Percolation/Bond percolation: Difference between revisions
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R Int(:seed >> 16) / Float(FF'FF) |
R Int(:seed >> 16) / Float(FF'FF) |
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| ⚫ | |||
T Grid |
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| ⚫ | |||
F (cell, hwall, vwall) |
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.cell = cell |
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.hwall = hwall |
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.vwall = vwall |
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V (M, nn, t) = (10, 10, 100) |
V (M, nn, t) = (10, 10, 100) |
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print(‘!) ’(‘ ’ * where)‘ ’f[1]) |
print(‘!) ’(‘ ’ * where)‘ ’f[1]) |
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F flood_fill(m, n, &cell, hwall, vwall) -> |
F flood_fill(m, n, &cell, hwall, vwall) -> Void |
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cell[n][m] = 1 |
cell[n][m] = 1 |
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I n < :nn - 1 & !hwall[n + 1][m] & !cell[n + 1][m] |
I n < :nn - 1 & !hwall[n + 1][m] & !cell[n + 1][m] |
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flood_fill(m, n + 1, &cell, hwall, vwall) |
flood_fill(m, n + 1, &cell, hwall, vwall) |
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E I n == :nn - 1 & !hwall[n + 1][m] |
E I n == :nn - 1 & !hwall[n + 1][m] |
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X PercolatedException((m, n + 1)) |
X.throw PercolatedException((m, n + 1)) |
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I m & !vwall[n][m] & !cell[n][m - 1] |
I m & !vwall[n][m] & !cell[n][m - 1] |
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flood_fill(m - 1, n, &cell, hwall, vwall) |
flood_fill(m - 1, n, &cell, hwall, vwall) |
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{{trans|Kotlin}} |
{{trans|Kotlin}} |
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{{libheader|Wren-fmt}} |
{{libheader|Wren-fmt}} |
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<syntaxhighlight lang=" |
<syntaxhighlight lang="wren">import "random" for Random |
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import "/fmt" for Fmt |
import "./fmt" for Fmt |
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var rand = Random.new() |
var rand = Random.new() |
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Latest revision as of 11:23, 18 May 2024
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.
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. Each is bounded by (horizontal) walls and ; (vertical) walls and
![{\displaystyle \mathrm{cell}[0..M-1, 0..N-1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba727d4b36cd2dc9c727d9c298fca9e8385ba7e)
![{\displaystyle \mathrm{cell}[m, n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/544a2d331a8a1e49efa6ff487758a126ddb92b6c)
![{\displaystyle \mathrm{hwall}[m, n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f996d6b44f00ce891d8b8b8145568e0e03d8378)
![{\displaystyle \mathrm{hwall}[m+1, n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71a74e6c34e6955924f250dd2f667d98c9fb383c)
![{\displaystyle \mathrm{vwall}[m, n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd40486387740e95c7d908502a1f7902a105722c)
![{\displaystyle \mathrm{vwall}[m, n+1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdc4523014bbb958b48965fb52277b75a8ece629)
Assume that the probability of any wall being present is a constant where
Except for the outer horizontal walls at and which are always present.
- The task
Simulate pouring a fluid onto the top surface () where the fluid will enter any empty cell it is adjacent to if there is no wall between where it currently is and the cell on the other side of the (missing) wall.
The fluid does not move beyond the horizontal constraints of the grid.
The fluid may move “up” within the confines of the grid of cells. If the fluid reaches a bottom cell that has a missing bottom wall then the fluid can be said to 'drip' out the bottom at that point.
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 fluid successfully percolating through a grid graphically.
Show all output on this page.
11l
UInt32 seed = 0
F nonrandom()
:seed = 1664525 * :seed + 1013904223
R Int(:seed >> 16) / Float(FF'FF)
T Grid = ([[Int]] cell, [[Int]] hwall, [[Int]] vwall)
V (M, nn, t) = (10, 10, 100)
T PercolatedException
(Int, Int) t
F (t)
.t = t
V HVF = ([‘ .’, ‘ _’], [‘:’, ‘|’], [‘ ’, ‘#’])
F newgrid(p)
V hwall = (0 .. :nn).map(n -> (0 .< :M).map(m -> Int(nonrandom() < @@p)))
V vwall = (0 .< :nn).map(n -> (0 .. :M).map(m -> (I m C (0, :M) {1} E Int(nonrandom() < @@p))))
V cell = (0 .< :nn).map(n -> (0 .< :M).map(m -> 0))
R Grid(cell, hwall, vwall)
F pgrid(grid, percolated)
V (cell, hwall, vwall) = grid
V (h, v, f) = :HVF
L(n) 0 .< :nn
print(‘ ’(0 .< :M).map(m -> @h[@hwall[@n][m]]).join(‘’))
print(‘#.) ’.format(n % 10)‘’(0 .. :M).map(m -> @v[@vwall[@n][m]]‘’@f[I m < :M {@cell[@n][m]} E 0]).join(‘’)[0 .< (len)-1])
V n = :nn
print(‘ ’(0 .< :M).map(m -> @h[@hwall[@n][m]]).join(‘’))
I percolated != (-1, -1)
V where = percolated[0]
print(‘!) ’(‘ ’ * where)‘ ’f[1])
F flood_fill(m, n, &cell, hwall, vwall) -> Void
cell[n][m] = 1
I n < :nn - 1 & !hwall[n + 1][m] & !cell[n + 1][m]
flood_fill(m, n + 1, &cell, hwall, vwall)
E I n == :nn - 1 & !hwall[n + 1][m]
X.throw PercolatedException((m, n + 1))
I m & !vwall[n][m] & !cell[n][m - 1]
flood_fill(m - 1, n, &cell, hwall, vwall)
I m < :M - 1 & !vwall[n][m + 1] & !cell[n][m + 1]
flood_fill(m + 1, n, &cell, hwall, vwall)
I n != 0 & !hwall[n][m] & !cell[n - 1][m]
flood_fill(m, n - 1, &cell, hwall, vwall)
F pour_on_top(Grid &grid) -> (Int, Int)?
V n = 0
X.try
L(m) 0 .< :M
I grid.hwall[n][m] == 0
flood_fill(m, n, &grid.cell, grid.hwall, grid.vwall)
X.catch PercolatedException ex
R ex.t
R N
V sample_printed = 0B
[Float = Int] pcount
L(p10) 11
V p = (10 - p10) / 10.0
pcount[p] = 0
L(tries) 0 .< t
V grid = newgrid(p)
(Int, Int)? percolated = pour_on_top(&grid)
I percolated != N
pcount[p]++
I !sample_printed
print("\nSample percolating #. x #. grid".format(M, nn))
pgrid(grid, percolated ? (-1, -1))
sample_printed = 1B
print("\n p: Fraction of #. tries that percolate through".format(t))
L(p, c) sorted(pcount.items())
print(‘#.1: #.’.format(p, c / Float(t)))- Output:
Sample percolating 10 x 10 grid
. _ _ _ . _ . _ _ _
0) |#:#:#| |#:#|#:#:#| |
. _ . _ _ _ _ _ . _
1) |#|#:#| | | | |#:#:#|
_ _ . _ . _ _ _ _ .
2) | |#:#| : : : | | |#|
. _ _ . _ _ _ _ . .
3) | | | : | : | | | :#|
_ . . _ _ _ . . _ .
4) | | : : | | |#:#| |#|
. _ _ _ _ _ . . . .
5) | : | | | : |#|#:#:#|
_ _ _ _ _ _ . _ _ _
6) | | | | | : :#: : | |
_ _ _ _ . . . . . .
7) | | | | | | |#:#:#| |
_ _ . . . _ _ _ . _
8) | : | | | | : | |#: |
. . . . _ . _ _ . .
9) | | | : | | | : |#| |
_ . _ _ . _ . . . .
!) #
p: Fraction of 100 tries that percolate through
0.0: 1
0.1: 1
0.2: 1
0.3: 0.99
0.4: 0.89
0.5: 0.49
0.6: 0.06
0.7: 0
0.8: 0
0.9: 0
1.0: 0
C
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
// cell states
#define FILL 1
#define RWALL 2 // right wall
#define BWALL 4 // bottom wall
typedef unsigned int c_t;
c_t *cells, *start, *end;
int m, n;
void make_grid(double p, int x, int y)
{
int i, j, thresh = RAND_MAX * p;
m = x, n = y;
// Allocate two addition rows to avoid checking bounds.
// Bottom row is also required by drippage
start = realloc(start, m * (n + 2) * sizeof(c_t));
cells = start + m;
for (i = 0; i < m; i++)
start[i] = BWALL | RWALL;
for (i = 0, end = cells; i < y; i++) {
for (j = x; --j; )
*end++ = (rand() < thresh ? BWALL : 0)
|(rand() < thresh ? RWALL : 0);
*end++ = RWALL | (rand() < thresh ? BWALL: 0);
}
memset(end, 0, sizeof(c_t) * m);
}
void show_grid(void)
{
int i, j;
for (j = 0; j < m; j++) printf("+--");
puts("+");
for (i = 0; i <= n; i++) {
putchar(i == n ? ' ' : '|');
for (j = 0; j < m; j++) {
printf((cells[i*m + j] & FILL) ? "[]" : " ");
putchar((cells[i*m + j] & RWALL) ? '|' : ' ');
}
putchar('\n');
if (i == n) return;
for (j = 0; j < m; j++)
printf((cells[i*m + j] & BWALL) ? "+--" : "+ ");
puts("+");
}
}
int fill(c_t *p)
{
if ((*p & FILL)) return 0;
*p |= FILL;
if (p >= end) return 1; // success: reached bottom row
return ( !(p[ 0] & BWALL) && fill(p + m) ) ||
( !(p[ 0] & RWALL) && fill(p + 1) ) ||
( !(p[-1] & RWALL) && fill(p - 1) ) ||
( !(p[-m] & BWALL) && fill(p - m) );
}
int percolate(void)
{
int i;
for (i = 0; i < m && !fill(cells + i); i++);
return i < m;
}
int main(void)
{
make_grid(.5, 10, 10);
percolate();
show_grid();
int cnt, i, p;
puts("\nrunning 10x10 grids 10000 times for each p:");
for (p = 1; p < 10; p++) {
for (cnt = i = 0; i < 10000; i++) {
make_grid(p / 10., 10, 10);
cnt += percolate();
//show_grid(); // don't
}
printf("p = %3g: %.4f\n", p / 10., (double)cnt / i);
}
free(start);
return 0;
}
- Output:
+--+--+--+--+--+--+--+--+--+--+
|[]|[] []|[] [] [] [] [] |
+ + + +--+--+--+--+ + +--+
|[]|[]|[]| | [] []| |
+ +--+ +--+--+--+ +--+ + +
|[] [] [] []| | []| |
+--+ +--+--+ +--+ +--+ +--+
| |[]| | | |[] |
+--+--+ + + + + +--+ + +
| | | | [] |
+--+ + +--+ +--+--+ + +--+
| | | |[] [] [] []| |
+ + + +--+ + +--+--+--+--+
| | | | [] | | | |
+--+ +--+--+--+ + + +--+ +
| | | | | |[]| |
+--+ + + + + +--+ + + +
| | | | [] []| | | | |
+--+ +--+--+ +--+ + + + +
| | | []| | |
+--+ +--+--+ + +--+--+ + +
[]
running 10x10 grids 10000 times for each p:
p = 0.1: 1.0000
p = 0.2: 1.0000
p = 0.3: 0.9958
p = 0.4: 0.9123
p = 0.5: 0.5014
p = 0.6: 0.0791
p = 0.7: 0.0037
p = 0.8: 0.0000
p = 0.9: 0.0000
C++
#include <cstdlib>
#include <cstring>
#include <iostream>
#include <string>
using namespace std;
class Grid {
public:
Grid(const double p, const int x, const int y) : m(x), n(y) {
const int thresh = static_cast<int>(RAND_MAX * p);
// Allocate two addition rows to avoid checking bounds.
// Bottom row is also required by drippage
start = new cell[m * (n + 2)];
cells = start + m;
for (auto i = 0; i < m; i++) start[i] = RBWALL;
end = cells;
for (auto i = 0; i < y; i++) {
for (auto j = x; --j;)
*end++ = (rand() < thresh ? BWALL : 0) | (rand() < thresh ? RWALL : 0);
*end++ = RWALL | (rand() < thresh ? BWALL : 0);
}
memset(end, 0u, sizeof(cell) * m);
}
~Grid() {
delete[] start;
cells = 0;
start = 0;
end = 0;
}
int percolate() const {
auto i = 0;
for (; i < m && !fill(cells + i); i++);
return i < m;
}
void show() const {
for (auto j = 0; j < m; j++)
cout << ("+-");
cout << '+' << endl;
for (auto i = 0; i <= n; i++) {
cout << (i == n ? ' ' : '|');
for (auto j = 0; j < m; j++) {
cout << ((cells[i * m + j] & FILL) ? "#" : " ");
cout << ((cells[i * m + j] & RWALL) ? '|' : ' ');
}
cout << endl;
if (i == n) return;
for (auto j = 0; j < m; j++)
cout << ((cells[i * m + j] & BWALL) ? "+-" : "+ ");
cout << '+' << endl;
}
}
private:
enum cell_state {
FILL = 1 << 0,
RWALL = 1 << 1, // right wall
BWALL = 1 << 2, // bottom wall
RBWALL = RWALL | BWALL // right/bottom wall
};
typedef unsigned int cell;
bool fill(cell* p) const {
if ((*p & FILL)) return false;
*p |= FILL;
if (p >= end) return true; // success: reached bottom row
return (!(p[0] & BWALL) && fill(p + m)) || (!(p[0] & RWALL) && fill(p + 1))
||(!(p[-1] & RWALL) && fill(p - 1)) || (!(p[-m] & BWALL) && fill(p - m));
}
cell* cells;
cell* start;
cell* end;
const int m;
const int n;
};
int main() {
const auto M = 10, N = 10;
const Grid grid(.5, M, N);
grid.percolate();
grid.show();
const auto C = 10000;
cout << endl << "running " << M << "x" << N << " grids " << C << " times for each p:" << endl;
for (auto p = 1; p < M; p++) {
auto cnt = 0, i = 0;
for (; i < C; i++)
cnt += Grid(p / static_cast<double>(M), M, N).percolate();
cout << "p = " << p / static_cast<double>(M) << ": " << static_cast<double>(cnt) / i << endl;
}
return EXIT_SUCCESS;
}
D
import std.stdio, std.random, std.array, std.range, std.algorithm;
struct Grid {
// Not enforced by runtime and type system:
// a Cell must contain only the flags bits.
alias Cell = uint;
enum : Cell { // Cell states (bit flags).
empty = 0,
filled = 1,
rightWall = 2,
bottomWall = 4
}
const size_t nc, nr;
Cell[] cells;
this(in size_t nRows, in size_t nCols) pure nothrow {
nr = nRows;
nc = nCols;
// Allocate two addition rows to avoid checking bounds.
// Bottom row is also required by drippage.
cells = new Cell[nc * (nr + 2)];
}
void initialize(in double prob, ref Xorshift rng) {
cells[0 .. nc] = bottomWall | rightWall; // First row.
uint pos = nc;
foreach (immutable r; 1 .. nr + 1) {
foreach (immutable c; 1 .. nc)
cells[pos++] = (uniform01 < prob ?bottomWall : empty) |
(uniform01 < prob ? rightWall : empty);
cells[pos++] = rightWall |
(uniform01 < prob ? bottomWall : empty);
}
cells[$ - nc .. $] = empty; // Last row.
}
bool percolate() pure nothrow @nogc {
bool fill(in size_t i) pure nothrow @nogc {
if (cells[i] & filled)
return false;
cells[i] |= filled;
if (i >= cells.length - nc)
return true; // Success: reached bottom row.
return (!(cells[i] & bottomWall) && fill(i + nc)) ||
(!(cells[i] & rightWall) && fill(i + 1)) ||
(!(cells[i - 1] & rightWall) && fill(i - 1)) ||
(!(cells[i - nc] & bottomWall) && fill(i - nc));
}
return iota(nc, nc + nc).any!fill;
}
void show() const {
writeln("+-".replicate(nc), '+');
foreach (immutable r; 1 .. nr + 2) {
write(r == nr + 1 ? ' ' : '|');
foreach (immutable c; 0 .. nc) {
immutable cell = cells[r * nc + c];
write((cell & filled) ? (r <= nr ? '#' : 'X') : ' ');
write((cell & rightWall) ? '|' : ' ');
}
writeln;
if (r == nr + 1)
return;
foreach (immutable c; 0 .. nc)
write((cells[r * nc + c] & bottomWall) ? "+-" : "+ ");
'+'.writeln;
}
}
}
void main() {
enum uint nr = 10, nc = 10; // N. rows and columns of the grid.
enum uint nTries = 10_000; // N. simulations for each probability.
enum uint nStepsProb = 10; // N. steps of probability.
auto rng = Xorshift(2);
auto g = Grid(nr, nc);
g.initialize(0.5, rng);
g.percolate;
g.show;
writefln("\nRunning %dx%d grids %d times for each p:",
nr, nc, nTries);
foreach (immutable p; 0 .. nStepsProb) {
immutable probability = p / double(nStepsProb);
uint nPercolated = 0;
foreach (immutable i; 0 .. nTries) {
g.initialize(probability, rng);
nPercolated += g.percolate;
}
writefln("p = %0.2f: %.4f",
probability, nPercolated / double(nTries));
}
}
- Output:
+-+-+-+-+-+-+-+-+-+-+
|#|#|#|#| | | |
+ +-+-+ +-+-+-+ +-+-+
|#| | # | | | | |
+ +-+-+ + +-+-+ + +-+
|#|# #|#| | | |
+ +-+ + +-+ + + +-+ +
|#|# #|#| | | | |
+-+ + + + +-+-+-+-+-+
|# # # # # | | | |
+ + + + + + + +-+ +-+
|#|# # #|# # # | |
+-+ + + +-+-+ + + + +
| |#|# #| | |# |
+-+-+-+-+ +-+ +-+-+-+
| | | # #| |
+-+-+-+ +-+ +-+-+-+ +
| | | # # # |
+ + +-+ +-+-+-+ +-+ +
| | | |# |
+ +-+ + + + +-+ + + +
X
Running 10x10 grids 10000 times for each p:
p = 0.00: 1.0000
p = 0.10: 1.0000
p = 0.20: 1.0000
p = 0.30: 0.9973
p = 0.40: 0.9177
p = 0.50: 0.5050
p = 0.60: 0.0880
p = 0.70: 0.0035
p = 0.80: 0.0001
p = 0.90: 0.0000
With LDC2 compiler this code runs in 0.26 seconds (almost two times faster than the C entry).
Go
package main
import (
"fmt"
"math/rand"
"strings"
"time"
)
func main() {
const (
m, n = 10, 10
t = 1000
minp, maxp, Δp = 0.1, 0.99, 0.1
)
// Purposely don't seed for a 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, %.3f\n", p, float64(count)/t)
}
}
type cell struct {
full bool
right, down bool // true if open to the right (x+1) or down (y+1)
}
type grid struct {
cell [][]cell // row first, i.e. [y][x]
}
func NewGrid(p float64, xsize, ysize int) *grid {
g := &grid{cell: make([][]cell, ysize)}
for y := range g.cell {
g.cell[y] = make([]cell, xsize)
for x := 0; x < xsize-1; x++ {
if rand.Float64() > p {
g.cell[y][x].right = true
}
if rand.Float64() > p {
g.cell[y][x].down = true
}
}
if rand.Float64() > p {
g.cell[y][xsize-1].down = true
}
}
return g
}
var (
full = map[bool]string{false: " ", true: "**"}
hopen = map[bool]string{false: "--", true: " "}
vopen = map[bool]string{false: "|", true: " "}
)
func (g *grid) String() string {
var buf strings.Builder
// Don't really need to call Grow but it helps avoid multiple
// reallocations if the size is large.
buf.Grow((len(g.cell) + 1) * len(g.cell[0]) * 7)
for _ = range g.cell[0] {
buf.WriteString("+")
buf.WriteString(hopen[false])
}
buf.WriteString("+\n")
for y := range g.cell {
buf.WriteString(vopen[false])
for x := range g.cell[y] {
buf.WriteString(full[g.cell[y][x].full])
buf.WriteString(vopen[g.cell[y][x].right])
}
buf.WriteByte('\n')
for x := range g.cell[y] {
buf.WriteString("+")
buf.WriteString(hopen[g.cell[y][x].down])
}
buf.WriteString("+\n")
}
ly := len(g.cell) - 1
for x := range g.cell[ly] {
buf.WriteByte(' ')
buf.WriteString(full[g.cell[ly][x].down && g.cell[ly][x].full])
}
return buf.String()
}
func (g *grid) Percolate() bool {
for x := range g.cell[0] {
if g.fill(x, 0) {
return true
}
}
return false
}
func (g *grid) fill(x, y int) bool {
if y >= len(g.cell) {
return true // Out the bottom
}
if g.cell[y][x].full {
return false // Allready filled
}
g.cell[y][x].full = true
if g.cell[y][x].down && g.fill(x, y+1) {
return true
}
if g.cell[y][x].right && g.fill(x+1, y) {
return true
}
if x > 0 && g.cell[y][x-1].right && g.fill(x-1, y) {
return true
}
if y > 0 && g.cell[y-1][x].down && g.fill(x, y-1) {
return true
}
return false
}
- Output:
+--+--+--+--+--+--+--+--+--+--+ |** ** **| | | | | | | + +--+ +--+--+ +--+--+--+ + |**| |** **| | | | | +--+ +--+ + + +--+ + +--+ | | **| | | +--+ +--+ +--+--+--+--+--+--+ | | ** **| | | +--+ + +--+ + +--+ +--+ + | |** ** **| | | + + +--+ +--+ + +--+ +--+ | | | | ** ** ** **| | | + +--+--+ + +--+--+ +--+--+ | |** ** **|**|**| |** ** **| + + + + + + +--+ +--+ + |** **|**|** ** **| |** ** **| + + +--+--+--+--+ + +--+ + |**|** ** **| | |**| |**| + +--+--+--+ + +--+--+--+--+ |** | | | | | + + + + +--+--+ +--+--+ + ** p=0.10, 1.000 p=0.20, 1.000 p=0.30, 0.998 p=0.40, 0.915 p=0.50, 0.502 p=0.60, 0.081 p=0.70, 0.002 p=0.80, 0.000 p=0.90, 0.000
Haskell
{-# LANGUAGE OverloadedStrings #-}
import Control.Monad
import Control.Monad.Random
import Data.Array.Unboxed
import Data.List
import Formatting
data Field = Field { f :: UArray (Int, Int) Char
, hWall :: UArray (Int, Int) Bool
, vWall :: UArray (Int, Int) Bool
}
-- Start percolating some seepage through a field.
-- Recurse to continue percolation with new seepage.
percolateR :: [(Int, Int)] -> Field -> (Field, [(Int,Int)])
percolateR [] (Field f h v) = (Field f h v, [])
percolateR seep (Field f h v) =
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
north (x,y) = if v ! (x ,y ) then [] else [(x ,y-1)]
south (x,y) = if v ! (x ,y+1) then [] else [(x ,y+1)]
west (x,y) = if h ! (x ,y ) then [] else [(x-1,y )]
east (x,y) = if h ! (x+1,y ) then [] else [(x+1,y )]
neighbors (x,y) = north(x,y) ++ south(x,y) ++ west(x,y) ++ east(x,y)
in percolateR
(concatMap neighbors validSeep)
(Field (f // map (\p -> (p,'.')) validSeep) h v)
-- Percolate a field; Return the percolated field.
percolate :: Field -> Field
percolate start@(Field f _ _) =
let ((_,_),(xHi,_)) = bounds f
(final, _) = percolateR [(x,0) | x <- [0..xHi]] start
in final
-- Generate a random field.
initField :: Int -> Int -> Double -> Rand StdGen Field
initField width height threshold = do
let f = listArray ((0,0), (width-1, height-1)) $ repeat ' '
hrnd <- fmap (<threshold) <$> getRandoms
let h0 = listArray ((0,0),(width, height-1)) hrnd
h1 = h0 // [((0,y), True) | y <- [0..height-1]] -- close left
h2 = h1 // [((width,y), True) | y <- [0..height-1]] -- close right
vrnd <- fmap (<threshold) <$> getRandoms
let v0 = listArray ((0,0),(width-1, height)) vrnd
v1 = v0 // [((x,0), True) | x <- [0..width-1]] -- close top
return $ Field f h2 v1
-- Assess whether or not percolation reached bottom of field.
leaks :: Field -> [Bool]
leaks (Field f _ v) =
let ((xLo,_),(xHi,yHi)) = bounds f
in [f!(x,yHi)=='.' && not (v!(x,yHi+1)) | x <- [xLo..xHi]]
-- Run test once; Return bool indicating success or failure.
oneTest :: Int -> Int -> Double -> Rand StdGen Bool
oneTest width height threshold =
or.leaks.percolate <$> initField width height threshold
-- Run test multple times; Return the number of tests that pass.
multiTest :: Int -> Int -> Int -> Double -> Rand StdGen Double
multiTest testCount width height threshold = do
results <- replicateM testCount $ oneTest width height threshold
let leakyCount = length $ filter id results
return $ fromIntegral leakyCount / fromIntegral testCount
-- Helper function for display
alternate :: [a] -> [a] -> [a]
alternate [] _ = []
alternate (a:as) bs = a : alternate bs as
-- Display a field with walls and leaks.
showField :: Field -> IO ()
showField field@(Field a h v) = do
let ((xLo,yLo),(xHi,yHi)) = bounds a
fLines = [ [ a!(x,y) | x <- [xLo..xHi]] | y <- [yLo..yHi]]
hLines = [ [ if h!(x,y) then '|' else ' ' | x <- [xLo..xHi+1]] | y <- [yLo..yHi]]
vLines = [ [ if v!(x,y) then '-' else ' ' | x <- [xLo..xHi]] | y <- [yLo..yHi+1]]
lattice = [ [ '+' | x <- [xLo..xHi+1]] | y <- [yLo..yHi+1]]
hDrawn = zipWith alternate hLines fLines
vDrawn = zipWith alternate lattice vLines
mapM_ putStrLn $ alternate vDrawn hDrawn
let leakLine = [ if l then '.' else ' ' | l <- leaks field]
putStrLn $ alternate (repeat ' ') leakLine
main :: IO ()
main = do
g <- getStdGen
let threshold = 0.45
(startField, g2) = runRand (initField 10 10 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]
let tests = sequence [multiTest testCount 10 10 v
| density <- densities,
let v = fromIntegral density / fromIntegral densityCount ]
let results = zip densities (evalRand tests g2)
mapM_ print [format ("p=" % int % "/" % int % " -> " % fixed 4) density densityCount x | (density,x) <- results]
- Output:
Unpercolated field with 0.45 threshold.
+-+-+-+-+-+-+-+-+-+-+
| | | | | | | |
+-+-+ +-+ + + + + +-+
| | | | | | | |
+ + +-+-+ + +-+-+ + +
| | | |
+ +-+-+-+ +-+-+ +-+ +
| | | | | |
+ +-+ + + +-+-+ + +-+
| | | | | |
+-+-+ + + + + +-+ + +
| | | | | | | |
+-+ + + + + + + +-+-+
| | | | |
+ + + + + +-+ +-+ + +
| | | | | | | | |
+ + + +-+-+-+-+-+ + +
| | | | |
+ +-+ +-+ +-+ + + +-+
| | | | | |
+ + + + +-+ +-+-+-+ +
Same field after percolation.
+-+-+-+-+-+-+-+-+-+-+
|. . . .|.|.|.|.|.|.|
+-+-+ +-+ + + + + +-+
| |.|. . . .|.|.|.|.|
+ + +-+-+ + +-+-+ + +
| |. . . . .|. . . .|
+ +-+-+-+ +-+-+ +-+ +
| |. . .|.|. . . .|.|
+ +-+ + + +-+-+ + +-+
| |. . .|. . . .|.|.|
+-+-+ + + + + +-+ + +
| |.|. .|.|.|.|. . .|
+-+ + + + + + + +-+-+
|. . . . .|. .|. .|.|
+ + + + + +-+ +-+ + +
|.|.|.|.|. . .| |.|.|
+ + + +-+-+-+-+-+ + +
|.|. . . .|. . .|. .|
+ +-+ +-+ +-+ + + +-+
|.| |.|. . . . . .| |
+ + + + +-+ +-+-+-+ +
. . . .
Results of running percolation test 10000 times with thresholds ranging from 0/10 to 10/10 .
"p=0/10 -> 1.0000"
"p=1/10 -> 1.0000"
"p=2/10 -> 1.0000"
"p=3/10 -> 0.9969"
"p=4/10 -> 0.9171"
"p=5/10 -> 0.5026"
"p=6/10 -> 0.0901"
"p=7/10 -> 0.0025"
"p=8/10 -> 0.0000"
"p=9/10 -> 0.0000"
"p=10/10 -> 0.0000"
Java
import java.util.Arrays;
import java.util.concurrent.ThreadLocalRandom;
public final class PercolationBond {
public static void main(String[] aArgs) {
System.out.println("Sample percolation with a " + COL_COUNT + " x " + ROW_COUNT + " grid:");
makeGrid(0.5);
percolate();
showGrid();
System.out.println("Using 10,000 repetitions for each probability p:");
for ( int p = 1; p <= 9; p++ ) {
int percolationCount = 0;
double probability = p / 10.0;
for ( int i = 0; i < 10_000; i++ ) {
makeGrid(probability);
if ( percolate() ) {
percolationCount += 1;
}
}
final double percolationProportion = (double) percolationCount / 10_000;
System.out.println(String.format("%s%.1f%s%.4f", "p = ", probability, ": ", percolationProportion));
}
}
private static void makeGrid(double aProbability) {
Arrays.fill(grid, 0);
for ( int i = 0; i < COL_COUNT; i++ ) {
grid[i] = LOWER_WALL | RIGHT_WALL;
}
endOfRow = COL_COUNT;
for ( int i = 0; i < ROW_COUNT; i++ ) {
for ( int j = COL_COUNT - 1; j >= 1; j-- ) {
final boolean chance1 = RANDOM.nextDouble() < aProbability;
final boolean chance2 = RANDOM.nextDouble() < aProbability;
grid[endOfRow++] = ( chance1 ? LOWER_WALL : 0 ) | ( chance2 ? RIGHT_WALL : 0 );
}
final boolean chance3 = RANDOM.nextDouble() < aProbability;
grid[endOfRow++] = RIGHT_WALL | ( chance3 ? LOWER_WALL : 0 );
}
}
private static void showGrid() {
for ( int j = 0; j < COL_COUNT; j++ ) {
System.out.print("+--");
}
System.out.println("+");
for ( int i = 0; i < ROW_COUNT; i++ ) {
System.out.print("|");
for ( int j = 0; j < COL_COUNT; j++ ) {
System.out.print( ( ( grid[i * COL_COUNT + j + COL_COUNT] & FILL ) != 0 ) ? "[]" : " " );
System.out.print( ( ( grid[i * COL_COUNT + j + COL_COUNT] & RIGHT_WALL ) != 0 ) ? "|" : " " );
}
System.out.println();
for ( int j = 0; j < COL_COUNT; j++ ) {
System.out.print( ( ( grid[i * COL_COUNT + j + COL_COUNT] & LOWER_WALL) != 0 ) ? "+--" : "+ " );
}
System.out.println("+");
}
System.out.print(" ");
for ( int j = 0; j < COL_COUNT; j++ ) {
System.out.print( ( ( grid[ROW_COUNT * COL_COUNT + j + COL_COUNT] & FILL ) != 0 ) ? "[]" : " " );
System.out.print( ( ( grid[ROW_COUNT * COL_COUNT + j + COL_COUNT] & RIGHT_WALL ) != 0 ) ? "|" : " " );
}
System.out.println(System.lineSeparator());
}
private static boolean fill(int aGridIndex) {
if ( ( grid[aGridIndex] & FILL ) != 0 ) {
return false;
}
grid[aGridIndex] |= FILL;
if ( aGridIndex >= endOfRow ) {
return true;
}
return ( ( ( grid[aGridIndex] & LOWER_WALL ) == 0 ) && fill(aGridIndex + COL_COUNT) ) ||
( ( ( grid[aGridIndex] & RIGHT_WALL ) == 0 ) && fill(aGridIndex + 1) ) ||
( ( ( grid[aGridIndex - 1] & RIGHT_WALL ) == 0 ) && fill(aGridIndex - 1) ) ||
( ( ( grid[aGridIndex - COL_COUNT] & LOWER_WALL ) == 0 ) && fill(aGridIndex - COL_COUNT) );
}
private static boolean percolate() {
int i = 0;
while ( i < COL_COUNT && ! fill(COL_COUNT + i) ) {
i++;
}
return i < COL_COUNT;
}
private static final int ROW_COUNT = 10;
private static final int COL_COUNT = 10;
private static int endOfRow = COL_COUNT;
private static int[] grid = new int[COL_COUNT * ( ROW_COUNT + 2 )];
private static final int FILL = 1;
private static final int RIGHT_WALL = 2;
private static final int LOWER_WALL = 4;
private static final ThreadLocalRandom RANDOM = ThreadLocalRandom.current();
}
- Output:
Sample percolation with a 10 x 10 grid:
+--+--+--+--+--+--+--+--+--+--+
|[] []|[]| | | |
+--+--+ +--+--+ +--+ +--+--+
| | |[] | | |
+ + + +--+ + +--+ + + +
| | []| | | |
+ +--+ +--+--+ + +--+--+--+
| |[] | | |
+ +--+ +--+--+--+ +--+--+--+
| | []| | | | | | |
+--+ + + +--+--+ + +--+ +
| |[] | | | | |
+--+ + + + +--+--+ + +--+
| [] []| | | |
+ + +--+--+ +--+--+--+ +--+
| |[] | | | | |
+ + +--+ +--+ + +--+ + +
| []| | | | | |
+ + +--+ +--+--+--+--+ + +
| |[] [] | |
+ +--+ +--+--+ +--+ +--+--+
[]
Using 10,000 repetitions for each probability p:
p = 0.1: 1.0000
p = 0.2: 0.9999
p = 0.3: 0.9973
p = 0.4: 0.9223
p = 0.5: 0.5011
p = 0.6: 0.0872
p = 0.7: 0.0022
p = 0.8: 0.0000
p = 0.9: 0.0000
Julia
using Printf, Distributions
struct Grid
cells::BitArray{2}
hwall::BitArray{2}
vwall::BitArray{2}
end
function Grid(p::AbstractFloat, m::Integer=10, n::Integer=10)
cells = fill(false, m, n)
hwall = rand(Bernoulli(p), m + 1, n)
vwall = rand(Bernoulli(p), m, n + 1)
vwall[:, 1] = true
vwall[:, end] = true
return Grid(cells, hwall, vwall)
end
function Base.show(io::IO, g::Grid)
H = (" .", " _")
V = (":", "|")
C = (" ", "#")
ind = findfirst(g.cells[end, :] .& .!g.hwall[end, :])
percolated = !iszero(ind)
println(io, "$(size(g.cells, 1))×$(size(g.cells, 2)) $(percolated ? "Percolated" : "Not percolated") grid")
for r in 1:size(g.cells, 1)
println(io, " ", join(H[w+1] for w in g.hwall[r, :]))
println(io, " $(r % 10)) ", join(V[w+1] * C[c+1] for (w, c) in zip(g.vwall[r, :], g.cells[r, :])))
end
println(io, " ", join(H[w+1] for w in g.hwall[end, :]))
if percolated
println(io, " !) ", " " ^ (ind - 1), '#')
end
end
function floodfill!(m::Integer, n::Integer, cells::AbstractMatrix{<:Integer},
hwall::AbstractMatrix{<:Integer}, vwall::AbstractMatrix{<:Integer})
# fill cells
cells[m, n] = true
percolated = false
# bottom
if m < size(cells, 1) && !hwall[m+1, n] && !cells[m+1, n]
percolated = percolated || floodfill!(m + 1, n, cells, hwall, vwall)
# The Bottom
elseif m == size(cells, 1) && !hwall[m+1, n]
return true
end
# left
if n > 1 && !vwall[m, n] && !cells[m, n-1]
percolated = percolated || floodfill!(m, n - 1, cells, hwall, vwall)
end
# right
if n < size(cells, 2) && !vwall[m, n+1] && !cells[m, n+1]
percolated = percolated || floodfill!(m, n + 1, cells, hwall, vwall)
end
# top
if m > 1 && !hwall[m, n] && !cells[m-1, n]
percolated = percolated || floodfill!(m - 1, n, cells, hwall, vwall)
end
return percolated
end
function pourontop!(g::Grid)
m, n = 1, 1
percolated = false
while !percolated && n ≤ size(g.cells, 2)
percolated = !g.hwall[m, n] && floodfill!(m, n, g.cells, g.hwall, g.vwall)
n += 1
end
return percolated
end
function main(probs, nrep::Integer=1000)
sampleprinted = false
pcount = zeros(Int, size(probs))
for (i, p) in enumerate(probs), _ in 1:nrep
g = Grid(p)
percolated = pourontop!(g)
if percolated
pcount[i] += 1
if !sampleprinted
println(g)
sampleprinted = true
end
end
end
return pcount ./ nrep
end
probs = collect(10:-1:0) ./ 10
percprobs = main(probs)
println("Fraction of 1000 tries that percolate through:")
for (pr, pp) in zip(probs, percprobs)
@printf("\tp = %.3f ⇒ freq. = %5.3f\n", pr, pp)
end
- Output:
10×10 Percolated grid
_ . . _ _ _ . _ . .
1) | |#:#| | : | : | :
_ _ . _ _ _ _ _ . _
2) | | |#| : : | : | |
_ _ . _ _ _ _ _ _ .
3) | | |#:#| : | : | :
. _ _ . _ _ _ . _ _
4) | | | :#: : | | | |
. _ _ . _ _ _ . _ _
5) | | : |#| | : | | :
_ . _ . _ _ . . . _
6) | | | |#| | | | | |
. . _ . _ _ _ . . .
7) | |#:#:#: | | : | |
_ . . _ . . . . _ _
8) | |#|#| | | : | | |
_ . . _ _ _ . _ _ _
9) |#:#|#| : : | : | |
. _ _ _ _ . _ _ _ .
0) |#: | : | | | : : |
. . _ _ _ _ . _ _ _
!) #
Fraction of 1000 tries that percolate through:
p = 1.000 ⇒ freq. = 0.000
p = 0.900 ⇒ freq. = 0.000
p = 0.800 ⇒ freq. = 0.000
p = 0.700 ⇒ freq. = 0.001
p = 0.600 ⇒ freq. = 0.064
p = 0.500 ⇒ freq. = 0.470
p = 0.400 ⇒ freq. = 0.895
p = 0.300 ⇒ freq. = 0.997
p = 0.200 ⇒ freq. = 1.000
p = 0.100 ⇒ freq. = 1.000
p = 0.000 ⇒ freq. = 1.000
Kotlin
// version 1.2.10
import java.util.Random
val rand = Random()
const val RAND_MAX = 32767
// cell states
const val FILL = 1
const val RWALL = 2 // right wall
const val BWALL = 4 // bottom wall
val x = 10
val y = 10
var grid = IntArray(x * (y + 2))
var cells = 0
var end = 0
var m = 0
var n = 0
fun makeGrid(p: Double) {
val thresh = (p * RAND_MAX).toInt()
m = x
n = y
grid.fill(0) // clears grid
for (i in 0 until m) grid[i] = BWALL or RWALL
cells = m
end = m
for (i in 0 until y) {
for (j in x - 1 downTo 1) {
val r1 = rand.nextInt(RAND_MAX + 1)
val r2 = rand.nextInt(RAND_MAX + 1)
grid[end++] = (if (r1 < thresh) BWALL else 0) or
(if (r2 < thresh) RWALL else 0)
}
val r3 = rand.nextInt(RAND_MAX + 1)
grid[end++] = RWALL or (if (r3 < thresh) BWALL else 0)
}
}
fun showGrid() {
for (j in 0 until m) print("+--")
println("+")
for (i in 0..n) {
print(if (i == n) " " else "|")
for (j in 0 until m) {
print(if ((grid[i * m + j + cells] and FILL) != 0) "[]" else " ")
print(if ((grid[i * m + j + cells] and RWALL) != 0) "|" else " ")
}
println()
if (i == n) return
for (j in 0 until m) {
print(if ((grid[i * m + j + cells] and BWALL) != 0) "+--" else "+ ")
}
println("+")
}
}
fun fill(p: Int): Boolean {
if ((grid[p] and FILL) != 0) return false
grid[p] = grid[p] or FILL
if (p >= end) return true // success: reached bottom row
return (((grid[p + 0] and BWALL) == 0) && fill(p + m)) ||
(((grid[p + 0] and RWALL) == 0) && fill(p + 1)) ||
(((grid[p - 1] and RWALL) == 0) && fill(p - 1)) ||
(((grid[p - m] and BWALL) == 0) && fill(p - m))
}
fun percolate(): Boolean {
var i = 0
while (i < m && !fill(cells + i)) i++
return i < m
}
fun main(args: Array<String>) {
makeGrid(0.5)
percolate()
showGrid()
println("\nrunning $x x $y grids 10,000 times for each p:")
for (p in 1..9) {
var cnt = 0
val pp = p / 10.0
for (i in 0 until 10_000) {
makeGrid(pp)
if (percolate()) cnt++
}
println("p = %3g: %.4f".format(pp, cnt.toDouble() / 10_000))
}
}
Sample output:
+--+--+--+--+--+--+--+--+--+--+
|[]|[] [] [] [] []| | | | |
+--+--+--+--+--+ +--+ + + +
| | | | []| |
+--+--+--+--+--+ + +--+ + +
| | | | |[] []| |
+ + + + + +--+--+--+--+--+
| | | [] [] []| | |
+--+--+ + +--+--+--+--+--+ +
| | |[] []| | | |
+--+--+ + + + +--+ +--+--+
| | | |[]|[]| | | |
+--+ +--+--+ +--+--+ + + +
| | | []| | | | |
+--+ + + + +--+--+ + + +
| | |[] []| | |
+ +--+--+ +--+ +--+ + +--+
| | [] | | | |
+ + +--+ + +--+--+--+--+ +
| [] | | | |
+ +--+--+ + +--+--+ +--+ +
[]
running 10 x 10 grids 10,000 times for each p:
p = 0.100000: 1.0000
p = 0.200000: 1.0000
p = 0.300000: 0.9968
p = 0.400000: 0.9184
p = 0.500000: 0.5047
p = 0.600000: 0.0828
p = 0.700000: 0.0034
p = 0.800000: 0.0000
p = 0.900000: 0.0000
Nim
import random, sequtils, strformat, tables
type
Cell = object
full: bool
right, down: bool # True if open to the right (x+1) or down (y+1).
Grid = seq[seq[Cell]] # Row first, i.e. [y][x].
proc newGrid(p: float; xsize, ysize: Positive): Grid =
result = newSeqWith(ysize, newSeq[Cell](xsize))
for row in result.mitems:
for x in 0..(xsize - 2):
if rand(1.0) > p: row[x].right = true
if rand(1.0) > p: row[x].down = true
if rand(1.0) > p: row[xsize - 1].down = true
const
Full = {false: " ", true: "()"}.toTable
HOpen = {false: "--", true: " "}.toTable
VOpen = {false: "|", true: " "}.toTable
proc `$`(grid: Grid): string =
# Preallocate result to avoid multiple reallocations.
result = newStringOfCap((grid.len + 1) * grid[0].len * 7)
for _ in 0..grid[0].high:
result.add '+'
result.add HOpen[false]
result.add "+\n"
for row in grid:
result.add VOpen[false]
for cell in row:
result.add Full[cell.full]
result.add VOpen[cell.right]
result.add '\n'
for cell in row:
result.add '+'
result.add HOpen[cell.down]
result.add "+\n"
for cell in grid[^1]:
result.add ' '
result.add Full[cell.down and cell.full]
proc fill(grid: var Grid; x, y: Natural): bool =
if y >= grid.len: return true # Out the bottom.
if grid[y][x].full: return false # Already filled.
grid[y][x].full = true
if grid[y][x].down and grid.fill(x, y + 1): return true
if grid[y][x].right and grid.fill(x + 1, y): return true
if x > 0 and grid[y][x - 1].right and grid.fill(x - 1, y): return true
if y > 0 and grid[y - 1][x].down and grid.fill(x, y - 1): return true
proc percolate(grid: var Grid): bool =
for x in 0..grid[0].high:
if grid.fill(x, 0): return true
const
M = 10
N = 10
T = 1000
MinP = 0.1
MaxP = 0.99
ΔP = 0.1
# Purposely don't seed for a repeatable example grid.
var grid = newGrid(0.4, M, N)
discard grid.percolate()
echo grid
echo ""
randomize()
var p = MinP
while p < MaxP:
var count = 0
for _ in 1..T:
var grid = newGrid(p, M, N)
if grid.percolate(): inc count
echo &"p = {p:.2f}: {count / T:.3f}"
p += ΔP
- Output:
+--+--+--+--+--+--+--+--+--+--+
|()|()|() () () () ()|()|() |
+ + +--+--+ + + +--+ +--+
|() ()| |() () ()|() ()|() |
+ +--+--+ + +--+ +--+ +--+
|() ()| |() ()|() ()| () |
+ + + +--+--+--+ +--+ +--+
|() ()| |() ()| () |
+ +--+--+--+ +--+--+--+ + +
|()| | ()| |
+--+ + +--+ + +--+--+ + +
| | | () ()| |
+--+--+ +--+ + + + +--+ +
| | |()| |
+--+--+--+ + + + + + +--+
| | |() () ()| |
+ +--+--+ + + +--+ + + +
| | () ()|()|()| | |
+--+ +--+ + +--+ + +--+ +
| | () |() ()| |
+ + +--+--+ + +--+--+ +--+
()
p = 0.10: 1.000
p = 0.20: 0.999
p = 0.30: 0.996
p = 0.40: 0.905
p = 0.50: 0.497
p = 0.60: 0.077
p = 0.70: 0.004
p = 0.80: 0.000
p = 0.90: 0.000
Perl
my @bond;
my $grid = 10;
my $water = '▒';
$D{$_} = $i++ for qw<DeadEnd Up Right Down Left>;
sub percolate {
generate(shift || 0.6);
fill(my $x = 1,my $y = 0);
my @stack;
while () {
if (my $dir = direction($x,$y)) {
push @stack, [$x,$y];
($x,$y) = move($dir, $x, $y)
} else {
return 0 unless @stack;
($x,$y) = @{pop @stack}
}
return 1 if $y == $#bond;
}
}
sub direction {
my($x, $y) = @_;
return $D{Down} if $bond[$y+1][$x ] =~ / /;
return $D{Left} if $bond[$y ][$x-1] =~ / /;
return $D{Right} if $bond[$y ][$x+1] =~ / /;
return $D{Up} if defined $bond[$y-1][$x ] && $bond[$y-1][$x] =~ / /;
return $D{DeadEnd}
}
sub move {
my($dir,$x,$y) = @_;
fill( $x,--$y), fill( $x,--$y) if $dir == $D{Up};
fill( $x,++$y), fill( $x,++$y) if $dir == $D{Down};
fill(--$x, $y), fill(--$x, $y) if $dir == $D{Left};
fill(++$x, $y), fill(++$x, $y) if $dir == $D{Right};
$x, $y
}
sub fill {
my($x, $y) = @_;
$bond[$y][$x] =~ s/ /$water/g
}
sub generate {
our($prob) = shift || 0.5;
@bond = ();
our $sp = ' ';
push @bond, ['│', ($sp, ' ') x ($grid-1), $sp, '│'],
['├', hx('┬'), h(), '┤'];
push @bond, ['│', vx( ), $sp, '│'],
['├', hx('┼'), h(), '┤'] for 1..$grid-1;
push @bond, ['│', vx( ), $sp, '│'],
['├', hx('┴'), h(), '┤'],
['│', ($sp, ' ') x ($grid-1), $sp, '│'];
sub hx { my($c)=@_; my @l; push @l, (h(),$c) for 1..$grid-1; return @l; }
sub vx { my @l; push @l, $sp, v() for 1..$grid-1; return @l; }
sub h { rand() < $prob ? $sp : '───' }
sub v { rand() < $prob ? ' ' : '│' }
}
print "Sample percolation at .6\n";
percolate(.6);
for my $row (@bond) {
my $line = '';
$line .= join '', $_ for @$row;
print "$line\n";
}
my $tests = 100;
print "Doing $tests trials at each porosity:\n";
my @table;
for my $p (1 .. 10) {
$p = $p/10;
my $total = 0;
$total += percolate($p) for 1..$tests;
printf "p = %0.1f: %0.2f\n", $p, $total / $tests
}
- Output:
Sample percolation at .6 │▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒ │ ├───┬───┬───┬▒▒▒┬ ┬ ┬ ┬───┬ ┬───┤ │ ▒▒▒▒▒▒▒ │ │ │ ├───┼───┼▒▒▒┼───┼───┼ ┼───┼───┼ ┼ ┤ │ │▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒│ │ │ │ ├───┼───┼───┼───┼▒▒▒┼▒▒▒┼ ┼───┼───┼───┤ │ │ │▒▒▒│▒▒▒▒▒▒▒ │ ├───┼───┼ ┼───┼───┼───┼▒▒▒┼ ┼ ┼ ┤ │ │ ▒▒▒▒▒▒▒ │ │ ├───┼───┼───┼ ┼ ┼▒▒▒┼───┼ ┼ ┼───┤ │ │ │▒▒▒ │ │ ├───┼ ┼───┼ ┼───┼▒▒▒┼ ┼ ┼───┼ ┤ │ │ ▒▒▒▒▒▒▒│ │ │ ├ ┼ ┼ ┼ ┼▒▒▒┼───┼ ┼───┼───┼ ┤ │ │ ▒▒▒│ │ │ ├ ┼ ┼ ┼ ┼▒▒▒┼───┼ ┼ ┼───┼ ┤ │ │ ▒▒▒│ │ │ ├───┼ ┼───┼ ┼▒▒▒┼ ┼ ┼ ┼ ┼ ┤ │ │ │▒▒▒ │ │ │ │ ├───┼───┼ ┼ ┼▒▒▒┼───┼ ┼ ┼ ┼───┤ │ │ ▒▒▒│ │ │ │ ├ ┴───┴ ┴ ┴▒▒▒┴ ┴───┴ ┴ ┴───┤ │ ▒▒▒ │ Doing 100 trials at each porosity: p = 0.1: 0.00 p = 0.2: 0.00 p = 0.3: 0.00 p = 0.4: 0.03 p = 0.5: 0.38 p = 0.6: 0.83 p = 0.7: 0.99 p = 0.8: 1.00 p = 0.9: 1.00 p = 1.0: 1.00
Phix
with javascript_semantics constant w = 10, h = 10 sequence wall = join(repeat("+",w+1),"---")&"\n", cell = join(repeat("|",w+1)," ")&"\n", grid procedure new_grid(atom p) grid = split(join(repeat(wall,h+1),cell),'\n') -- now knock down some walls for i=1 to length(grid)-1 do integer jstart = 5-mod(i,2)*3, jlimit = length(grid[i])-3 -- (ie 2..38 on odd lines, 5..37 on even) for j=jstart to jlimit by 4 do if rnd()>p then grid[i][j..j+2] = " " end if end for end for end procedure function percolate(integer x=0, y=0) if x=0 then for j=3 to length(grid[1])-2 by 4 do if grid[1][j]=' ' and percolate(1,j) then return true end if end for elsif grid[x][y]=' ' then grid[x][y] = '*' if (x=length(grid)-1) or ( grid[x+1][y]=' ' and percolate(x+1,y)) or (y>6 and grid[x][y-2]=' ' and percolate(x,y-4)) or (y<36 and grid[x][y+2]=' ' and percolate(x,y+4)) or (x>1 and grid[x-1][y]=' ' and percolate(x-1,y)) then return true end if end if return false end function constant LIM=1000 for p=0 to 10 do integer count = 0 for t=1 to LIM do new_grid(p/10) count += percolate() end for printf(1,"p=%.1f: %5.3f\n",{p/10,count/LIM}) end for puts(1,"sample grid for p=0.6:\n") new_grid(0.6) {} = percolate() printf(1,"%s\n",{join(grid,'\n')})
- Output:
p=0.0: 1.000 p=0.1: 1.000 p=0.2: 1.000 p=0.3: 0.997 p=0.4: 0.897 p=0.5: 0.434 p=0.6: 0.067 p=0.7: 0.003 p=0.8: 0.000 p=0.9: 0.000 p=1.0: 0.000 sample grid for p=0.6: +---+---+ * +---+ * + * +---+---+ * + * + | * * * | | * * | * | | * * | +---+---+ * + +---+ * + * +---+---+ * + | * * | * | | | * | * | * * * | +---+ * + * +---+---+---+ * +---+ * +---+ | | * | * | * | * | * * * * * | + + * + * + * + * + * +---+ * + * +---+ | | * * * * | * * | * * * | + + * + * +---+ * +---+ * +---+ * + * + | | * | * * * | * * | * * | * | +---+ * +---+ * +---+---+---+---+ * + * + | | * | | * * * * | * * | * | + +---+ +---+ * +---+---+---+---+ * + | | | | | * | | | * | + +---+---+ +---+---+---+---+---+---+ | | | | | | | | | | +---+---+---+ + +---+ + +---+---+ | | | | | | | | | + +---+ +---+---+ +---+---+---+---+ | | | | | | | | | | +---+---+ + + +---+---+---+---+ +
Python
from collections import namedtuple
from random import random
from pprint import pprint as pp
Grid = namedtuple('Grid', 'cell, hwall, vwall')
M, N, t = 10, 10, 100
class PercolatedException(Exception): pass
HVF = [(' .', ' _'), (':', '|'), (' ', '#')] # Horiz, vert, fill chars
def newgrid(p):
hwall = [[int(random() < p) for m in range(M)]
for n in range(N+1)]
vwall = [[(1 if m in (0, M) else int(random() < p)) for m in range(M+1)]
for n in range(N)]
cell = [[0 for m in range(M)]
for n in range(N)]
return Grid(cell, hwall, vwall)
def pgrid(grid, percolated=None):
cell, hwall, vwall = grid
h, v, f = HVF
for n in range(N):
print(' ' + ''.join(h[hwall[n][m]] for m in range(M)))
print('%i) ' % (n % 10) + ''.join(v[vwall[n][m]] + f[cell[n][m] if m < M else 0]
for m in range(M+1))[:-1])
n = N
print(' ' + ''.join(h[hwall[n][m]] for m in range(M)))
if percolated:
where = percolated.args[0][0]
print('!) ' + ' ' * where + ' ' + f[1])
def pour_on_top(grid):
cell, hwall, vwall = grid
n = 0
try:
for m in range(M):
if not hwall[n][m]:
flood_fill(m, n, cell, hwall, vwall)
except PercolatedException as ex:
return ex
return None
def flood_fill(m, n, cell, hwall, vwall):
# fill cell
cell[n][m] = 1
# bottom
if n < N - 1 and not hwall[n + 1][m] and not cell[n+1][m]:
flood_fill(m, n+1, cell, hwall, vwall)
# THE bottom
elif n == N - 1 and not hwall[n + 1][m]:
raise PercolatedException((m, n+1))
# left
if m and not vwall[n][m] and not cell[n][m - 1]:
flood_fill(m-1, n, cell, hwall, vwall)
# right
if m < M - 1 and not vwall[n][m + 1] and not cell[n][m + 1]:
flood_fill(m+1, n, cell, hwall, vwall)
# top
if n and not hwall[n][m] and not cell[n-1][m]:
flood_fill(m, n-1, cell, hwall, vwall)
if __name__ == '__main__':
sample_printed = False
pcount = {}
for p10 in range(11):
p = (10 - p10) / 10.0 # count down so sample print is interesting
pcount[p] = 0
for tries in range(t):
grid = newgrid(p)
percolated = pour_on_top(grid)
if percolated:
pcount[p] += 1
if not sample_printed:
print('\nSample percolating %i x %i grid' % (M, N))
pgrid(grid, percolated)
sample_printed = True
print('\n p: Fraction of %i tries that percolate through' % t )
pp({p:c/float(t) for p, c in pcount.items()})
- Output:
In the Ascii art, cells are either a space or a hash and are surrounded by either '_', '|' for intact walls and '.' and ':' for missing (leaky) walls.
The bottom-most line starting '!)' shows where the fluid can drip out from. (The percolation stops when one route through the bottom is found).
Sample percolating 10 x 10 grid
_ _ . _ . _ _ . _ _
0) | |#:#:#|#| | :#| | |
_ _ . _ _ _ . . _ _
1) | | |#:#| | | |#| : |
_ _ _ . _ . . . . _
2) | | |#:#| : | |#: | |
_ _ _ _ . . _ . . .
3) | : : | | | : |#: | |
_ _ . _ . . _ . _ _
4) | : : : | | | |#: : |
_ _ _ . _ _ _ . . _
5) | : | | : | | :#| | |
_ _ . . _ _ _ . _ .
6) | : | | : | |#:#:#| |
_ . _ _ . _ _ _ . .
7) | : | : | : | | |#: |
_ _ _ . . _ _ . . _
8) | | : | | | |#:#:#: |
_ _ _ . . . . _ _ .
9) | : : | : : :#: | : |
. _ . _ . . . . _ _
!) #
p: Fraction of 100 tries that percolate through
{0.0: 1.0,
0.1: 1.0,
0.2: 1.0,
0.3: 1.0,
0.4: 0.9,
0.5: 0.47,
0.6: 0.06,
0.7: 0.0,
0.8: 0.0,
0.9: 0.0,
1.0: 0.0}
Note the abrupt cut-off in percolation at around p = 0.5 which is to be expected.
Racket
#lang racket
(define has-left-wall? (lambda (x) (bitwise-bit-set? x 0)))
(define has-right-wall? (lambda (x) (bitwise-bit-set? x 1)))
(define has-top-wall? (lambda (x) (bitwise-bit-set? x 2)))
(define has-bottom-wall? (lambda (x) (bitwise-bit-set? x 3)))
(define has-fluid? (lambda (x) (bitwise-bit-set? x 4)))
(define (walls->cell l? r? t? b?)
(+ (if l? 1 0) (if r? 2 0) (if t? 4 0) (if b? 8 0)))
(define (bonded-percol-grid M N p)
(define rv (make-vector (* M N)))
(for* ((idx (in-range (* M N))))
(define left-wall?
(or (zero? (modulo idx M))
(has-right-wall? (vector-ref rv (sub1 idx)))))
(define right-wall?
(or (= (modulo idx M) (sub1 M))
(< (random) p)))
(define top-wall?
(if (< idx M) (< (random) p)
(has-bottom-wall? (vector-ref rv (- idx M)))))
(define bottom-wall? (< (random) p))
(define cell-value
(walls->cell left-wall? right-wall? top-wall? bottom-wall?))
(vector-set! rv idx cell-value))
rv)
(define (display-percol-grid M . vs)
(define N (/ (vector-length (car vs)) M))
(define-syntax-rule (tab-eol m)
(when (= m (sub1 M)) (printf "\t")))
(for ((n N))
(for* ((v vs) (m M))
(when (zero? m) (printf "+"))
(printf
(match (vector-ref v (+ (* n M) m))
((? has-top-wall?) "-+")
((? has-fluid?) "#+")
(else ".+")))
(tab-eol m))
(newline)
(for* ((v vs) (m M))
(when (zero? m) (printf "|"))
(printf
(match (vector-ref v (+ (* n M) m))
((and (? has-fluid?) (? has-right-wall?)) "#|")
((? has-right-wall?) ".|")
((? has-fluid?) "##")
(else "..")))
(tab-eol m))
(newline))
(for* ((v vs) (m M))
(when (zero? m) (printf "+"))
(printf
(match (vector-ref v (+ (* (sub1 M) M) m))
((? has-bottom-wall?) "-+")
((? has-fluid?) "#+")
(else ".+")))
(tab-eol m))
(newline))
(define (find-bonded-grid-t/b-path M v)
(define N (/ (vector-length v) M))
(define (flood-cell idx)
(cond
[(= (quotient idx M) N) #t] ; wootiments!
[(has-fluid? (vector-ref v idx)) #f] ; been here
[else (define cell (vector-ref v idx))
(vector-set! v idx (bitwise-ior cell 16))
(or (and (not (has-bottom-wall? cell)) (flood-cell (+ idx M)))
(and (not (has-left-wall? cell)) (flood-cell (- idx 1)))
(and (not (has-right-wall? cell)) (flood-cell (+ idx 1)))
(and (not (has-top-wall? cell))
(>= idx M) ; not top row
(flood-cell (- idx M))))]))
(for/first ((m (in-range M))
#:unless (has-top-wall? (vector-ref v m))
#:when (flood-cell m)) #t))
(define t (make-parameter 1000))
(define (experiment p)
(/ (for*/sum ((sample (in-range (t)))
(v (in-value (bonded-percol-grid 10 10 p)))
#:when (find-bonded-grid-t/b-path 10 v)) 1)
(t)))
(define (main)
(for ((tenths (in-range 0 (add1 10))))
(define p (/ tenths 10))
(define e (experiment p))
(printf "proportion of grids that percolate p=~a : ~a (~a)~%"
p e (real->decimal-string e 5))))
(module+ test
(define (make/display/flood/display-bonded-grid M N p attempts (atmpt 1))
(define v (bonded-percol-grid M N p))
(define v+ (vector-copy v))
(cond [(or (find-bonded-grid-t/b-path M v+) (= attempts 0))
(define v* (vector-copy v+))
(define (flood-bonded-grid)
(when (find-bonded-grid-t/b-path M v*)
(flood-bonded-grid)))
(flood-bonded-grid)
(display-percol-grid M v v+ v*)
(printf "After ~a attempt(s)~%~%" atmpt)]
[else
(make/display/flood/display-bonded-grid
M N p (sub1 attempts) (add1 atmpt))]))
(make/display/flood/display-bonded-grid 10 10 0 20)
(make/display/flood/display-bonded-grid 10 10 .25 20)
(make/display/flood/display-bonded-grid 10 10 .50 20)
(make/display/flood/display-bonded-grid 10 10 .75 20000))
- Output:
Welcome to DrRacket, version 5.3.5 [3m]. Language: racket [custom]; memory limit: 1024 MB. +.+.+.+.+.+.+.+.+.+.+ +#+.+.+.+.+.+.+.+.+.+ +#+#+#+#+#+#+#+#+#+#+ |...................| |##.................| |###################| +.+.+.+.+.+.+.+.+.+.+ +#+.+.+.+.+.+.+.+.+.+ +#+#+#+#+#+#+#+#+#+#+ |...................| |##.................| |###################| +.+.+.+.+.+.+.+.+.+.+ +#+.+.+.+.+.+.+.+.+.+ +#+#+#+#+#+#+#+#+#+#+ |...................| |##.................| |###################| +.+.+.+.+.+.+.+.+.+.+ +#+.+.+.+.+.+.+.+.+.+ +#+#+#+#+#+#+#+#+#+#+ |...................| |##.................| |###################| +.+.+.+.+.+.+.+.+.+.+ +#+.+.+.+.+.+.+.+.+.+ +#+#+#+#+#+#+#+#+#+#+ |...................| |##.................| |###################| +.+.+.+.+.+.+.+.+.+.+ +#+.+.+.+.+.+.+.+.+.+ +#+#+#+#+#+#+#+#+#+#+ |...................| |##.................| |###################| +.+.+.+.+.+.+.+.+.+.+ +#+.+.+.+.+.+.+.+.+.+ +#+#+#+#+#+#+#+#+#+#+ |...................| |##.................| |###################| +.+.+.+.+.+.+.+.+.+.+ +#+.+.+.+.+.+.+.+.+.+ +#+#+#+#+#+#+#+#+#+#+ |...................| |##.................| |###################| +.+.+.+.+.+.+.+.+.+.+ +#+.+.+.+.+.+.+.+.+.+ +#+#+#+#+#+#+#+#+#+#+ |...................| |##.................| |###################| +.+.+.+.+.+.+.+.+.+.+ +#+.+.+.+.+.+.+.+.+.+ +#+#+#+#+#+#+#+#+#+#+ |...................| |##.................| |###################| +.+.+.+.+.+.+.+.+.+.+ +#+.+.+.+.+.+.+.+.+.+ +#+#+#+#+#+#+#+#+#+#+ After 1 attempt(s) +.+-+-+.+.+.+-+.+.+.+ +#+-+-+.+.+.+-+.+.+.+ +#+-+-+#+#+#+-+#+#+#+ |...................| |##.................| |##..###############| +.+-+.+-+.+.+-+-+-+.+ +#+-+.+-+.+.+-+-+-+.+ +#+-+#+-+#+#+-+-+-+#+ |.................|.| |##...............|.| |##..##..####.....|#| +.+-+.+.+.+.+-+.+.+.+ +#+-+.+.+.+.+-+.+.+.+ +#+-+#+.+#+#+-+.+.+#+ |.............|.....| |##...........|.....| |######..#####|....#| +.+.+.+.+.+.+.+.+.+.+ +#+.+.+.+.+.+.+.+.+.+ +#+#+#+.+#+#+#+.+.+#+ |.....|...|.|.......| |##...|...|.|.......| |#####|..#|#|##....#| +.+.+.+.+.+.+.+-+-+.+ +#+.+.+.+.+.+.+-+-+.+ +#+#+#+#+#+#+#+-+-+#+ |.|.............|...| |#|.............|...| |#|############.|..#| +.+-+-+.+-+.+.+.+.+.+ +#+-+-+.+-+.+.+.+.+.+ +#+-+-+#+-+#+#+.+.+#+ |...................| |##.................| |##....##..####....#| +.+.+-+.+.+.+.+-+-+.+ +#+.+-+.+.+.+.+-+-+.+ +#+.+-+#+.+#+#+-+-+#+ |...|...|...........| |##.|...|...........| |##.|###|..####..###| +.+.+.+-+.+.+.+.+.+.+ +#+#+.+-+.+.+.+.+.+.+ +#+#+#+-+.+#+#+.+#+#+ |...|...|.........|.| |###|...|.........|.| |###|##.|..####..#|#| +-+.+.+-+-+.+.+.+.+-+ +-+#+.+-+-+.+.+.+.+-+ +-+#+#+-+-+#+#+.+#+-+ |.....|.........|...| |..##.|.........|...| |..###|....####.|###| +.+.+.+.+.+.+.+.+.+.+ +.+#+.+.+.+.+.+.+.+.+ +.+#+#+.+.+#+#+#+#+#+ |.........|.......|.| |..##.....|.......|.| |..####...|#######|#| +.+.+.+-+.+.+-+.+-+.+ +.+#+.+-+.+.+-+.+-+.+ +.+#+#+-+.+#+-+#+-+#+ After 1 attempt(s) +.+.+.+.+-+-+.+-+.+.+ +#+#+#+#+-+-+.+-+.+.+ +#+#+#+#+-+-+#+-+#+#+ |.........|.|.|...|.| |########.|.|.|...|.| |########.|.|#|###|#| +.+-+-+.+-+-+-+.+.+-+ +#+-+-+#+-+-+-+.+.+-+ +#+-+-+#+-+-+-+#+#+-+ |...|...|...|.|.|.|.| |###|..#|...|.|.|.|.| |###|..#|...|.|#|#|.| +-+-+.+.+.+.+-+.+-+.+ +-+-+.+#+#+.+-+.+-+.+ +-+-+.+#+#+.+-+#+-+.+ |.|.|.|...|.|.|.|...| |.|.|.|###|.|.|.|...| |.|.|.|###|.|.|#|...| +.+-+.+-+.+.+.+-+.+-+ +.+-+.+-+#+.+.+-+.+-+ +.+-+.+-+#+.+.+-+.+-+ |.|...|...|.|.....|.| |.|...|###|.|.....|.| |.|...|###|.|.....|.| +.+-+.+.+.+-+-+.+.+.+ +.+-+.+#+#+-+-+.+.+.+ +.+-+.+#+#+-+-+.+.+.+ |.|...|.|.....|.....| |.|...|#|####.|.....| |.|...|#|####.|.....| +-+.+-+.+-+.+-+.+-+-+ +-+.+-+#+-+#+-+#+-+-+ +-+.+-+#+-+#+-+#+-+-+ |.|.|.....|.....|...| |.|.|#####|#####|...| |.|.|#####|#####|...| +-+-+.+.+.+.+-+.+-+-+ +-+-+#+#+#+#+-+#+-+-+ +-+-+#+#+#+#+-+#+-+-+ |...|.|.....|.......| |...|#|#####|..##...| |...|#|#####|..##...| +-+-+-+-+-+-+-+.+-+-+ +-+-+-+-+-+-+-+#+-+-+ +-+-+-+-+-+-+-+#+-+-+ |.|...|.|.|.......|.| |.|...|.|.|######.|.| |.|...|.|.|######.|.| +.+-+-+-+.+.+-+.+.+.+ +.+-+-+-+.+#+-+#+.+.+ +.+-+-+-+.+#+-+#+.+.+ |.|...|.......|.|.|.| |.|...|....##.|#|.|.| |.|...|....##.|#|.|.| +.+.+-+.+.+.+-+-+-+-+ +.+.+-+.+.+#+-+-+-+-+ +.+.+-+.+.+#+-+-+-+-+ |.|.........|.....|.| |.|........#|.....|.| |.|........#|.....|.| +-+.+-+-+-+.+.+.+-+.+ +-+.+-+-+-+#+.+.+-+.+ +-+.+-+-+-+#+.+.+-+.+ After 2 attempt(s) +-+-+-+-+-+-+.+-+-+.+ +-+-+-+-+-+-+#+-+-+.+ +-+-+-+-+-+-+#+-+-+#+ |.|.|...|.|.|.|.|...| |.|.|...|.|.|#|.|...| |.|.|...|.|.|#|.|###| +-+-+-+-+-+-+.+-+-+-+ +-+-+-+-+-+-+#+-+-+-+ +-+-+-+-+-+-+#+-+-+-+ |.|.|.|...|.|...|.|.| |.|.|.|...|.|##.|.|.| |.|.|.|...|.|##.|.|.| +.+.+.+.+.+-+.+-+-+.+ +.+.+.+.+.+-+#+-+-+.+ +.+.+.+.+.+-+#+-+-+.+ |.|.|.|.|...|.|...|.| |.|.|.|.|...|#|...|.| |.|.|.|.|...|#|...|.| +.+-+.+-+-+-+.+-+.+.+ +.+-+.+-+-+-+#+-+.+.+ +.+-+.+-+-+-+#+-+.+.+ |...|...|.|.|...|.|.| |...|...|.|.|###|.|.| |...|...|.|.|###|.|.| +-+-+-+-+-+-+-+.+-+-+ +-+-+-+-+-+-+-+#+-+-+ +-+-+-+-+-+-+-+#+-+-+ |.|.......|.....|.|.| |.|.......|#####|.|.| |.|.......|#####|.|.| +.+-+-+-+.+.+-+-+-+-+ +.+-+-+-+.+#+-+-+-+-+ +.+-+-+-+.+#+-+-+-+-+ |.|.|.|.|.|.|.|.....| |.|.|.|.|.|#|.|.....| |.|.|.|.|.|#|.|.....| +-+-+-+-+-+.+.+-+.+-+ +-+-+-+-+-+#+.+-+.+-+ +-+-+-+-+-+#+.+-+.+-+ |...|.|.|.|.|.|.|.|.| |...|.|.|.|#|.|.|.|.| |...|.|.|.|#|.|.|.|.| +.+.+.+-+-+.+.+-+.+-+ +.+.+.+-+-+#+#+-+.+-+ +.+.+.+-+-+#+#+-+.+-+ |.|.|.|.|.|...|.|...| |.|.|.|.|.|###|.|...| |.|.|.|.|.|###|.|...| +-+-+-+-+-+-+.+.+.+-+ +-+-+-+-+-+-+#+.+.+-+ +-+-+-+-+-+-+#+.+.+-+ |.|.|.|.|.|...|...|.| |.|.|.|.|.|###|...|.| |.|.|.|.|.|###|...|.| +-+-+.+-+-+.+-+-+-+.+ +-+-+.+-+-+#+-+-+-+.+ +-+-+.+-+-+#+-+-+-+.+ |.|.|.|...|...|.|...| |.|.|.|...|###|.|...| |.|.|.|...|###|.|...| +-+-+.+-+.+-+.+.+.+-+ +-+-+.+-+.+-+#+.+.+-+ +-+-+.+-+.+-+#+.+.+-+ After 4611 attempt(s) > (main) proportion of grids that percolate p=0 : 1 (1.00000) proportion of grids that percolate p=1/10 : 1 (1.00000) proportion of grids that percolate p=1/5 : 1 (1.00000) proportion of grids that percolate p=3/10 : 199/200 (0.99500) proportion of grids that percolate p=2/5 : 179/200 (0.89500) proportion of grids that percolate p=1/2 : 451/1000 (0.45100) proportion of grids that percolate p=3/5 : 29/500 (0.05800) proportion of grids that percolate p=7/10 : 1/1000 (0.00100) proportion of grids that percolate p=4/5 : 0 (0.00000) proportion of grids that percolate p=9/10 : 0 (0.00000) proportion of grids that percolate p=1 : 0 (0.00000)
Raku
(formerly Perl 6)
Starts "filling" from the top left. Fluid flow favours directions in Down, Left, Right, Up order. I interpreted p to be porosity, so small p mean low permeability, large p means high permeability.
my @bond;
my $grid = 10;
my $geom = $grid - 1;
my $water = '▒';
enum Direction <DeadEnd Up Right Down Left>;
say 'Sample percolation at .6';
percolate .6;
.join.say for @bond;
say "\n";
my $tests = 100;
say "Doing $tests trials at each porosity:";
for .1, .2 ... 1 -> $p {
printf "p = %0.1f: %0.2f\n", $p, (sum percolate($p) xx $tests) / $tests
}
sub percolate ( $prob ) {
generate $prob;
my @stack;
my $current = [1;0];
$current.&fill;
loop {
if my $dir = direction( $current ) {
@stack.push: $current;
$current = move $dir, $current
}
else {
return False unless @stack;
$current = @stack.pop
}
return True if $current[1] == +@bond - 1
}
sub direction( [$x, $y] ) {
( Down if @bond[$y + 1][$x].contains: ' ' ) ||
( Left if @bond[$y][$x - 1].contains: ' ' ) ||
( Right if @bond[$y][$x + 1].contains: ' ' ) ||
( Up if @bond[$y - 1][$x].defined && @bond[$y - 1][$x].contains: ' ' ) ||
DeadEnd
}
sub move ( $dir, @cur ) {
my ( $x, $y ) = @cur;
given $dir {
when Up { [$x,--$y].&fill xx 2 }
when Down { [$x,++$y].&fill xx 2 }
when Left { [--$x,$y].&fill xx 2 }
when Right { [++$x,$y].&fill xx 2 }
}
[$x, $y]
}
sub fill ( [$x, $y] ) { @bond[$y;$x].=subst(' ', $water, :g) }
}
sub generate ( $prob = .5 ) {
@bond = ();
my $sp = ' ';
append @bond, [flat '│', ($sp, ' ') xx $geom, $sp, '│'],
[flat '├', (h(), '┬') xx $geom, h(), '┤'];
append @bond, [flat '│', ($sp, v()) xx $geom, $sp, '│'],
[flat '├', (h(), '┼') xx $geom, h(), '┤'] for ^$geom;
append @bond, [flat '│', ($sp, v()) xx $geom, $sp, '│'],
[flat '├', (h(), '┴') xx $geom, h(), '┤'],
[flat '│', ($sp, ' ') xx $geom, $sp, '│'];
sub h () { rand < $prob ?? $sp !! '───' }
sub v () { rand < $prob ?? ' ' !! '│' }
}
- Output:
Sample percolation at .6 │▒▒▒ │ ├▒▒▒┬ ┬───┬ ┬ ┬ ┬ ┬ ┬───┬ ┤ │▒▒▒▒▒▒▒ │ │ │ ├───┼▒▒▒┼ ┼ ┼ ┼ ┼ ┼───┼ ┼ ┤ │▒▒▒▒▒▒▒▒▒▒▒│ │ │ │ │ │ │ ├▒▒▒┼───┼▒▒▒┼ ┼───┼ ┼───┼ ┼ ┼ ┤ │▒▒▒│▒▒▒▒▒▒▒▒▒▒▒ │ │ │ ├▒▒▒┼───┼───┼▒▒▒┼ ┼ ┼───┼ ┼ ┼ ┤ │▒▒▒│ ▒▒▒│ │ │ │ │ ├───┼ ┼ ┼▒▒▒┼───┼ ┼ ┼ ┼ ┼───┤ │ │▒▒▒ │ │ ├ ┼───┼ ┼▒▒▒┼───┼───┼───┼───┼ ┼───┤ │ │▒▒▒│ │ ├───┼ ┼───┼▒▒▒┼───┼───┼───┼───┼ ┼───┤ │▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒ │ │ │ ├▒▒▒┼▒▒▒┼───┼▒▒▒┼───┼ ┼───┼ ┼ ┼ ┤ │▒▒▒│▒▒▒▒▒▒▒│▒▒▒▒▒▒▒│ │ ├▒▒▒┼───┼───┼───┼───┼───┼ ┼ ┼ ┼ ┤ │▒▒▒▒▒▒▒ │ │ │ │ ├▒▒▒┼▒▒▒┼───┼───┼ ┼───┼───┼ ┼ ┼ ┤ │▒▒▒│▒▒▒ │ │ │ ├───┴▒▒▒┴ ┴ ┴ ┴───┴ ┴ ┴ ┴───┤ │ ▒▒▒ │ Doing 100 trials at each porosity: p = 0.1: 0.00 p = 0.2: 0.00 p = 0.3: 0.00 p = 0.4: 0.05 p = 0.5: 0.42 p = 0.6: 0.92 p = 0.7: 1.00 p = 0.8: 1.00 p = 0.9: 1.00 p = 1.0: 1.00
Swift
let randMax = 32767.0
let filled = 1
let rightWall = 2
let bottomWall = 4
final class Percolate {
let height: Int
let width: Int
private var grid: [Int]
private var end: Int
init(height: Int, width: Int) {
self.height = height
self.width = width
self.end = width
self.grid = [Int](repeating: 0, count: width * (height + 2))
}
private func fill(at p: Int) -> Bool {
guard grid[p] & filled == 0 else { return false }
grid[p] |= filled
guard p < end else { return true }
return (((grid[p + 0] & bottomWall) == 0) && fill(at: p + width)) ||
(((grid[p + 0] & rightWall) == 0) && fill(at: p + 1)) ||
(((grid[p - 1] & rightWall) == 0) && fill(at: p - 1)) ||
(((grid[p - width] & bottomWall) == 0) && fill(at: p - width))
}
func makeGrid(porosity p: Double) {
grid = [Int](repeating: 0, count: width * (height + 2))
end = width
let thresh = Int(randMax * p)
for i in 0..<width {
grid[i] = bottomWall | rightWall
}
for _ in 0..<height {
for _ in stride(from: width - 1, through: 1, by: -1) {
let r1 = Int.random(in: 0..<Int(randMax)+1)
let r2 = Int.random(in: 0..<Int(randMax)+1)
grid[end] = (r1 < thresh ? bottomWall : 0) | (r2 < thresh ? rightWall : 0)
end += 1
}
let r3 = Int.random(in: 0..<Int(randMax)+1)
grid[end] = rightWall | (r3 < thresh ? bottomWall : 0)
end += 1
}
}
@discardableResult
func percolate() -> Bool {
var i = 0
while i < width && !fill(at: width + i) {
i += 1
}
return i < width
}
func showGrid() {
for _ in 0..<width {
print("+--", terminator: "")
}
print("+")
for i in 0..<height {
print(i == height ? " " : "|", terminator: "")
for j in 0..<width {
print(grid[i * width + j + width] & filled != 0 ? "[]" : " ", terminator: "")
print(grid[i * width + j + width] & rightWall != 0 ? "|" : " ", terminator: "")
}
print()
guard i != height else { return }
for j in 0..<width {
print(grid[i * width + j + width] & bottomWall != 0 ? "+--" : "+ ", terminator: "")
}
print("+")
}
}
}
let p = Percolate(height: 10, width: 10)
p.makeGrid(porosity: 0.5)
p.percolate()
p.showGrid()
print("Running \(p.height) x \(p.width) grid 10,000 times for each porosity")
for factor in 1...10 {
var count = 0
let porosity = Double(factor) / 10.0
for _ in 0..<10_000 {
p.makeGrid(porosity: porosity)
if p.percolate() {
count += 1
}
}
print("p = \(porosity): \(Double(count) / 10_000.0)")
}
- Output:
+--+--+--+--+--+--+--+--+--+--+ |[]| | | | | | | + + +--+--+ + +--+ + + + |[] [] | | | | +--+ +--+--+ +--+--+ +--+ + | [] [] []| | | | + +--+--+ + +--+--+ + + + | [] []| | | +--+--+ +--+ + + + +--+ + | [] [] | | | | + +--+--+ +--+--+--+--+ +--+ | | | |[] | | | + + +--+ +--+--+--+--+--+--+ | | |[] [] [] | | +--+--+--+--+--+ + +--+--+ + | | | |[]| | | | + + + + +--+ +--+ + +--+ | | | | [] []| | | | +--+--+--+--+ +--+--+--+--+ + | | | [] | | | +--+--+ +--+ +--+ +--+--+ + Running 10 x 10 grid 10,000 times for each porosity p = 0.1: 1.0 p = 0.2: 1.0 p = 0.3: 0.9968 p = 0.4: 0.9125 p = 0.5: 0.4959 p = 0.6: 0.0858 p = 0.7: 0.004 p = 0.8: 0.0 p = 0.9: 0.0 p = 1.0: 0.0
Tcl
package require Tcl 8.6
# Structure the bond percolation system as a class
oo::class create BondPercolation {
variable hwall vwall cells M N
constructor {width height probability} {
set M $height
set N $width
for {set i 0} {$i <= $height} {incr i} {
for {set j 0;set walls {}} {$j < $width} {incr j} {
lappend walls [expr {rand() < $probability}]
}
lappend hwall $walls
}
for {set i 0} {$i <= $height} {incr i} {
for {set j 0;set walls {}} {$j <= $width} {incr j} {
lappend walls [expr {$j==0 || $j==$width || rand() < $probability}]
}
lappend vwall $walls
}
set cells [lrepeat $height [lrepeat $width 0]]
}
method print {{percolated ""}} {
set nw [string length $M]
set grid $cells
if {$percolated ne ""} {
lappend grid [lrepeat $N 0]
lset grid end $percolated 1
}
foreach hws $hwall vws [lrange $vwall 0 end-1] r $grid {
incr row
puts -nonewline [string repeat " " [expr {$nw+2}]]
foreach w $hws {
puts -nonewline [if {$w} {subst "+-"} {subst "+ "}]
}
puts "+"
puts -nonewline [format "%-*s" [expr {$nw+2}] [expr {
$row>$M ? $percolated eq "" ? " " : ">" : "$row)"
}]]
foreach v $vws c $r {
puts -nonewline [if {$v==1} {subst "|"} {subst " "}]
puts -nonewline [if {$c==1} {subst "#"} {subst " "}]
}
puts ""
}
}
method percolate {} {
try {
for {set i 0} {$i < $N} {incr i} {
if {![lindex $hwall 0 $i]} {
my FloodFill $i 0
}
}
return ""
} trap PERCOLATED n {
return $n
}
}
method FloodFill {x y} {
# fill cell
lset cells $y $x 1
# bottom
if {![lindex $hwall [expr {$y+1}] $x]} {
if {$y == $N-1} {
# THE bottom
throw PERCOLATED $x
}
if {$y < $N-1 && ![lindex $cells [expr {$y+1}] $x]} {
my FloodFill $x [expr {$y+1}]
}
}
# left
if {![lindex $vwall $y $x] && ![lindex $cells $y [expr {$x-1}]]} {
my FloodFill [expr {$x-1}] $y
}
# right
if {![lindex $vwall $y [expr {$x+1}]] && ![lindex $cells $y [expr {$x+1}]]} {
my FloodFill [expr {$x+1}] $y
}
# top
if {$y>0 && ![lindex $hwall $y $x] && ![lindex $cells [expr {$y-1}] $x]} {
my FloodFill $x [expr {$y-1}]
}
}
}
# Demonstrate one run
puts "Sample percolation, 10x10 p=0.5"
BondPercolation create bp 10 10 0.5
bp print [bp percolate]
bp destroy
puts ""
# Collect some aggregate 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 [BondPercolation new 10 10 $p]
if {[$bp percolate] ne ""} {
incr tot
}
$bp destroy
}
puts [format "p=%.2f: %2.1f%%" $p [expr {$tot*100./$tries}]]
}
}}
- Output:
Sample percolation, 10x10 p=0.5
+ + +-+-+-+ +-+ +-+ +
1) |# | | | | |
+ +-+ + + +-+ + + +-+
2) |#| | | | |
+ + +-+ +-+ +-+ + +-+
3) |# # #|# # #| | | |
+ +-+ + +-+ +-+ +-+ +
4) |#|# # #| |# | |
+-+ + + +-+ +-+-+ +-+
5) |# # # #| |# | | |
+-+-+-+-+ + + + +-+-+
6) | | | |#| | |
+-+-+-+-+-+ + +-+-+ +
7) | | | | |# | |
+ +-+ +-+-+ +-+ +-+ +
8) | | # | |
+ +-+-+ +-+ + + + + +
9) | # |
+ + +-+-+ + +-+-+ + +
10) | | | # | | |
+ + + + + + +-+ +-+ +
> #
Percentage of tries that percolate, varying p
p=0.00: 100.0%
p=0.10: 100.0%
p=0.20: 100.0%
p=0.30: 100.0%
p=0.40: 86.0%
p=0.50: 50.0%
p=0.60: 6.0%
p=0.70: 0.0%
p=0.80: 0.0%
p=0.90: 0.0%
p=1.00: 0.0%
Wren
import "random" for Random
import "./fmt" for Fmt
var rand = Random.new()
var RAND_MAX = 32767
// cell states
var FILL = 1
var RWALL = 2 // right wall
var BWALL = 4 // bottom wall
var x = 10
var y = 10
var grid = List.filled(x * (y + 2), 0)
var cells = 0
var end = 0
var m = 0
var n = 0
var makeGrid = Fn.new { |p|
var thresh = (p * RAND_MAX).truncate
m = x
n = y
for (i in 0...grid.count) grid[i] = 0 // clears grid
for (i in 0...m) grid[i] = BWALL | RWALL
cells = m
end = m
for (i in 0...y) {
for (j in x - 1..1) {
var r1 = rand.int(RAND_MAX + 1)
var r2 = rand.int(RAND_MAX + 1)
grid[end] = ((r1 < thresh) ? BWALL : 0) |
((r2 < thresh) ? RWALL : 0)
end = end + 1
}
var r3 = rand.int(RAND_MAX + 1)
grid[end] = RWALL | ((r3 < thresh) ? BWALL : 0)
end = end + 1
}
}
var showGrid = Fn.new {
for (j in 0...m) System.write("+--")
System.print("+")
for (i in 0..n) {
System.write((i == n) ? " " : "|")
for (j in 0...m) {
System.write(((grid[i * m + j + cells] & FILL) != 0) ? "[]" : " ")
System.write(((grid[i * m + j + cells] & RWALL) != 0) ? "|" : " ")
}
System.print()
if (i == n) return
for (j in 0...m) {
System.write(((grid[i * m + j + cells] & BWALL) != 0) ? "+--" : "+ ")
}
System.print("+")
}
}
var fill // recursive
fill = Fn.new { |p|
if ((grid[p] & FILL) != 0) return false
grid[p] = grid[p] | FILL
if (p >= end) return true // success: reached bottom row
return (((grid[p + 0] & BWALL) == 0) && fill.call(p + m)) ||
(((grid[p + 0] & RWALL) == 0) && fill.call(p + 1)) ||
(((grid[p - 1] & RWALL) == 0) && fill.call(p - 1)) ||
(((grid[p - m] & BWALL) == 0) && fill.call(p - m))
}
var percolate = Fn.new {
var i = 0
while (i < m && !fill.call(cells + i)) i = i + 1
return i < m
}
makeGrid.call(0.5)
percolate.call()
showGrid.call()
System.print("\nRunning %(x) x %(y) grids 10,000 times for each p:")
for (p in 1..9) {
var cnt = 0
var pp = p / 10
for (i in 0...10000) {
makeGrid.call(pp)
if (percolate.call()) cnt = cnt + 1
}
Fmt.print("p = $3g: $.4f", pp, cnt / 10000)
}
- Output:
Sample run:
+--+--+--+--+--+--+--+--+--+--+
|[]|[] []|[] [] []|[] []|[] []|
+--+ + +--+ +--+--+ +--+ +
| [] []|[] []|[] [] []|[] []|
+--+--+--+--+--+--+--+--+ + +
| | | | |[] [] [] [] []|
+--+--+ + + +--+--+--+ +--+
| | |[] [] [] []| |
+--+ +--+ + + + +--+ + +
| | |[]|[]|[] []| |
+ +--+--+ + + +--+--+--+--+
| | | |[]| | |
+--+ + + +--+ +--+--+ +--+
| | |[]| | |
+ + + +--+--+ + +--+--+--+
| |[] []| |
+ +--+--+--+ +--+ +--+--+--+
| | [] [] []| | |
+--+--+--+--+--+--+ +--+--+ +
| | |[] []| | | |
+ +--+--+--+ + +--+ + +--+
[]
Running 10 x 10 grids 10,000 times for each p:
p = 0.1 : 1.0000
p = 0.2 : 0.9999
p = 0.3 : 0.9970
p = 0.4 : 0.9120
p = 0.5 : 0.5022
p = 0.6 : 0.0829
p = 0.7 : 0.0026
p = 0.8 : 0.0000
p = 0.9 : 0.0000
zkl
// cell states
const FILLED=1; // and odd
const RWALL =2; // right wall
const BWALL =4; // bottom wall
fcn P(p,wall){ (0.0).random(1)<p and wall or 0 }
fcn makeGrid(m,n,p){
// Allocate two addition rows to avoid checking bounds.
// Bottom row is also required by drippage
grid:=Data(m*(n+2));
do(m){ grid.write(BWALL + RWALL); } // grid is topped with walls
do(n){
do(m-1){ grid.write( P(p,BWALL) + P(p,RWALL) ) }
grid.write(RWALL + P(p,BWALL)); // right border is all right wall, as is left border
}
do(m){ grid.write(0); } // for drips off the bottom of grid
grid
}
fcn show(grid,m,n){ n+=1;
println("+--"*m,"+");
foreach i in ([1..n]){ y:=i*m;
print(i==n and " " or "|"); // bottom row is special, otherwise always have left wall
foreach j in (m){ c:=grid[y + j];
print(c.bitAnd(FILLED) and "**" or " ", c.bitAnd(RWALL)and"|"or" ");
}
println();
if(i==n) return(); // nothing under the bottom row
foreach j in (m){ print((grid[y + j].bitAnd(BWALL)) and "+--" or "+ "); }
println("+");
}
}
fcn fill(grid,x,m){
if(grid[x].isOdd) return(False); // aka .bitAnd(FILLED) aka already been here
grid[x]+=FILLED;
if(x+m>=grid.len()) return(True); // success: reached bottom row
return(( not grid[x] .bitAnd(BWALL) and fill(grid,x + m,m) ) or // down
( not grid[x] .bitAnd(RWALL) and fill(grid,x + 1,m) ) or // right
( not grid[x - 1].bitAnd(RWALL) and fill(grid,x - 1,m) ) or // left
( not grid[x - m].bitAnd(BWALL) and fill(grid,x - m,m) )); // up
}
fcn percolate(grid,m){
i:=0; while(i<m and not fill(grid,i+m,m)){ i+=1; } // pour juice on top row
return(i<m); // percolated through the grid?
}grid:=makeGrid(10,10,0.40);
println("Did liquid percolate: ",percolate(grid,10)); show(grid,10,10);
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(10,10,p),10); }
"p=%.1f: %.4f".fmt(p, cnt/10000).println();
}- Output:
Did liquid percolate: True
+--+--+--+--+--+--+--+--+--+--+
|** ** | | |
+--+ +--+--+ + + + + + +
| **| | | | |
+ + + +--+ +--+--+ +--+--+
| ** ** | |
+--+--+ + + + +--+ + +--+
| |**| | | |
+ + + + +--+ + +--+ + +
| ** **| |** **| | |
+ +--+--+ +--+ + +--+ + +
| | |**| |**|** ** |
+ + + + +--+ +--+ + + +
| | |** ** ** **|** |
+--+--+--+--+ +--+--+ +--+--+
| | |** **| ** | |
+ + +--+ + + + + +--+--+
| |** **| |**| | |
+ +--+ +--+--+--+--+ + + +
| | | **| |
+ + + + + + +--+ + + +
**
Running 10,000 tests for each case:
p=0.0: 1.0000
p=0.1: 1.0000
p=0.2: 1.0000
p=0.3: 0.9978
p=0.4: 0.9163
p=0.5: 0.5017
p=0.6: 0.0890
p=0.7: 0.0033
p=0.8: 0.0000
p=0.9: 0.0000
p=1.0: 0.0000