swap +! @ 4 = if 2swap else 2drop then else 2drop then ;
: inc-north 2dup north increase ;
: inc-east 2dup east increase ;
: inc-south 2dup south increase ;
: inc-west 2dup west increase ;
: inc-all inc-north inc-east inc-south inc-west 2drop ;
: simulate prepare begin stack-full? while 2dup 2>r reduce 2r> inc-all repeat drop to-pgm ." written to " filename type cr ;
 
simulate bye
Output:

sandpile with 5000 grains of sand: ./sandpile.fs 61 5000: [1]
sandpile with 50000 grains of sand: ./sandpile.fs 201 50000: [2]
sandpile with 500000 grains of sand: ./sandpile.fs 601 500000: [3]

Go[edit]

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:
                
                
                
                
                
       ░        
      ▓░▓       
     ░░ ░░      
      ▓░▓       
       ░        
                
                
                
                
                
                
       

Haskell[edit]

Works with: GHC version 8.8.1
Library: base version 4.13.0.0
Library: array version 0.5.4.0
Library: mtl version 2.2.2


Using a custom monad to make the code cleaner.

{-# LANGUAGE FlexibleContexts           #-}
{-# LANGUAGE GeneralizedNewtypeDeriving #-}
{-# LANGUAGE ScopedTypeVariables #-}
 
module Rosetta.AbelianSandpileModel.ST
( simulate
, test
, toPGM
) where
 
import Control.Monad.Reader (asks, MonadReader (..), ReaderT, runReaderT)
import Control.Monad.ST (runST, ST)
import Control.Monad.State (evalStateT, forM_, lift, MonadState (..), StateT, modify, when)
import Data.Array.ST (freeze, readArray, STUArray, thaw, writeArray)
import Data.Array.Unboxed (array, assocs, bounds, UArray, (!))
import Data.Word (Word32)
import System.IO (hPutStr, hPutStrLn, IOMode (WriteMode), withFile)
import Text.Printf (printf)
 
type Point = (Int, Int)
type ArrayST s = STUArray s Point Word32
type ArrayU = UArray Point Word32
 
newtype M s a = M (ReaderT (S s) (StateT [Point] (ST s)) a)
deriving (Functor, Applicative, Monad, MonadReader (S s), MonadState [Point])
 
data S s = S
{ bMin :: !Point
, bMax :: !Point
, arr :: !(ArrayST s)
}
 
runM :: M s a -> S s -> [Point]-> ST s a
runM (M m) = evalStateT . runReaderT m
 
liftST :: ST s a -> M s a
liftST = M . lift . lift
 
simulate :: ArrayU -> ArrayU
simulate a = runST $ simulateST a
 
simulateST :: forall s. ArrayU -> ST s ArrayU
simulateST a = do
let (p1, p2) = bounds a
s = [p | (p, c) <- assocs a, c >= 4]
b <- thaw a :: ST s (ArrayST s)
let st = S { bMin = p1
, bMax = p2
, arr = b
}
runM simulateM st s
 
simulateM :: forall s. M s ArrayU
simulateM = do
ps <- get
case ps of
[] -> asks arr >>= liftST . freeze
p : ps' -> do
c <- changeArr p $ \x -> x - 4
when (c < 4) $ put ps'

forM_ [north, east, south, west] $ inc . ($ p)
simulateM
 
changeArr :: Point -> (Word32 -> Word32) -> M s Word32
changeArr p f = do
a <- asks arr
oldC <- liftST $ readArray a p
let newC = f oldC
liftST $ writeArray a p newC
return newC
 
inc :: Point -> M s ()
inc p = do
b <- inBounds p
when b $ do
c <- changeArr p succ
when (c == 4) $ modify $ (p :)
 
inBounds :: Point -> M s Bool
inBounds p = do
st <- ask
return $ p >= bMin st && p <= bMax st
 
north, east, south, west :: Point -> Point
north (x, y) = (x, y + 1)
east (x, y) = (x + 1, y)
south (x, y) = (x, y - 1)
west (x, y) = (x - 1, y)
 
toPGM :: ArrayU -> FilePath -> IO ()
toPGM a fp = withFile fp WriteMode $ \h -> do
let ((x1, y1), (x2, y2)) = bounds a
width = x2 - x1 + 1
height = y2 - y1 + 1
hPutStrLn h "P2"
hPutStrLn h $ show width ++ " " ++ show height
hPutStrLn h "3"
forM_ [y1 .. y2] $ \y -> do
forM_ [x1 .. x2] $ \x -> do
let c = min 3 $ a ! (x, y)
hPutStr h $ show c ++ " "
hPutStrLn h ""
 
initArray :: Int -> Word32 -> ArrayU
initArray size height = array
((-size, -size), (size, size))
[((x, y), if x == 0 && y == 0 then height else 0) | x <- [-size .. size], y <- [-size .. size]]
 
test :: Int -> Word32 -> IO ()
test size height = do
printf "size = %d, height = %d\n" size height
let a = initArray size height
b = simulate a
fp = printf "sandpile_%d_%d.pgm" size height
toPGM b fp
putStrLn $ "wrote image to " ++ fp
Output:

sandpile with 1000 grains of sand: test 15 1000: [4]
sandpile with 10000 grains of sand: test 40 10000: [5]
sandpile with 100000 grains of sand: test 150 100000: [6]
sandpile with 1000000 grains of sand: test 400 1000000: [7]

Julia[edit]

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, j2
end
 
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::Int
end
 
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 -1
end
 
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:

Link to PNG output file for N=100000 ie. AbelSand.move(100000)
Link to PNG output file (run time >90 min) for N=1000000 (move(1000000))

Pascal[edit]

Works with: Free Pascal
The main optimization was to spread the sand immediatly.
mul := val DIV 4;//not only := val -4 
so that only (sand mod 4) stays in place.runtime for abelian(1e6) down to 1min 20 secs from 9 min

Memorizing the used colums of the rows has little effect when choosing the right size of the grid.Only 11 secs for abelian(1e6) -> 1min 9sec
Python shows 64 too.

 
program Abelian2;
{$IFDEF FPC}
{$MODE DELPHI}{$OPTIMIZATION ON,ALL}{$CODEALIGN proc=16}{$ALIGN 16}
{$ELSE}
{$APPTYPE CONSOLE}
{$ENDIF}
uses
SysUtils;
 
type
Tlimit = record
lmtLow,LmtHigh : LongWord;
end;
TRowlimits = array of Tlimit;
tOneRow = pLongWord;
tGrid = array of LongWord;
 
var
Grid: tGrid;
Rowlimits:TRowlimits;
s : AnsiString;
maxval,maxCoor : NativeUint;
 
function CalcMaxCoor(maxVal : NativeUint):NativeUint;
// maxVal = 10000;maxCoor = 77-2;// maxCoor*maxCoor *1,778; 0.009sec
// maxVal = 100000;maxCoor = 236-2;// maxCoor*maxCoor *1.826; 0.825sec
// maxVal = 1000000;maxCoor = 732-2;// maxCoor*maxCoor *1.877; 74 sec
Begin
result := trunc(sqrt(maxval/1.75))+3;
end;
 
procedure clear;
begin
setlength(Grid,0);
setlength(Rowlimits,0);
s := '';
end;
 
procedure InitGrid(var G:tGrid;InitVal:NativeUint);
var
row,middle: nativeINt;
begin
// setlength(Rowlimits,0); setlength(G,0);
MaxCoor := CalcMaxCoor(InitVal);
setlength(G,sqr(maxCoor));
setlength(Rowlimits,maxCoor);
fillchar(G[0],length(G)*SizeOf(G[0]),#0);
 
middle := (maxCoor) div 2;
Grid[middle*maxcoor+middle] := InitVal;
For row := 1 to maxCoor do
with Rowlimits[row] do
Begin
lmtLow := middle;
lmtHigh := middle;
end;
 
with Rowlimits[middle] do
Begin
lmtLow := middle;
lmtHigh := middle;
end;
end;
procedure OutGridPPM(const G:tGrid;maxValue : NativeUint);
const
color : array[0..3] of array[0..2] of Byte =
//R,G,B)
((0,0,0),
(255,0,0),
(0,255,0),
(0,0,255));
var
f :text;
pActRow: tOneRow;
col,row,sIdx,value : NativeInt;
Begin
Assignfile(f,'ppm/Grid_'+IntToStr(maxValue)+'.ppm');
rewrite(f);
write(f,Format('P6 %d %d %d ',[maxCoor-1,maxCoor-1,255]));
setlength(s,(maxCoor-1)*3);
pActRow :=@G[0];
For row := maxCoor-2 downto 0 do
Begin
inc(pActRow,maxCoor);
sIdx := 1;
For col := 1 to maxCoor-1 do
Begin
value := pActRow[col];
s[sIdx] := CHR(color[value,0]);
s[sIdx+1] := CHR(color[value,1]);
s[sIdx+2] := CHR(color[value,2]);
inc(sIdx,3);
end;
write(f,s);
end;
CloseFile(f);
end;
 
procedure OutGrid(const G:tGrid);
//output of grid and test, if no sand is lost
var
pActRow: tOneRow;
col,row,sum,value : NativeUint;
Begin
setlength(s,maxcoor-1);
pActRow := @G[0];
sum := 0;
For row := maxCoor-1 downto 1 do
Begin
inc(pActRow,maxcoor);
For col := 1 to maxCoor-1 do
Begin
value := pActRow[col];
// IF value>=4 then writeln(row:5,col:5,value:13);
s[col] := chr(value+48);
inc(sum,value);
end;
if maxCoor <80 then
writeln(s);
end;
writeln('columns ',maxcoor-1,' checksum ',maxVal,' ?=? ',sum);
{
For row := 1 to maxCoor do
with Rowlimits[row] do
writeln(lmtLow:10,lmtHigh:10);
* }

end;
 
procedure Evolution(var G:tGrid);
var
pActRow,pRowBefore,pRowAfter : tOneRow;
col,row,mul,val,done : NativeUint;
begin
repeat
pRowBefore := @G[0];
pActRow := @G[maxcoor];
pRowAfter := @G[2*maxcoor];
done := 0;
For row := maxCoor-1 downto 1 do
Begin
with RowLimits[row] do
Begin
while (LmtLow >1) AND (pActRow[lmtLow]<> 0) do
dec(lmtLow);
while (lmtHigh < maxCoor) AND (pActRow[lmtHigh]<> 0) do
inc(lmtHigh);
For col := lmtLow to lmtHigh do
Begin
val := pActRow[col];
IF val >=4 then
Begin
mul := val DIV 4;
done := val;
inc(pRowBefore[col],mul);
inc(pActRow[col-1],mul);
pActRow[col] := val-4*Mul;
inc(pActRow[col+1],mul);
inc(pRowAfter[col],mul);
end;
end;
pRowBefore:= pActRow;
pActRow := pRowAfter;
inc(pRowAfter,maxcoor);
end;
end;
until done=0;
end;
 
procedure OneTurn(count:NativeUint);
begin
Writeln(' Test abelian sandpile( ',count,' )');
MaxVal := count;
InitGrid(Grid,count);
Evolution(Grid);
OutGrid(Grid);
OutGridPPM(Grid,count);
clear;
end;
 
BEGIN
OneTurn(4);
OneTurn(16);
OneTurn(64);
OneTurn(1000);
OneTurn(10000);
OneTurn(100000);
END.
 
Output:
 Test abelian sandpile( 4 )
010
101
010
columns 3 checksum 4 ?=? 4
 Test abelian sandpile( 16 )
00100
02120
11011
02120
00100
columns 5 checksum 16 ?=? 16
 Test abelian sandpile( 64 )
00121000
02222200
12222210
22202220
12222210
02222200
00121000
00000000
columns 8 checksum 64 ?=? 64
 Test abelian sandpile( 1000 )
0000000001111111000000000
0000000130233320310000000
0000013223313133223100000
0000213222130312223120000
0002220123332333210222000
0011223233123213323221100
0033032313221223132303300
0122123203311133023212210
0322231023333333201322230
1032333332231322333332301
1231312332232322332131321
1313322133322233312233131
1330231131220221311320331
1313322133322233312233131
1231312332232322332131321
1032333332231322333332301
0322231023333333201322230
0122123203311133023212210
0033032313221223132303300
0011223233123213323221100
0002220123332333210222000
0000213222130312223120000
0000013223313133223100000
0000000130233320310000000
0000000001111111000000000
columns 25 checksum 1000 ?=? 1000
 Test abelian sandpile( 10000 )
--shortened
columns 77 checksum 10000 ?=? 10000
 Test abelian sandpile( 100000 )
columns 241 checksum 100000 ?=? 100000

real    0m0,815s

Perl[edit]

#!/usr/bin/perl
 
use strict; # http://www.rosettacode.org/wiki/Abelian_sandpile_model
use warnings;
 
my ($high, $wide) = split ' ', qx(stty size);
my $mask = "\0" x $wide . ("\0" . "\177" x ($wide - 2) . "\0") x ($high - 5) .
"\0" x $wide;
my $pile = $mask =~ s/\177/ rand() < 0.02 ? chr 64 + rand 20 : "\0" /ger;
 
for ( 1 .. 1e6 )
{
print "\e[H", $pile =~ tr/\0-\177/ 1-~/r, "\n$_";
my $add = $pile =~ tr/\1-\177/\0\0\0\200/r; # set high bit for >=4
$add =~ /\200/ or last;
$pile =~ tr/\4-\177/\0-\173/; # subtract 4 if >=4
for ("\0$add", "\0" x $wide . $add, substr($add, 1), substr $add, $wide)
{
$pile |= $_;
$pile =~ tr/\200-\377/\1-\176/; # add one to each neighbor of >=4
$pile &= $mask;
}
select undef, undef, undef, 0.1; # comment out for full speed
}

Perl 6[edit]

Works with: Rakudo version 2019.07.1

Defaults to a stack of 1000 and showing progress. Pass in a custom stack size if desired and -hide-progress to run without displaying progress (much faster.)

sub cleanup { print "\e[0m\e[?25h\n"; exit(0) }
 
signal(SIGINT).tap: { cleanup(); exit(0) }
 
unit sub MAIN ($stack = 1000, :$hide-progress = False );
 
my @color = "\e[38;2;0;0;0m█",
"\e[38;2;255;0;0m█",
"\e[38;2;255;255;0m█",
"\e[38;2;0;0;255m█",
"\e[38;2;255;255;255m█"
;
 
my ($h, $w) = qx/stty size/.words».Int;
my $buf = $w * $h;
my @buffer = 0 xx $buf;
my $done;
 
@buffer[$w * ($h div 2) + ($w div 2) - 1] = $stack;
 
print "\e[?25l\e[48;5;232m";
 
repeat {
$done = True;
loop (my int $row; $row < $h; $row = $row + 1) {
my int $rs = $row * $w; # row start
my int $re = $rs + $w; # row end
loop (my int $idx = $rs; $idx < $re; $idx = $idx + 1) {
if @buffer[$idx] >= 4 {
++@buffer[ $idx - $w ] if $row > 0;
++@buffer[ $idx - 1 ] if $idx - 1 > $rs;
++@buffer[ $idx + $w ] if $row < $h - 1;
++@buffer[ $idx + 1 ] if $idx + 1 < $buf;
@buffer[ $idx ] -= 4;
$done = False;
}
}
}
unless $hide-progress {
print join '', @buffer.map( { @color[$_ min 4] })
}
} until $done;
 
print join '', @buffer.map( { @color[$_ min 4] });
 
cleanup;

Passing in 2048 as a stack size results in: Abelian-sandpile-model-perl6.png (offsite .png image)

Phix[edit]

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.exw
include pGUI.e
 
Ihandle dlg, canvas
cdCanvas 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][2] += 1
end for
elsif x=length(board[1]) 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[$])
for i=1 to length(moves) do
moves[i][1] += 1
end for
elsif y=length(board) then -- extend down
board = append(board,board[1])
end if
if length(moves)=0 then exit end if
{y,x} = moves[$]
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[1])/2)+1
drop(y,x)
return IUP_DEFAULT
end 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[1]) 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_DEFAULT
end 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_DEFAULT
end 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[edit]

 
import numpy as np
import 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[edit]

// 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:

                
                
                
                
                
       ░        
      ▒░▒       
     ░░ ░░      
      ▒░▒       
       ░