Percolation/Site percolation

From Rosetta Code
Task
Percolation/Site percolation
You are encouraged to solve this task according to the task description, using any language you may know.

Percolation Simulation
This is a simulation of aspects of mathematical percolation theory.

For other percolation simulations, see Category:Percolation Simulations, or:
1D finite grid simulation
Mean run density
2D finite grid simulations

Site percolation | Bond percolation | Mean cluster density

Given an rectangular array of cells numbered assume is horizontal and is downwards.

Assume that the probability of any cell being filled is a constant where

The task

Simulate creating the array of cells with probability and then testing if there is a route through adjacent filled cells from any on row to any on row , i.e. testing for site percolation.

Given repeat the percolation times to estimate the proportion of times that the fluid can percolate to the bottom for any given .

Show how the probability of percolating through the random grid changes with going from to in increments and with the number of repetitions to estimate the fraction at any given as .

Use an grid of cells for all cases.

Optionally depict a percolation through a cell grid graphically.

Show all output on this page.

C[edit]

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
 
char *cell, *start, *end;
int m, n;
 
void make_grid(int x, int y, double p)
{
int i, j, thresh = p * RAND_MAX;
 
m = x, n = y;
end = start = realloc(start, (x+1) * (y+1) + 1);
 
memset(start, 0, m + 1);
 
cell = end = start + m + 1;
for (i = 0; i < n; i++) {
for (j = 0; j < m; j++)
*end++ = rand() < thresh ? '+' : '.';
*end++ = '\n';
}
 
end[-1] = 0;
end -= ++m; // end is the first cell of bottom row
}
 
int ff(char *p) // flood fill
{
if (*p != '+') return 0;
 
*p = '#';
return p >= end || ff(p+m) || ff(p+1) || ff(p-1) || ff(p-m);
}
 
int percolate(void)
{
int i;
for (i = 0; i < m && !ff(cell + i); i++);
return i < m;
}
 
int main(void)
{
make_grid(15, 15, .5);
percolate();
 
puts("15x15 grid:");
puts(cell);
 
puts("\nrunning 10,000 tests for each case:");
 
double p;
int ip, i, cnt;
for (ip = 0; ip <= 10; ip++) {
p = ip / 10.;
for (cnt = i = 0; i < 10000; i++) {
make_grid(15, 15, p);
cnt += percolate();
}
printf("p=%.1f:  %.4f\n", p, cnt / 10000.);
}
 
return 0;
}
Output:
15x15 grid:
.#...##.#.#.#..
...+.###.####.#
...+..#.+...#.#
+..+..##..#####
+...+.#....##..
.+..+.##..##.+.
....+.#...##..+
..+.+.#####.++.
+++....#.###.++
.+.+.#.#.##....
..++.####...++.
+.+.+.##..+++..
+..+.+..+.....+
..........++..+
.+.+.++++.+...+

running 10,000 tests for each case:
p=0.0:  0.0000
p=0.1:  0.0000
p=0.2:  0.0000
p=0.3:  0.0000
p=0.4:  0.0032
p=0.5:  0.0902
p=0.6:  0.5771
p=0.7:  0.9587
p=0.8:  0.9996
p=0.9:  1.0000
p=1.0:  1.0000
Translation of: D
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
#include <string.h>
#include <stdbool.h>
 
#define N_COLS 15
#define N_ROWS 15
 
// Probability granularity 0.0, 0.1, ... 1.0
#define N_STEPS 11
 
// Simulation tries
#define N_TRIES 100
 
typedef unsigned char Cell;
enum { EMPTY_CELL = ' ',
FILLED_CELL = '#',
VISITED_CELL = '.' };
typedef Cell Grid[N_ROWS][N_COLS];
 
void initialize(Grid grid, const double probability) {
for (size_t r = 0; r < N_ROWS; r++)
for (size_t c = 0; c < N_COLS; c++) {
const double rnd = rand() / (double)RAND_MAX;
grid[r][c] = (rnd < probability) ? EMPTY_CELL : FILLED_CELL;
}
}
 
void show(Grid grid) {
char line[N_COLS + 3];
memset(&line[0], '-', N_COLS + 2);
line[0] = '+';
line[N_COLS + 1] = '+';
line[N_COLS + 2] = '\0';
 
printf("%s\n", line);
for (size_t r = 0; r < N_ROWS; r++) {
putchar('|');
for (size_t c = 0; c < N_COLS; c++)
putchar(grid[r][c]);
puts("|");
}
printf("%s\n", line);
}
 
bool walk(Grid grid, const size_t r, const size_t c) {
const size_t bottom = N_ROWS - 1;
grid[r][c] = VISITED_CELL;
 
if (r < bottom && grid[r + 1][c] == EMPTY_CELL) { // Down.
if (walk(grid, r + 1, c))
return true;
} else if (r == bottom)
return true;
 
if (c && grid[r][c - 1] == EMPTY_CELL) // Left.
if (walk(grid, r, c - 1))
return true;
 
if (c < N_COLS - 1 && grid[r][c + 1] == EMPTY_CELL) // Right.
if (walk(grid, r, c + 1))
return true;
 
if (r && grid[r - 1][c] == EMPTY_CELL) // Up.
if (walk(grid, r - 1, c))
return true;
 
return false;
}
 
bool percolate(Grid grid) {
const size_t startR = 0;
for (size_t c = 0; c < N_COLS; c++)
if (grid[startR][c] == EMPTY_CELL)
if (walk(grid, startR, c))
return true;
return false;
}
 
typedef struct {
double prob;
size_t count;
} Counter;
 
int main() {
const double probability_step = 1.0 / (N_STEPS - 1);
Counter counters[N_STEPS];
for (size_t i = 0; i < N_STEPS; i++)
counters[i] = (Counter){ i * probability_step, 0 };
 
bool sample_shown = false;
static Grid grid;
srand(time(NULL));
 
for (size_t i = 0; i < N_STEPS; i++) {
for (size_t t = 0; t < N_TRIES; t++) {
initialize(grid, counters[i].prob);
if (percolate(grid)) {
counters[i].count++;
if (!sample_shown) {
printf("Percolating sample (%dx%d,"
" probability =%5.2f):\n",
N_COLS, N_ROWS, counters[i].prob);
show(grid);
sample_shown = true;
}
}
}
}
 
printf("\nFraction of %d tries that percolate through:\n", N_TRIES);
for (size_t i = 0; i < N_STEPS; i++)
printf("%1.1f %1.3f\n", counters[i].prob,
counters[i].count / (double)N_TRIES);
 
return 0;
}
 
Output:
Percolating sample (15x15, probability = 0.40):
+---------------+
|###.  #  # #  #|
|###.. #  ##### |
|   #. ######  #|
|###....  ######|
|######.  ### # |
| #####.######  |
|#......... ##  |
|...#...##.# ## |
|##.#...##.### #|
| ###..# #. #   |
|# #######. # ##|
|   # ##...#### |
| ##  # .#####  |
|#######.##  ###|
|# ##   .## # # |
+---------------+

Fraction of 100 tries that percolate through:
0.0 0.000
0.1 0.000
0.2 0.000
0.3 0.000
0.4 0.010
0.5 0.070
0.6 0.630
0.7 0.970
0.8 1.000
0.9 1.000
1.0 1.000

D[edit]

Translation of: Python
import std.stdio, std.random, std.array, std.datetime;
 
enum size_t nCols = 15,
nRows = 15,
nSteps = 11, // Probability granularity.
nTries = 20_000; // Simulation tries.
 
enum Cell : char { empty = ' ', filled = '#', visited = '.' }
alias Grid = Cell[nCols][nRows];
 
void initialize(ref Grid grid, in double probability, ref Xorshift rng) {
foreach (ref row; grid)
foreach (ref cell; row)
cell = (rng.uniform01 < probability) ? Cell.empty : Cell.filled;
}
 
void show(in ref Grid grid) @safe {
writefln("%(|%(%c%)|\n%)|", grid);
}
 
bool percolate(ref Grid grid) pure nothrow @safe @nogc {
bool walk(in size_t r, in size_t c) nothrow @safe @nogc {
enum bottom = nRows - 1;
grid[r][c] = Cell.visited;
 
if (r < bottom && grid[r + 1][c] == Cell.empty) { // Down.
if (walk(r + 1, c))
return true;
} else if (r == bottom)
return true;
 
if (c && grid[r][c - 1] == Cell.empty) // Left.
if (walk(r, c - 1))
return true;
 
if (c < nCols - 1 && grid[r][c + 1] == Cell.empty) // Right.
if (walk(r, c + 1))
return true;
 
if (r && grid[r - 1][c] == Cell.empty) // Up.
if (walk(r - 1, c))
return true;
 
return false;
}
 
enum startR = 0;
foreach (immutable c; 0 .. nCols)
if (grid[startR][c] == Cell.empty)
if (walk(startR, c))
return true;
return false;
}
 
void main() {
static struct Counter {
double prob;
size_t count;
}
 
StopWatch sw;
sw.start;
 
enum probabilityStep = 1.0 / (nSteps - 1);
Counter[nSteps] counters;
foreach (immutable i, ref co; counters)
co.prob = i * probabilityStep;
 
Grid grid;
bool sampleShown = false;
auto rng = Xorshift(unpredictableSeed);
 
foreach (ref co; counters) {
foreach (immutable _; 0 .. nTries) {
grid.initialize(co.prob, rng);
if (grid.percolate) {
co.count++;
if (!sampleShown) {
writefln("Percolating sample (%dx%d, probability =%5.2f):",
nCols, nRows, co.prob);
grid.show;
sampleShown = true;
}
}
}
}
sw.stop;
 
writefln("\nFraction of %d tries that percolate through:", nTries);
foreach (const co; counters)
writefln("%1.3f %1.3f", co.prob, co.count / double(nTries));
 
writefln("\nSimulations and grid printing performed" ~
" in %3.2f seconds.", sw.peek.msecs / 1000.0);
}
Output:
Percolating sample (15x15, probability = 0.40):
|#.###.##..#. # |
|#.###.# ###.  #|
|#.##..#####. ##|
|## ####  ...# #|
|# # #  ##.#..##|
|### # ## .#####|
|   ######.## ##|
|    ## #..###  |
|#### ##..##### |
|#   ###...  #  |
|### ## ##.   # |
|# ###  ##. ### |
|## ##### . ####|
|# ## #  #. ####|
|####### #.## ##|

Fraction of 20000 tries that percolate through:
0.000 0.000
0.100 0.000
0.200 0.000
0.300 0.000
0.400 0.004
0.500 0.090
0.600 0.565
0.700 0.958
0.800 1.000
0.900 1.000
1.000 1.000

Simulations and grid printing performed in 0.70 seconds.

Fortran[edit]

Please see sample compilation and program execution in comments at top of program. Thank you. This example demonstrates recursion and integer constants of a specific kind.

 
! loosely translated from python.
! compilation: gfortran -Wall -std=f2008 thisfile.f08
 
!$ a=site && gfortran -o $a -g -O0 -Wall -std=f2008 $a.f08 && $a
!100 trials per
!Fill Fraction goal(%) simulated through paths(%)
! 0 0
! 10 0
! 20 0
! 30 0
! 40 0
! 50 6
!
!
! b b b b h j m m m
! b b b b b h h m m m m m
! b b b h h h m
! b h h h h h h h
! b b h h h h h h h h h
! b b b h h h h h h h h h h
! b b @ h h h h h h h
! @ @ h h h h h h h h
! @ @ @ @ h h h h
! @ @ @ @ h h h h h h
! @ @ @ h h h h h h h
! @ @ @ h h h h h h
! @ h h h h h h
! @ h h h h h h h
! @ @ h h h h h h h h h h
! 60 59
! 70 97
! 80 100
! 90 100
! 100 100
 
program percolation_site
implicit none
integer, parameter :: m=15,n=15,t=100
!integer, parameter :: m=2,n=2,t=8
integer(kind=1), dimension(m, n) :: grid
real :: p
integer :: i, ip, trial, successes
logical :: success, unseen, q
data unseen/.true./
write(6,'(i3,a11)') t,' trials per'
write(6,'(a21,a30)') 'Fill Fraction goal(%)','simulated through paths(%)'
do ip=0, 10
p = ip/10.0
successes = 0
do trial = 1, t
call newgrid(grid, p)
success = .false.
do i=1, m
q = walk(grid, i) ! deliberately compute all paths
success = success .or. q
end do
if ((ip == 6) .and. unseen) then
call display(grid)
unseen = .false.
end if
successes = successes + merge(1, 0, success)
end do
write(6,'(9x,i3,24x,i3)')ip*10,nint(100*real(successes)/real(t))
end do
 
contains
 
logical function walk(grid, start)
integer(kind=1), dimension(m,n), intent(inout) :: grid
integer, intent(in) :: start
walk = rwalk(grid, 1, start, int(start+1,1))
end function walk
 
recursive function rwalk(grid, i, j, k) result(through)
logical :: through
integer(kind=1), dimension(m,n), intent(inout) :: grid
integer, intent(in) :: i, j
integer(kind=1), intent(in) :: k
logical, dimension(4) :: q
!out of bounds
through = .false.
if (i < 1) return
if (m < i) return
if (j < 1) return
if (n < j) return
!visited or non-pore
if (1_1 /= grid(i, j)) return
!update grid and recurse with neighbors. deny 'shortcircuit' evaluation
grid(i, j) = k
q(1) = rwalk(grid,i+0,j+1,k)
q(2) = rwalk(grid,i+0,j-1,k)
q(3) = rwalk(grid,i+1,j+0,k)
q(4) = rwalk(grid,i-1,j+0,k)
!newly discovered outlet
through = (i == m) .or. any(q)
end function rwalk
 
subroutine newgrid(grid, probability)
implicit none
real :: probability
integer(kind=1), dimension(m,n), intent(out) :: grid
real, dimension(m,n) :: harvest
call random_number(harvest)
grid = merge(1_1, 0_1, harvest < probability)
end subroutine newgrid
 
subroutine display(grid)
integer(kind=1), dimension(m,n), intent(in) :: grid
integer :: i, j, k, L
character(len=n*2) :: lineout
write(6,'(/)')
lineout = ' '
do i=1,m
do j=1,n
k = j+j
L = grid(i,j)+1
lineout(k:k) = ' @abcdefghijklmnopqrstuvwxyz'(L:L)
end do
write(6,*) lineout
end do
end subroutine display
 
end program percolation_site
 

Go[edit]

package main
 
import (
"bytes"
"fmt"
"math/rand"
"time"
)
 
func main() {
const (
m, n = 15, 15
t = 1e4
minp, maxp, Δp = 0, 1, 0.1
)
 
rand.Seed(2) // Fixed seed for repeatable example grid
g := NewGrid(.5, m, n)
g.Percolate()
fmt.Println(g)
 
rand.Seed(time.Now().UnixNano()) // could pick a better seed
for p := float64(minp); p < maxp; p += Δp {
count := 0
for i := 0; i < t; i++ {
g := NewGrid(p, m, n)
if g.Percolate() {
count++
}
}
fmt.Printf("p=%.2f, %.4f\n", p, float64(count)/t)
}
}
 
const (
full = '.'
used = '#'
empty = ' '
)
 
type grid struct {
cell [][]byte // row first, i.e. [y][x]
}
 
func NewGrid(p float64, xsize, ysize int) *grid {
g := &grid{cell: make([][]byte, ysize)}
for y := range g.cell {
g.cell[y] = make([]byte, xsize)
for x := range g.cell[y] {
if rand.Float64() < p {
g.cell[y][x] = full
} else {
g.cell[y][x] = empty
}
}
}
return g
}
 
func (g *grid) String() string {
var buf bytes.Buffer
// Don't really need to call Grow but it helps avoid multiple
// reallocations if the size is large.
buf.Grow((len(g.cell) + 2) * (len(g.cell[0]) + 3))
 
buf.WriteByte('+')
for _ = range g.cell[0] {
buf.WriteByte('-')
}
buf.WriteString("+\n")
 
for y := range g.cell {
buf.WriteByte('|')
buf.Write(g.cell[y])
buf.WriteString("|\n")
}
 
buf.WriteByte('+')
ly := len(g.cell) - 1
for x := range g.cell[ly] {
if g.cell[ly][x] == used {
buf.WriteByte(used)
} else {
buf.WriteByte('-')
}
}
buf.WriteByte('+')
return buf.String()
}
 
func (g *grid) Percolate() bool {
for x := range g.cell[0] {
if g.use(x, 0) {
return true
}
}
return false
}
 
func (g *grid) use(x, y int) bool {
if y < 0 || x < 0 || x >= len(g.cell[0]) || g.cell[y][x] != full {
return false // Off the edges, empty, or used
}
g.cell[y][x] = used
if y+1 == len(g.cell) {
return true // We're on the bottom
}
 
// Try down, right, left, up in that order.
return g.use(x, y+1) ||
g.use(x+1, y) ||
g.use(x-1, y) ||
g.use(x, y-1)
}
Output:
+---------------+
|####  ###.  .. |
| ##   # #   . .|
| ###   #### .. |
|### ##### #..  |
|   ### # ## .. |
|# ##     # . ..|
|### .    #.. . |
| ##      ##. ..|
| ## .. .. # .. |
| ##    . .#....|
|##   .. .##  . |
|# .  . . # .   |
| ..   . .#. .. |
|. . .... #  .. |
| . ..  . # .. .|
+---------#-----+
p=0.00, 0.0000
p=0.10, 0.0000
p=0.20, 0.0000
p=0.30, 0.0000
p=0.40, 0.0040
p=0.50, 0.0980
p=0.60, 0.5641
p=0.70, 0.9583
p=0.80, 0.9995
p=0.90, 1.0000
p=1.00, 1.0000

Haskell[edit]

{-# LANGUAGE OverloadedStrings #-}
import Control.Monad
import Control.Monad.Random
import Data.Array.Unboxed
import Data.List
import Formatting
 
type Field = UArray (Int, Int) Char
 
-- Start percolating some seepage through a field.
-- Recurse to continue percolation with new seepage.
percolateR :: [(Int, Int)] -> Field -> (Field, [(Int,Int)])
percolateR [] f = (f, [])
percolateR seep f =
let ((xLo,yLo),(xHi,yHi)) = bounds f
validSeep = filter (\p@(x,y) -> x >= xLo
&& x <= xHi
&& y >= yLo
&& y <= yHi
&& f!p == ' ') $ nub $ sort seep
 
neighbors (x,y) = [(x,y-1), (x,y+1), (x-1,y), (x+1,y)]
 
in percolateR
(concatMap neighbors validSeep)
(f // map (\p -> (p,'.')) validSeep)
 
-- Percolate a field. Return the percolated field.
percolate :: Field -> Field
percolate start =
let ((_,_),(xHi,_)) = bounds start
(final, _) = percolateR [(x,0) | x <- [0..xHi]] start
in final
 
-- Generate a random field.
initField :: Int -> Int -> Double -> Rand StdGen Field
initField w h threshold = do
frnd <- fmap (\rv -> if rv<threshold then ' ' else '#') <$> getRandoms
return $ listArray ((0,0), (w-1, h-1)) frnd
 
-- Get a list of "leaks" from the bottom of a field.
leaks :: Field -> [Bool]
leaks f =
let ((xLo,_),(xHi,yHi)) = bounds f
in [f!(x,yHi)=='.'| x <- [xLo..xHi]]
 
-- Run test once; Return bool indicating success or failure.
oneTest :: Int -> Int -> Double -> Rand StdGen Bool
oneTest w h threshold =
or.leaks.percolate <$> initField w h threshold
 
-- Run test multple times; Return the number of tests that pass.
multiTest :: Int -> Int -> Int -> Double -> Rand StdGen Double
multiTest testCount w h threshold = do
results <- replicateM testCount $ oneTest w h threshold
let leakyCount = length $ filter id results
return $ fromIntegral leakyCount / fromIntegral testCount
 
-- Display a field with walls and leaks.
showField :: Field -> IO ()
showField a = do
let ((xLo,yLo),(xHi,yHi)) = bounds a
mapM_ print [ [ a!(x,y) | x <- [xLo..xHi]] | y <- [yLo..yHi]]
 
main :: IO ()
main = do
g <- getStdGen
let w = 15
h = 15
threshold = 0.6
(startField, g2) = runRand (initField w h threshold) g
 
putStrLn ("Unpercolated field with " ++ show threshold ++ " threshold.")
putStrLn ""
showField startField
 
putStrLn ""
putStrLn "Same field after percolation."
putStrLn ""
showField $ percolate startField
 
let testCount = 10000
densityCount = 10
 
putStrLn ""
putStrLn ( "Results of running percolation test " ++ show testCount
++ " times with thresholds ranging from 0/" ++ show densityCount
++ " to " ++ show densityCount ++ "/" ++ show densityCount ++ " .")
 
let densities = [0..densityCount]
tests = sequence [multiTest testCount w h v
| density <- densities,
let v = fromIntegral density / fromIntegral densityCount ]
results = zip densities (evalRand tests g2)
mapM_ print [format ("p=" % int % "/" % int % " -> " % fixed 4) density densityCount x | (density,x) <- results]
Output:
Unpercolated field with 0.6 threshold.

"  ### # # # ## "
"### # ##  # #  "
"  #####  #   ##"
"#    # ##  #   "
"###         #  "
" ### ### #     "
"  ### #  ### ##"
"   # ## #    ##"
" #  #   # #  ##"
"### ## #       "
"   ## #      ##"
" #    # ## ## #"
" ### ##  ##    "
"#### # #  ## ##"
"   #    #    # "

Same field after percolation.

"..###.#.#.#.##."
"### #.##..#.#.."
"..#####..#...##"
"#....#.##..#..."
"###.........#.."
" ###.###.#....."
"  ### #..###.##"
"   # ##.#....##"
" #  #...#.#..##"
"### ##.#......."
"   ## #......##"
" #    #.##.##.#"
" ### ##..##...."
"#### # #..##.##"
"   #    #....# "

Results of running percolation test 10000 times with thresholds ranging from 0/10 to 10/10 .
"p=0/10 -> 0.0000"
"p=1/10 -> 0.0000"
"p=2/10 -> 0.0000"
"p=3/10 -> 0.0000"
"p=4/10 -> 0.0028"
"p=5/10 -> 0.0910"
"p=6/10 -> 0.5684"
"p=7/10 -> 0.9572"
"p=8/10 -> 0.9997"
"p=9/10 -> 1.0000"
"p=10/10 -> 1.0000"

J[edit]

One approach:

groups=:[: +/\ 2 </\ 0 , *
ooze=: [ >. [ +&* [ * [: ; groups@[ <@(* * 2 < >./)/. +
percolate=: ooze/\.@|.^:2^:_@(* (1 + # {. 1:))
 
trial=: percolate@([ >: ]?@$0:)
simulate=: %@[ * [: +/ (2 e. {:)@trial&15 15"0@#

Example Statistics:

   ,.'  P THRU';(, 100&simulate)"0 (i.%<:)11
┌────────┐
│ P THRU│
├────────┤
0 0
0.1 0
0.2 0
0.3 0
0.4 0.01
0.5 0.09
0.6 0.61
0.7 0.97
0.8 1
0.9 1
1 1
└────────┘

Worked sample:

   1j1 #"1 ' .#'{~ percolate 0.6>:?15 15$0
# # # # # # # #
# # # # # # # # # # # #
# # # # # # # #
# # # # # # # # #
# . # # # # # #
# # # # # # # # # #
# # # # # # # # # # # # #
# # # # # # # # # #
. # #
. . # # # #
. . . . # # # # # # # # #
. . . . # # # # # # #
. . . # . # # #
. . . . . . . # # .
. . . . . . . . # #

An explanation with examples would be somewhat longer than the implementation.

Alternative implementation (with an incompatible internal API):

 
any =: +./
all =: *./
 
quickCheck =: [: all [: (any"1) 2 *./\ ] NB. a complete path requires connections between all row pairs
 
percolate =: 15 15&$: : (dyad define) NB. returns 0 iff blocked Use: (N, M) percolate P
NB. make a binary grid
GRID =: y (> ?@($&0)) x
 
NB. compute the return value
if. -. quickCheck GRID do. 0 return. end.
STARTING_SITES =. 0 ,. ({. GRID) # i. {: x NB. indexes of 1 in head row of GRID
any STARTING_SITES check GRID
)
 
 
NB. use local copy of GRID. Too slow.
check =: dyad define"1 2 NB. return 1 iff through path found use: START check GRID
GRID =. y
LOCATION =. x
if. 0 (= #) LOCATION do. 0 return. end. NB. no starting point? 0
if. LOCATION any@:((>: , 0 > [) $) GRID do. 0 return. end. NB. off grid? 0
INDEX =. <LOCATION
if. 1 ~: INDEX { GRID do. 0 return. end. NB. fail. either already looked here or non-path
if. (>: {. LOCATION) = (# GRID) do. 1 return. end. NB. Success! (display GRID here)
G =: GRID =. INDEX (>:@:{)`[`]}GRID
any GRID check~ LOCATION +"1 (, -)0 1,:1 0
)
 
NB. use global GRID.
check =: dyad define"1 2 NB. return 1 iff through path found use: START check GRID
LOCATION =. x
if. 0 (= #) LOCATION do. 0 return. end. NB. no starting point? 0
if. LOCATION any@:((>: , 0 > [) $) GRID do. 0 return. end. NB. off grid? 0
INDEX =. <LOCATION
if. 1 ~: INDEX { GRID do. 0 return. end. NB. fail. either already looked here or non-path
if. (>: {. LOCATION) = (# GRID) do. 1 return. end. NB. Success! (display GRID here)
GRID =: INDEX (>:@:{)`[`]}GRID
any GRID check~ LOCATION +"1 (, -)0 1,:1 0
)
 
simulate =: 100&$: : ([ %~ [: +/ [: percolate"0 #) NB. return fraction of connected cases. Use: T simulate P
 
   ,. '   P  THRU' ; (, 100x&simulate)"0 (i. % <:)11x
+-----------+
|   P  THRU |
+-----------+
|   0      0|
|1r10      0|
| 1r5      0|
|3r10      0|
| 2r5  1r100|
| 1r2   1r20|
| 3r5  31r50|
|7r10 97r100|
| 4r5      1|
|9r10      1|
|   1      1|
+-----------+
   


   NB. example

   simulate 0.6
0.51

   GRID  NB. the final grid of the 100 simulated cases.
2 2 2 2 0 2 2 2 2 0 2 2 0 0 2
2 0 0 2 0 0 2 0 2 0 2 0 0 1 0
2 0 1 0 2 2 0 2 2 0 2 2 0 1 0
2 2 0 0 0 2 2 0 2 0 2 0 0 0 1
0 2 2 2 2 2 2 0 2 0 2 0 0 1 1
0 2 0 2 0 0 0 0 0 0 0 1 1 0 1
0 0 0 0 1 0 0 1 1 1 0 1 1 0 1
1 1 1 1 1 1 1 0 0 0 1 1 0 1 1
0 0 1 1 1 0 1 1 0 0 1 1 1 0 1
0 1 0 1 1 1 1 1 0 0 1 1 1 1 1
1 1 1 1 0 1 1 0 1 1 0 0 1 1 1
0 1 1 1 0 1 1 0 0 0 1 1 1 1 1
0 0 0 0 1 0 1 1 1 1 1 0 0 1 0
1 1 1 1 0 1 1 1 1 1 0 1 0 0 0
1 0 1 1 1 1 1 0 0 1 1 1 1 1 1

   
   (0 ,. 0 6 10 14) check GRID  NB. show possible starting points all fail
0 0 0 0
   
   1j1#"1 GRID { '#',~u: 32 16bb7 NB. sample paths with unicode pepper.
# # # #   # # # #   # #     #             
#     #     #   #   #     ·              
#   ·   # #   # #   # #   ·             
# #       # #   #   #       ·            
  # # # # # #   #   #     · ·           
  #   #               · ·   ·          
        ·     · · ·   · ·   ·      
· · · · · · ·       · ·   · ·  
    · · ·   · ·     · · ·   ·    
  ·   · · · · ·     · · · · ·  
· · · ·   · ·   · ·     · · ·  
  · · ·   · ·       · · · · ·   
        ·   · · · · ·     ·        
· · · ·   · · · · ·   ·         
·   · · · · ·     · · · · · · 

Perl 6[edit]

Works with: Rakudo version 2017.02
my $block = '▒';
my $water = '+';
my $pore = ' ';
my $grid = 15;
my @site;
 
enum Direction <DeadEnd Up Right Down Left>;
 
say 'Sample percolation at .6';
percolate(.6);
.join.say for @site;
say "\n";
 
my $tests = 1000;
say "Doing $tests trials at each porosity:";
for .1, .2 ... 1 -> $p {
printf "p = %0.1f: %0.3f\n", $p, (sum percolate($p) xx $tests) / $tests
}
 
sub infix:<deq> ( $a, $b ) { $a.defined && ($a eq $b) }
 
sub percolate ( $prob = .6 ) {
@site[0] = [$pore xx $grid];
@site[$grid + 1] = [$pore xx $grid];
 
for ^$grid X 1..$grid -> ($x, $y) {
@site[$y;$x] = rand < $prob ?? $pore !! $block
}
@site[0;0] = $water;
 
my @stack;
my $current = [0;0];
 
loop {
if my $dir = direction( $current ) {
@stack.push: $current;
$current = move( $dir, $current )
}
else {
return 0 unless @stack;
$current = @stack.pop
}
return 1 if $current[1] > $grid
}
 
sub direction( [$x, $y] ) {
(Down if @site[$y + 1][$x] deq $pore) ||
(Left if @site[$y][$x - 1] deq $pore) ||
(Right if @site[$y][$x + 1] deq $pore) ||
(Up if @site[$y - 1][$x] deq $pore) ||
DeadEnd
}
 
sub move ( $dir, @cur ) {
my ( $x, $y ) = @cur;
given $dir {
when Up { @site[--$y][$x] = $water }
when Down { @site[++$y][$x] = $water }
when Left { @site[$y][--$x] = $water }
when Right { @site[$y][++$x] = $water }
}
[$x, $y]
}
}
Output:
Sample percolation at .6
++++           
▒▒▒+ ▒ ▒ ▒ ▒ ▒▒
 ▒▒++ ▒▒   ▒▒  
   ▒+   ▒▒ ▒ ▒▒
▒▒ ▒++++▒ ▒▒   
 ▒ ▒+▒▒+▒   ▒  
  ▒++▒++ ▒▒▒ ▒ 
  ▒▒▒ +▒       
▒▒ ▒ ▒++ ▒   ▒▒
▒▒▒▒▒▒▒+▒▒▒    
▒   ▒  +   ▒   
 ▒▒   ▒+ ▒  ▒ ▒
▒  ▒ ▒▒+    ▒  
▒▒ ▒ ▒++▒   ▒  
   ▒  +▒ ▒▒  ▒▒
▒  ▒▒▒+    ▒▒ ▒
      +        


Doing 1000 trials at each porosity:
p = 0.1: 0.000
p = 0.2: 0.000
p = 0.3: 0.000
p = 0.4: 0.005
p = 0.5: 0.096
p = 0.6: 0.573
p = 0.7: 0.959
p = 0.8: 0.999
p = 0.9: 1.000
p = 1.0: 1.000

Python[edit]

from random import random
import string
from pprint import pprint as pp
 
M, N, t = 15, 15, 100
 
cell2char = ' #' + string.ascii_letters
NOT_VISITED = 1 # filled cell not walked
 
class PercolatedException(Exception): pass
 
def newgrid(p):
return [[int(random() < p) for m in range(M)] for n in range(N)] # cell
 
def pgrid(cell, percolated=None):
for n in range(N):
print( '%i) ' % (n % 10)
+ ' '.join(cell2char[cell[n][m]] for m in range(M)))
if percolated:
where = percolated.args[0][0]
print('!) ' + ' ' * where + cell2char[cell[n][where]])
 
def check_from_top(cell):
n, walk_index = 0, 1
try:
for m in range(M):
if cell[n][m] == NOT_VISITED:
walk_index += 1
walk_maze(m, n, cell, walk_index)
except PercolatedException as ex:
return ex
return None
 
 
def walk_maze(m, n, cell, indx):
# fill cell
cell[n][m] = indx
# down
if n < N - 1 and cell[n+1][m] == NOT_VISITED:
walk_maze(m, n+1, cell, indx)
# THE bottom
elif n == N - 1:
raise PercolatedException((m, indx))
# left
if m and cell[n][m - 1] == NOT_VISITED:
walk_maze(m-1, n, cell, indx)
# right
if m < M - 1 and cell[n][m + 1] == NOT_VISITED:
walk_maze(m+1, n, cell, indx)
# up
if n and cell[n-1][m] == NOT_VISITED:
walk_maze(m, n-1, cell, indx)
 
if __name__ == '__main__':
sample_printed = False
pcount = {}
for p10 in range(11):
p = p10 / 10.0
pcount[p] = 0
for tries in range(t):
cell = newgrid(p)
percolated = check_from_top(cell)
if percolated:
pcount[p] += 1
if not sample_printed:
print('\nSample percolating %i x %i, p = %5.2f grid\n' % (M, N, p))
pgrid(cell, percolated)
sample_printed = True
print('\n p: Fraction of %i tries that percolate through\n' % t )
 
pp({p:c/float(t) for p, c in pcount.items()})
Output:

The Ascii art grid of cells has blanks for cells that were not filled. Filled cells start off as the '#', hash character and are changed to a succession of printable characters by successive tries to navigate from the top, (top - left actually), filled cell to the bottom.

The '!)' row shows where the percolation finished and you can follow the letter backwards from that row, (letter 'c' in this case), to get the route. The program stops after finding its first route through.

Sample percolating 15 x 15, p =  0.40 grid

0)    a a a       b   c #        
1)    a a   #         c c   #   #
2)        # #   # #     c # # #  
3)  #   #       # # #   c        
4)    #     #         c c c c c c
5)  # # # # # #         c   c   c
6)        # # #         c   c   c
7)  #   #     # #     #   #   # c
8)  #   # #     #   #       c c c
9)    #       #         #   c    
0)  #       #   # # # #   c c # #
1)      #     #   #     # c      
2)  #     # # # # #   c c c   c  
3)  #   # # #         c   c c c  
4)      #           # c         #
!)                    c

 p: Fraction of 100 tries that percolate through

{0.0: 0.0,
 0.1: 0.0,
 0.2: 0.0,
 0.3: 0.0,
 0.4: 0.01,
 0.5: 0.11,
 0.6: 0.59,
 0.7: 0.94,
 0.8: 1.0,
 0.9: 1.0,
 1.0: 1.0}

Note the abrupt change in percolation at around p = 0.6. These abrupt changes are expected.

Racket[edit]

#lang racket
(require racket/require (only-in racket/fixnum for*/fxvector))
(require (filtered-in (lambda (name) (regexp-replace #rx"unsafe-" name ""))
racket/unsafe/ops))
 
(define cell-empty 0)
(define cell-filled 1)
(define cell-wall 2)
(define cell-visited 3)
(define cell-exit 4)
 
(define ((percol->generator p)) (if (< (random) p) cell-filled cell-empty))
 
(define t (make-parameter 1000))
 
(define ((make-percol-grid M N) p)
(define p->10 (percol->generator p))
(define M+1 (fx+ 1 M))
(define M+2 (fx+ 2 M))
(for*/fxvector
#:length (fx* N M+2)
((n (in-range N)) (m (in-range M+2)))
(cond
[(fx= 0 m) cell-wall]
[(fx= m M+1) cell-wall]
[else (p->10)])))
 
(define (cell->str c) (substring " #|+*" c (fx+ 1 c)))
 
(define ((draw-percol-grid M N) g)
(define M+2 (fx+ M 2))
(for ((row N))
(for ((col (in-range M+2)))
(define idx (fx+ (fx* M+2 row) col))
(printf "~a" (cell->str (fxvector-ref g idx))))
(newline)))
 
(define ((percolate-percol-grid?! M N) g)
(define M+2 (fx+ M 2))
(define N-1 (fx- N 1))
(define max-idx (fx* N M+2))
(define (inner-percolate g idx)
(define row (fxquotient idx M+2))
(cond
((fx< idx 0) #f)
((fx>= idx max-idx) #f)
((fx= N-1 row) (fxvector-set! g idx cell-exit) #t)
((fx= cell-filled (fxvector-ref g idx))
(fxvector-set! g idx cell-visited)
(or
 ; gravity first (thanks Mr Newton)
(inner-percolate g (fx+ idx M+2))
 ; stick-to-the-left
(inner-percolate g (fx- idx 1))
(inner-percolate g (fx+ idx 1))
 ; go uphill only if we have to!
(inner-percolate g (fx- idx M+2))))
(else #f)))
(for/first ((m (in-range 1 M+2)) #:when (inner-percolate g m)) g))
 
(define make-15x15-grid (make-percol-grid 15 15))
(define draw-15x15-grid (draw-percol-grid 15 15))
(define perc-15x15-grid?! (percolate-percol-grid?! 15 15))
 
(define (display-sample-percolation p)
(printf "Percolation sample: p=~a~%" p)
(for*/first
((i (in-naturals))
(g (in-value (make-15x15-grid 0.6)))
#:when (perc-15x15-grid?! g))
(draw-15x15-grid g))
(newline))
 
(display-sample-percolation 0.4)
 
(for ((p (sequence-map (curry * 1/10) (in-range 0 (add1 10)))))
(define n-percolated-grids
(for/sum
((i (in-range (t))) #:when (perc-15x15-grid?! (make-15x15-grid p))) 1))
(define proportion-percolated (/ n-percolated-grids (t)))
(printf "p=~a\t->\t~a~%" p (real->decimal-string proportion-percolated 4)))
Output:
Percolation sample: p=0.4
|+++ ++++  + +++|
| +++ ++ #     +|
|   +  ++   ##++|
| ##    +  ###+ |
| ###### + #+++#|
|  ##### +  +  #|
|## # # +++++## |
|### # ++ +++#  |
|##  ## +++++#  |
|# ###   ++ +  #|
| ## ## +++   ##|
|##  ##  +++ # #|
|###   #   +### |
|####  ####+  # |
|# ## #    *#  #|

p=0	->	0.0000
p=1/10	->	0.0000
p=1/5	->	0.0000
p=3/10	->	0.0000
p=2/5	->	0.0030
p=1/2	->	0.1110
p=3/5	->	0.5830
p=7/10	->	0.9530
p=4/5	->	1.0000
p=9/10	->	1.0000
p=1	->	1.0000

Tcl[edit]

Works with: Tcl version 8.6
package require Tcl 8.6
 
oo::class create SitePercolation {
variable cells w h
constructor {width height probability} {
set w $width
set h $height
for {set cells {}} {[llength $cells] < $h} {lappend cells $row} {
for {set row {}} {[llength $row] < $w} {lappend row $cell} {
set cell [expr {rand() < $probability}]
}
}
}
method print {out} {
array set map {0 "#" 1 " " -1 .}
puts "+[string repeat . $w]+"
foreach row $cells {
set s "|"
foreach cell $row {
append s $map($cell)
}
puts [append s "|"]
}
set outline [lrepeat $w "-"]
foreach index $out {
lset outline $index "."
}
puts "+[join $outline {}]+"
}
method percolate {} {
for {set work {}; set i 0} {$i < $w} {incr i} {
if {[lindex $cells 0 $i]} {lappend work 0 $i}
}
try {
my Fill $work
return {}
} trap PERCOLATED x {
return [list $x]
}
}
method Fill {queue} {
while {[llength $queue]} {
set queue [lassign $queue y x]
if {$y >= $h} {throw PERCOLATED $x}
if {$y < 0 || $x < 0 || $x >= $w} continue
if {[lindex $cells $y $x]<1} continue
lset cells $y $x -1
lappend queue [expr {$y+1}] $x [expr {$y-1}] $x
lappend queue $y [expr {$x-1}] $y [expr {$x+1}]
}
}
}
 
# Demonstrate one run
puts "Sample percolation, 15x15 p=0.6"
SitePercolation create bp 15 15 0.6
bp print [bp percolate]
bp destroy
puts ""
 
# Collect statistics
apply {{} {
puts "Percentage of tries that percolate, varying p"
set tries 100
for {set pint 0} {$pint <= 10} {incr pint} {
set p [expr {$pint * 0.1}]
set tot 0
for {set i 0} {$i < $tries} {incr i} {
set bp [SitePercolation new 15 15 $p]
if {[$bp percolate] ne ""} {
incr tot
}
$bp destroy
}
puts [format "p=%.2f: %2.1f%%" $p [expr {$tot*100./$tries}]]
}
}}
Output:
Sample percolation, 15x15 p=0.6
+...............+
|.##...###.##...|
|.#.#####.####..|
|............##.|
|....###.###.#..|
|.#.##..#....#..|
|#.........#..#.|
|..#...##.##....|
|#.#.#....##...#|
|###.....#.#...#|
|.....##........|
|.#.#..## ......|
|  #..## # .##.#|
| # #.#  ####...|
|# #  # #  ##...|
| ###   ##  # . |
+-------------.-+

Percentage of tries that percolate, varying p
p=0.00: 0.0%
p=0.10: 0.0%
p=0.20: 0.0%
p=0.30: 0.0%
p=0.40: 0.0%
p=0.50: 6.0%
p=0.60: 54.0%
p=0.70: 98.0%
p=0.80: 100.0%
p=0.90: 100.0%
p=1.00: 100.0%

zkl[edit]

Translation of: C
fcn makeGrid(m,n,p){
grid:=Data((m+1)*(n+1)); // first row and right edges are buffers
grid.write(" "*m); grid.write("\r");
do(n){
do(m){ grid.write(((0.0).random(1)<p) and "+" or "."); } // cell is porous or not
grid.write("\n");
}
grid
}
fcn ff(grid,x,m){ // walk across row looking for a porous cell
if(grid[x]!=43) return(0); // '+' == 43 ASCII == porous
grid[x]="#";
return(x+m>=grid.len() or
ff(grid,x+m,m) or ff(grid,x+1,m) or ff(grid,x-1,m) or ff(grid,x-m,m));
}
fcn percolate(grid,m){
x:=m+1; i:=0; while(i<m and not ff(grid,x,m)){ x+=1; i+=1; }
return(i<m); // percolated through the grid?
}
 
grid:=makeGrid(15,15,0.60);
println("Did liquid percolate: ",percolate(grid,15));
println("15x15 grid:\n",grid.text);
 
println("Running 10,000 tests for each case:");
foreach p in ([0.0 .. 1.0, 0.1]){
cnt:=0.0; do(10000){ cnt+=percolate(makeGrid(15,15,p),15); }
"p=%.1f:  %.4f".fmt(p, cnt/10000).println();
}
Output:
Did liquid percolate: True
15x15 grid:
.###.##.#++..++
......+###..+.+
+...+...##..+++
++..+.+.#+.+.++
..+++###..+..++
.+.##..++.+..++
.+#.+..++++++..
+####+..+....++
.#.#..+..++.+.+
#.#++++.+++.+++
+#++..+.+.+.+++
#######..++++++
#.##.#+++...+..
+.#.#+++.++.+++
+.+#+.++..+..++

Running 10,000 tests for each case:
p=0.0:  0.0000
p=0.1:  0.0000
p=0.2:  0.0000
p=0.3:  0.0000
p=0.4:  0.0006
p=0.5:  0.0304
p=0.6:  0.2989
p=0.7:  0.8189
p=0.8:  0.9903
p=0.9:  1.0000
p=1.0:  1.0000