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{{task|Percolation Simulations}}{{Percolation Simulation}} Let be a 2D boolean square matrix of values of either 1 or 0 where the probability of any value being 1 is , (and of 0 is therefore ). We define a ''cluster'' of 1's as being a group of 1's connected vertically or horizontally (i.e., using the [[wp:Von Neumann neighborhood|Von Neumann neighborhood rule]]) and bounded by either or by the limits of the matrix. Let the number of such clusters in such a randomly constructed matrix be .
Percolation theory states that (the mean cluster density) will satisfy as tends to infinity. For , is found numerically to approximate ...
;Task
Show the effect of varying on the accuracy of simulated for and for values of up to at least . Any calculation of for finite is subject to randomness, so an approximation should be computed as the average of runs, where ≥ .
For extra credit, graphically show clusters in a , grid.
Show your output here.
;See also
- [http://mathworld.wolfram.com/s-Cluster.html s-Cluster] on Wolfram mathworld.
C
#include <stdio.h> #include <stdlib.h> int *map, w, ww; void make_map(double p) { int i, thresh = RAND_MAX * p; i = ww = w * w; map = realloc(map, i * sizeof(int)); while (i--) map[i] = -(rand() < thresh); } char alpha[] = "+.ABCDEFGHIJKLMNOPQRSTUVWXYZ" "abcdefghijklmnopqrstuvwxyz"; #define ALEN ((int)(sizeof(alpha) - 3)) void show_cluster(void) { int i, j, *s = map; for (i = 0; i < w; i++) { for (j = 0; j < w; j++, s++) printf(" %c", *s < ALEN ? alpha[1 + *s] : '?'); putchar('\n'); } } void recur(int x, int v) { if (x >= 0 && x < ww && map[x] == -1) { map[x] = v; recur(x - w, v); recur(x - 1, v); recur(x + 1, v); recur(x + w, v); } } int count_clusters(void) { int i, cls; for (cls = i = 0; i < ww; i++) { if (-1 != map[i]) continue; recur(i, ++cls); } return cls; } double tests(int n, double p) { int i; double k; for (k = i = 0; i < n; i++) { make_map(p); k += (double)count_clusters() / ww; } return k / n; } int main(void) { w = 15; make_map(.5); printf("width=15, p=0.5, %d clusters:\n", count_clusters()); show_cluster(); printf("\np=0.5, iter=5:\n"); for (w = 1<<2; w <= 1<<14; w<<=2) printf("%5d %9.6f\n", w, tests(5, .5)); free(map); return 0; }
{{out}}
width=15, p=0.5, 23 clusters:
A . . . B . C C C C . D . E .
A . . B B . . . . . . . . . .
A . . . . . F . . G . H . . I
. . J J J . . K K . L . M . I
. J J . . . K K K K . M M . .
. . . . K K . K . K . M . N .
O O . K K K . K . . . . N N N
. O O . K K K K K . P . N . .
Q . . K K . . . K . P . . . .
. R . K K . . K K . P . . S .
. . K K . . . . K . P . . . K
K K K K K . . K K . . T . . K
K . K . . . U . K . . T . . .
K . K K K K . K K K . T . . .
. . K . K . V . K K . . . W .
p=0.5, iter=5:
4 0.125000
16 0.083594
64 0.064453
256 0.066864
1024 0.065922
4096 0.065836
16384 0.065774
D
{{trans|python}}
import std.stdio, std.algorithm, std.random, std.math, std.array, std.range, std.ascii; alias Cell = ubyte; alias Grid = Cell[][]; enum Cell notClustered = 1; // Filled cell, but not in a cluster. Grid initialize(Grid grid, in double prob, ref Xorshift rng) nothrow { foreach (row; grid) foreach (ref cell; row) cell = Cell(rng.uniform01 < prob); return grid; } void show(in Grid grid) { immutable static cell2char = " #" ~ letters; writeln('+', "-".replicate(grid.length), '+'); foreach (row; grid) { write('|'); row.map!(c => c < cell2char.length ? cell2char[c] : '@').write; writeln('|'); } writeln('+', "-".replicate(grid.length), '+'); } size_t countClusters(bool justCount=false)(Grid grid) pure nothrow @safe @nogc { immutable side = grid.length; static if (justCount) enum Cell clusterID = 2; else Cell clusterID = 1; void walk(in size_t r, in size_t c) nothrow @safe @nogc { grid[r][c] = clusterID; // Fill grid. if (r < side - 1 && grid[r + 1][c] == notClustered) // Down. walk(r + 1, c); if (c < side - 1 && grid[r][c + 1] == notClustered) // Right. walk(r, c + 1); if (c > 0 && grid[r][c - 1] == notClustered) // Left. walk(r, c - 1); if (r > 0 && grid[r - 1][c] == notClustered) // Up. walk(r - 1, c); } size_t nClusters = 0; foreach (immutable r; 0 .. side) foreach (immutable c; 0 .. side) if (grid[r][c] == notClustered) { static if (!justCount) clusterID++; nClusters++; walk(r, c); } return nClusters; } double clusterDensity(Grid grid, in double prob, ref Xorshift rng) { return grid.initialize(prob, rng).countClusters!true / double(grid.length ^^ 2); } void showDemo(in size_t side, in double prob, ref Xorshift rng) { auto grid = new Grid(side, side); grid.initialize(prob, rng); writefln("Found %d clusters in this %d by %d grid:\n", grid.countClusters, side, side); grid.show; } void main() { immutable prob = 0.5; immutable nIters = 5; auto rng = Xorshift(unpredictableSeed); showDemo(15, prob, rng); writeln; foreach (immutable i; iota(4, 14, 2)) { immutable side = 2 ^^ i; auto grid = new Grid(side, side); immutable density = nIters .iota .map!(_ => grid.clusterDensity(prob, rng)) .sum / nIters; writefln("n_iters=%3d, p=%4.2f, n=%5d, sim=%7.8f", nIters, prob, side, density); } }
{{out}}
Found 26 clusters in this 15 by 15 grid:
+---------------+
| AA B CCCC |
|AA D E F CC G |
| DDD FF CC H|
| I D FF J K |
| L FF JJJJ |
|L LLL J M|
|LLLLLL JJJ MM|
|L LL L N J M|
|LL O P J M|
|LLL QQ R JJ S |
|LL T RR J SSS|
| L U V JJ S|
| WW XX JJ YY |
| XXX JJ YY|
|ZZ XXX JJ |
+---------------+
n_iters= 5, p=0.50, n= 16, sim=0.09765625
n_iters= 5, p=0.50, n= 64, sim=0.07260742
n_iters= 5, p=0.50, n= 256, sim=0.06679993
n_iters= 5, p=0.50, n= 1024, sim=0.06609497
n_iters= 5, p=0.50, n= 4096, sim=0.06580237
Increasing the index i to 15:
n_iters= 5, p=0.50, n=32768, sim=0.06578374
EchoLisp
We use the canvas bit-map as 2D-matrix. For extra-extra credit, a 800x800 nice cluster tapestry image is shown here : http://www.echolalie.org/echolisp/images/rosetta-clusters-800.png.
(define-constant BLACK (rgb 0 0 0.6))
(define-constant WHITE -1)
;; sets pixels to clusterize to WHITE
;; returns bit-map vector
(define (init-C n p )
(plot-size n n)
(define C (pixels->int32-vector )) ;; get canvas bit-map
(pixels-map (lambda (x y) (if (< (random) p) WHITE BLACK )) C)
C )
;; random color for new cluster
(define (new-color)
(hsv->rgb (random) 0.9 0.9))
;; make-region predicate
(define (in-cluster C x y)
(= (pixel-ref C x y) WHITE))
;; paint all adjacents to (x0,y0) with new color
(define (make-cluster C x0 y0)
(pixel-set! C x0 y0 (new-color))
(make-region in-cluster C x0 y0))
;; task
(define (make-clusters (n 400) (p 0.5))
(define Cn 0)
(define C null)
(for ((t 5)) ;; 5 iterations
(plot-clear)
(set! C (init-C n p))
(for* ((x0 n) (y0 n))
#:when (= (pixel-ref C x0 y0) WHITE)
(set! Cn (1+ Cn))
(make-cluster C x0 y0)))
(writeln 'n n 'Cn Cn 'density (// Cn (* n n) 5) )
(vector->pixels C)) ;; to screen
{{out}}
n 100 Cn 3420 density 0.0684
n 400 Cn 53246 density 0.0665575
n 600 Cn 118346 density 0.06574778
n 800 Cn 212081 density 0.0662753125
n 1000 Cn 330732 density 0.0661464
Factor
USING: combinators formatting generalizations kernel math
math.matrices random sequences ;
IN: rosetta-code.mean-cluster-density
CONSTANT: p 0.5
CONSTANT: iterations 5
: rand-bit-matrix ( n probability -- matrix )
dupd [ random-unit > 1 0 ? ] curry make-matrix ;
: flood-fill ( x y matrix -- )
3dup ?nth ?nth 1 = [
[ [ -1 ] 3dip nth set-nth ] [
{
[ [ 1 + ] 2dip ]
[ [ 1 - ] 2dip ]
[ [ 1 + ] dip ]
[ [ 1 - ] dip ]
} [ flood-fill ] map-compose 3cleave
] 3bi
] [ 3drop ] if ;
: count-clusters ( matrix -- Cn )
0 swap dup dim matrix-coordinates flip concat [
first2 rot 3dup ?nth ?nth 1 = [ flood-fill 1 + ]
[ 3drop ] if
] with each ;
: mean-cluster-density ( matrix -- mcd )
[ count-clusters ] [ dim first sq / ] bi ;
: simulate ( n -- avg-mcd )
iterations swap [ p rand-bit-matrix mean-cluster-density ]
curry replicate sum iterations / ;
: main ( -- )
{ 4 64 256 1024 4096 } [
[ iterations p ] dip dup simulate
"iterations = %d p = %.1f n = %4d sim = %.5f\n" printf
] each ;
MAIN: main
{{out}}
iterations = 5 p = 0.5 n = 4 sim = 0.13750
iterations = 5 p = 0.5 n = 64 sim = 0.07437
iterations = 5 p = 0.5 n = 256 sim = 0.06786
iterations = 5 p = 0.5 n = 1024 sim = 0.06621
iterations = 5 p = 0.5 n = 4096 sim = 0.06589
Go
{{trans|Python}}
package main import ( "fmt" "math/rand" "time" ) var ( n_range = []int{4, 64, 256, 1024, 4096} M = 15 N = 15 ) const ( p = .5 t = 5 NOT_CLUSTERED = 1 cell2char = " #abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ" ) func newgrid(n int, p float64) [][]int { g := make([][]int, n) for y := range g { gy := make([]int, n) for x := range gy { if rand.Float64() < p { gy[x] = 1 } } g[y] = gy } return g } func pgrid(cell [][]int) { for n := 0; n < N; n++ { fmt.Print(n%10, ") ") for m := 0; m < M; m++ { fmt.Printf(" %c", cell2char[cell[n][m]]) } fmt.Println() } } func cluster_density(n int, p float64) float64 { cc := clustercount(newgrid(n, p)) return float64(cc) / float64(n) / float64(n) } func clustercount(cell [][]int) int { walk_index := 1 for n := 0; n < N; n++ { for m := 0; m < M; m++ { if cell[n][m] == NOT_CLUSTERED { walk_index++ walk_maze(m, n, cell, walk_index) } } } return walk_index - 1 } func walk_maze(m, n int, cell [][]int, indx int) { cell[n][m] = indx if n < N-1 && cell[n+1][m] == NOT_CLUSTERED { walk_maze(m, n+1, cell, indx) } if m < M-1 && cell[n][m+1] == NOT_CLUSTERED { walk_maze(m+1, n, cell, indx) } if m > 0 && cell[n][m-1] == NOT_CLUSTERED { walk_maze(m-1, n, cell, indx) } if n > 0 && cell[n-1][m] == NOT_CLUSTERED { walk_maze(m, n-1, cell, indx) } } func main() { rand.Seed(time.Now().Unix()) cell := newgrid(N, .5) fmt.Printf("Found %d clusters in this %d by %d grid\n\n", clustercount(cell), N, N) pgrid(cell) fmt.Println() for _, n := range n_range { M = n N = n sum := 0. for i := 0; i < t; i++ { sum += cluster_density(n, p) } sim := sum / float64(t) fmt.Printf("t=%3d p=%4.2f n=%5d sim=%7.5f\n", t, p, n, sim) } }
{{out}}
Found 29 clusters in this 15 by 15 grid
0) a a a b c
1) d e a c c f
2) g e e e
3) h e i j k
4) l h m m n
5) l o p n n n n
6) q o o o r n
7) o o r r s
8) t t o u r s s s s
9) t t v u u r r s s s s
0) t v u r r s
1) w x r r y y
2) z x r r r r r y
3) A z z B r r r r
4) A A A z C r r r r
t= 5 p=0.50 n= 4 sim=0.16250
t= 5 p=0.50 n= 64 sim=0.07334
t= 5 p=0.50 n= 256 sim=0.06710
t= 5 p=0.50 n= 1024 sim=0.06619
t= 5 p=0.50 n= 4096 sim=0.06585
J
The first thing this task seems to need is some mechanism of identifying "clusters", using "percolation". We can achieve this by assigning every "1" in a matrix a unique integer value and then defining an operation which combines two numbers - doing nothing unless the second one (the one on the right) is non-zero. If it is non-zero we pick the larger of the two values. (*@[ * >.)
Once we have this, we can identify clusters by propagating information in a single direction through the matrix using this operation, rotating the matrix 90 degrees, and then repeating this combination of operations four times. And, finally, by keeping at this until there's nothing more to be done.
congeal=: |.@|:@((*@[*>.)/\.)^:4^:_
Example:
?6 6$0
M
1 0 0 0 0 0
0 0 0 1 0 0
0 0 0 1 0 0
1 1 0 0 0 0
0 0 0 1 0 1
1 1 0 1 1 0
M*p:i.$M
2 0 0 0 0 0
0 0 0 29 0 0
0 0 0 53 0 0
67 71 0 0 0 0
0 0 0 107 0 113
127 131 0 139 149 0
congeal M*p:i.$M
2 0 0 0 0 0
0 0 0 53 0 0
0 0 0 53 0 0
71 71 0 0 0 0
0 0 0 149 0 113
131 131 0 149 149 0
We did not have to use primes there - any mechanism for assigning distinct positive integers to the 1s would work. And, in fact, it might be nice if - once we found our clusters - we assigned the smallest distinct positive integers to the clusters. This would allow us to use simple indexing to map the array to characters.
idclust=: $ $ [: (~. i.])&.(0&,)@,@congeal ] * 1 + i.@$
Example use:
idclust M
1 0 0 0 0 0
0 0 0 2 0 0
0 0 0 2 0 0
3 3 0 0 0 0
0 0 0 4 0 5
6 6 0 4 4 0
(idclust M) {'.ABCDEFG'
A.....
...B..
...B..
CC....
...D.E
FF.DD.
Now we just need a measure of cluster density. Formally cluster density seems to be defined as the number of clusters divided by the total number of elements of the matrix. Thus:
K=: (%&#~ }.@~.)&,
Example use:
K idclust M
0.1666667
So we can create a word that performs a simulation experiment, given a probability getting a 1 and the number of rows (and columns) of our square matrix M.
0 ?@$~ ,~
Example use:
0.4 experiment 6
0.1666667
0.4 experiment 6
0.1944444
The task wants us to perform at least five trials for sizes up to 1000 by 1000 with probability of 1 being 0.5:
trials=: 0.5&experiment"0@#
Example use:
6 trials 3
0.1111111 0.1111111 0.2222222 0.1111111 0.1111111 0.3333333
6 trials 10
0.16 0.12 0.09 0.1 0.1 0.03
6 trials 30
0.05666667 0.1033333 0.08222222 0.07444444 0.08333333 0.07666667
6 trials 100
0.069 0.0678 0.0666 0.0677 0.0653 0.0739
6 trials 300
0.06563333 0.06663333 0.06713333 0.06727778 0.06658889 0.06664444
6 trials 1000
0.066079 0.066492 0.065847 0.065943 0.066318 0.065998
Now for averages (these are different trials from the above):
mean=: +/%#
mean 8 trials 3
0.1805556
mean 8 trials 10
0.0875
mean 8 trials 30
0.07486111
mean 8 trials 100
0.0690625
mean 8 trials 300
0.06749861
mean 8 trials 1000
0.06616738
Finally, for the extra credit (thru taken from the [[Loops/Downward_for#J|Loops/Downward for]] task):
thru=: <./ + i.@(+*)@-~
(idclust 0.5 > 0 ?@$~ 15 15) {'.', 'A' thru&.(a.&i.) 'Z'
A.......B..C...
AAAA...D..E.F..
A..A.G.D.D.FFF.
AA..H..DDD.FF.I
AAA...J...FFF..
..AAAA.A.K...AA
LL.A...A..A.AAA
.L.A..AAA.AAAAA
..AA.AAA.AAA.A.
AA.AAAAAA....A.
A.AAAA.AAAA.AA.
AAA...AAA.AAAAA
..AA..A.A...AAA
.M.A.AA.AA..AA.
.MM..A.N..O..A.
'''Collected definitions'''
congeal=: |.@|:@((*@[*>.)/\.)^:4^:_
idclust=: $ $ [: (~. i.])&.(0&,)@,@congeal ] * 1 + i.@$
K=: (%&#~ }.@~.)&,
experiment=: K@ idclust@: > 0 ?@$~ ,~
trials=: 0.5&experiment"0@#
mean=:+/ % #
thru=: <./ + i.@(+*)@-~
'''Extra Credit'''
M=: (* 1+i.@$)?15 15$2
M
0 2 3 4 0 6 0 8 0 10 11 12 0 0 15
0 0 18 19 20 0 22 0 0 0 0 0 28 29 0
31 32 0 34 35 36 37 38 0 0 0 42 0 0 45
0 0 48 49 0 51 0 0 54 55 0 57 58 0 0
61 62 63 64 0 0 67 0 69 0 71 72 0 74 0
0 0 78 79 0 0 82 0 84 85 86 87 88 0 0
0 92 0 94 0 0 0 0 99 100 101 0 103 0 105
106 107 108 0 0 111 0 0 114 115 116 0 0 0 0
0 0 0 124 125 126 127 0 0 0 0 0 133 134 135
0 0 138 0 0 141 0 143 144 145 0 0 0 0 150
0 152 153 154 0 0 0 158 0 160 0 162 163 164 165
0 167 168 169 170 0 172 173 0 175 176 177 0 0 180
181 182 183 0 0 186 0 188 189 190 191 192 0 194 195
196 197 198 0 200 201 202 0 0 205 0 207 0 0 0
211 212 213 0 0 0 217 218 0 220 221 0 0 224 0
congeal M
0 94 94 94 0 6 0 8 0 12 12 12 0 0 15
0 0 94 94 94 0 94 0 0 0 0 0 29 29 0
32 32 0 94 94 94 94 94 0 0 0 116 0 0 45
0 0 94 94 0 94 0 0 116 116 0 116 116 0 0
94 94 94 94 0 0 82 0 116 0 116 116 0 74 0
0 0 94 94 0 0 82 0 116 116 116 116 116 0 0
0 108 0 94 0 0 0 0 116 116 116 0 116 0 105
108 108 108 0 0 141 0 0 116 116 116 0 0 0 0
0 0 0 141 141 141 141 0 0 0 0 0 221 221 221
0 0 213 0 0 141 0 221 221 221 0 0 0 0 221
0 213 213 213 0 0 0 221 0 221 0 221 221 221 221
0 213 213 213 213 0 221 221 0 221 221 221 0 0 221
213 213 213 0 0 218 0 221 221 221 221 221 0 221 221
213 213 213 0 218 218 218 0 0 221 0 221 0 0 0
213 213 213 0 0 0 218 218 0 221 221 0 0 224 0
(~.@, i. ])congeal M
0 1 1 1 0 2 0 3 0 4 4 4 0 0 5
0 0 1 1 1 0 1 0 0 0 0 0 6 6 0
7 7 0 1 1 1 1 1 0 0 0 8 0 0 9
0 0 1 1 0 1 0 0 8 8 0 8 8 0 0
1 1 1 1 0 0 10 0 8 0 8 8 0 11 0
0 0 1 1 0 0 10 0 8 8 8 8 8 0 0
0 12 0 1 0 0 0 0 8 8 8 0 8 0 13
12 12 12 0 0 14 0 0 8 8 8 0 0 0 0
0 0 0 14 14 14 14 0 0 0 0 0 15 15 15
0 0 16 0 0 14 0 15 15 15 0 0 0 0 15
0 16 16 16 0 0 0 15 0 15 0 15 15 15 15
0 16 16 16 16 0 15 15 0 15 15 15 0 0 15
16 16 16 0 0 17 0 15 15 15 15 15 0 15 15
16 16 16 0 17 17 17 0 0 15 0 15 0 0 0
16 16 16 0 0 0 17 17 0 15 15 0 0 18 0
Julia
{{works with|Julia|0.6}} {{trans|Python}}
using Distributions newgrid(p::Float64, r::Int, c::Int=r) = rand(Bernoulli(p), r, c) function walkmaze!(grid::Matrix{Int}, r::Int, c::Int, indx::Int) NOT_CLUSTERED = 1 # const N, M = size(grid) dirs = [[1, 0], [-1, 0], [0, 1], [0, -1]] # fill cell grid[r, c] = indx # check for each direction for d in dirs rr, cc = (r, c) .+ d if checkbounds(Bool, grid, rr, cc) && grid[rr, cc] == NOT_CLUSTERED walkmaze!(grid, rr, cc, indx) end end end function clustercount!(grid::Matrix{Int}) NOT_CLUSTERED = 1 # const walkind = 1 for r in 1:size(grid, 1), c in 1:size(grid, 2) if grid[r, c] == NOT_CLUSTERED walkind += 1 walkmaze!(grid, r, c, walkind) end end return walkind - 1 end clusterdensity(p::Float64, n::Int) = clustercount!(newgrid(p, n)) / n ^ 2 function printgrid(G::Matrix{Int}) LETTERS = vcat(' ', '#', 'A':'Z', 'a':'z') for r in 1:size(G, 1) println(r % 10, ") ", join(LETTERS[G[r, :] .+ 1], ' ')) end end G = newgrid(0.5, 15) @printf("Found %i clusters in this %i×%i grid\n\n", clustercount!(G), size(G, 1), size(G, 2)) printgrid(G) println() const nrange = 2 .^ (4:2:12) const p = 0.5 const nrep = 5 for n in nrange sim = mean(clusterdensity(p, n) for _ in 1:nrep) @printf("nrep = %2i p = %.2f dim = %-13s sim = %.5f\n", nrep, p, "$n × $n", sim) end
{{out}}
Found 20 clusters in this 15×15 grid
1) A B C C D D D
2) E F D D
3) G F D D D D D D
4) G G H F I D D D J
5) G G K L D J
6) G G G G M N N
7) G G G G G G O O O N
8) G G O N N N
9) P P P G G G N N
0) P P P P G Q Q Q N
1) P Q Q Q Q N N
2) P N N N N
3) P P P R S N
4) P R R R S S N N
5) R R R S T N
nrep = 5 p = 0.50 dim = 16 × 16 sim = 0.07500
nrep = 5 p = 0.50 dim = 64 × 64 sim = 0.07178
nrep = 5 p = 0.50 dim = 256 × 256 sim = 0.06690
nrep = 5 p = 0.50 dim = 1024 × 1024 sim = 0.06609
nrep = 5 p = 0.50 dim = 4096 × 4096 sim = 0.06588
Kotlin
{{trans|C}}
// version 1.2.10 import java.util.Random val rand = Random() const val RAND_MAX = 32767 lateinit var map: IntArray var w = 0 var ww = 0 const val ALPHA = "+.ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz" const val ALEN = ALPHA.length - 3 fun makeMap(p: Double) { val thresh = (p * RAND_MAX).toInt() ww = w * w var i = ww map = IntArray(i) while (i-- != 0) { val r = rand.nextInt(RAND_MAX + 1) if (r < thresh) map[i] = -1 } } fun showCluster() { var k = 0 for (i in 0 until w) { for (j in 0 until w) { val s = map[k++] val c = if (s < ALEN) ALPHA[1 + s] else '?' print(" $c") } println() } } fun recur(x: Int, v: Int) { if ((x in 0 until ww) && map[x] == -1) { map[x] = v recur(x - w, v) recur(x - 1, v) recur(x + 1, v) recur(x + w, v) } } fun countClusters(): Int { var cls = 0 for (i in 0 until ww) { if (map[i] != -1) continue recur(i, ++cls) } return cls } fun tests(n: Int, p: Double): Double { var k = 0.0 for (i in 0 until n) { makeMap(p) k += countClusters().toDouble() / ww } return k / n } fun main(args: Array<String>) { w = 15 makeMap(0.5) val cls = countClusters() println("width = 15, p = 0.5, $cls clusters:") showCluster() println("\np = 0.5, iter = 5:") w = 1 shl 2 while (w <= 1 shl 13) { val t = tests(5, 0.5) println("%5d %9.6f".format(w, t)) w = w shl 1 } }
Sample output:
width = 15, p = 0.5, 23 clusters:
A . B . C C . . . . D D . D .
. B B B . . D D D . D D . D D
. B . B B B . . D D D . . D .
. . E . . B . B . . D D D D .
F . . B B B B B . G . D . D D
. . B B . B . . H . D D . D D
. B B . . . I . H H . . D D .
B B B . . J . K . . L . D . B
B . . M . . K K K . . D D . B
B B . . . . . K K . . D . B B
B . . N N . . . . . . . O . .
B . . . . P . . O O O O O . .
. . Q . . P . . . O O O O . .
. . . R . . . . S . O . . T T
. U . . . V . . . . . W . . .
p = 0.5, iter = 5:
4 0.112500
8 0.121875
16 0.075000
32 0.068750
64 0.068164
128 0.065625
256 0.067093
512 0.065815
1024 0.065863
2048 0.065815
4096 0.065764
8192 0.065766
Perl
{{trans|Perl 6}}
$fill = 'x'; $D{$_} = $i++ for qw<DeadEnd Up Right Down Left>; sub deq { defined $_[0] && $_[0] eq $_[1] } sub perctest { my($grid) = @_; generate($grid); my $block = 1; for my $y (0..$grid-1) { for my $x (0..$grid-1) { fill($x, $y, $block++) if $perc[$y][$x] eq $fill } } ($block - 1) / $grid**2; } sub generate { my($grid) = @_; for my $y (0..$grid-1) { for my $x (0..$grid-1) { $perc[$y][$x] = rand() < .5 ? '.' : $fill; } } } sub fill { my($x, $y, $block) = @_; $perc[$y][$x] = $block; my @stack; while (1) { if (my $dir = direction( $x, $y )) { push @stack, [$x, $y]; ($x,$y) = move($dir, $x, $y, $block) } else { return unless @stack; ($x,$y) = @{pop @stack}; } } } sub direction { my($x, $y) = @_; return $D{Down} if deq($perc[$y+1][$x ], $fill); return $D{Left} if deq($perc[$y ][$x-1], $fill); return $D{Right} if deq($perc[$y ][$x+1], $fill); return $D{Up} if deq($perc[$y-1][$x ], $fill); return $D{DeadEnd}; } sub move { my($dir,$x,$y,$block) = @_; $perc[--$y][ $x] = $block if $dir == $D{Up}; $perc[++$y][ $x] = $block if $dir == $D{Down}; $perc[ $y][ --$x] = $block if $dir == $D{Left}; $perc[ $y][ ++$x] = $block if $dir == $D{Right}; ($x, $y) } my $K = perctest(15); for my $row (@perc) { printf "%3s", $_ for @$row; print "\n"; } printf "𝘱 = 0.5, 𝘕 = 15, 𝘒 = %.4f\n\n", $K; $trials = 5; for $N (10, 30, 100, 300, 1000) { my $total = 0; $total += perctest($N) for 1..$trials; printf "𝘱 = 0.5, trials = $trials, 𝘕 = %4d, 𝘒 = %.4f\n", $N, $total / $trials; }
{{out}}
1 1 1 . . . . 2 2 2 . . . . .
. 1 . 1 1 1 . 2 2 . 2 2 2 . 3
. 1 . . 1 . 2 2 2 2 2 2 . . 3
1 1 1 . 1 . 2 2 . . . . 4 4 .
1 1 1 . 1 . . 2 . . . . . . 1
1 1 1 1 1 . . 2 . . 5 . 6 . .
1 1 . . 1 1 . 2 . 7 . . . 1 1
1 . . . 1 1 . 2 2 . . 8 8 . 1
. 9 9 9 . 1 . . 2 2 . . . 1 1
. . 9 9 . . 10 . . . 11 . 12 . .
. 9 9 . 13 13 . 13 . 14 . . 12 . .
15 . . 13 13 13 13 13 . . . 16 . 17 .
15 . . 13 . 13 . 13 13 . . 16 16 . .
. 18 . . 13 13 13 13 . . . . . 19 19
1 . 1 . . 13 . . . . 20 . 19 19 .
𝘱 = 0.5, 𝘕 = 15, 𝘒 = 0.0889
𝘱 = 0.5, trials = 5, 𝘕 = 10, 𝘒 = 0.0980
𝘱 = 0.5, trials = 5, 𝘕 = 30, 𝘒 = 0.0738
𝘱 = 0.5, trials = 5, 𝘕 = 100, 𝘒 = 0.0670
𝘱 = 0.5, trials = 5, 𝘕 = 300, 𝘒 = 0.0660
𝘱 = 0.5, trials = 5, 𝘕 = 1000, 𝘒 = 0.0661
Perl 6
{{works with|Rakudo|2017.02}}
my @perc;
my $fill = 'x';
enum Direction <DeadEnd Up Right Down Left>;
my $𝘒 = perctest(15);
.fmt("%-2s").say for @perc;
say "𝘱 = 0.5, 𝘕 = 15, 𝘒 = $𝘒\n";
my $trials = 5;
for 10, 30, 100, 300, 1000 -> $𝘕 {
my $𝘒 = ( [+] perctest($𝘕) xx $trials ) / $trials;
say "𝘱 = 0.5, trials = $trials, 𝘕 = $𝘕, 𝘒 = $𝘒";
}
sub infix:<deq> ( $a, $b ) { $a.defined && ($a eq $b) }
sub perctest ( $grid ) {
generate $grid;
my $block = 1;
for ^$grid X ^$grid -> ($y, $x) {
fill( [$x, $y], $block++ ) if @perc[$y; $x] eq $fill
}
($block - 1) / $grid²;
}
sub generate ( $grid ) {
@perc = ();
@perc.push: [ ( rand < .5 ?? '.' !! $fill ) xx $grid ] for ^$grid;
}
sub fill ( @cur, $block ) {
@perc[@cur[1]; @cur[0]] = $block;
my @stack;
my $current = @cur;
loop {
if my $dir = direction( $current ) {
@stack.push: $current;
$current = move $dir, $current, $block
}
else {
return unless @stack;
$current = @stack.pop
}
}
sub direction( [$x, $y] ) {
( Down if @perc[$y + 1][$x] deq $fill ) ||
( Left if @perc[$y][$x - 1] deq $fill ) ||
( Right if @perc[$y][$x + 1] deq $fill ) ||
( Up if @perc[$y - 1][$x] deq $fill ) ||
DeadEnd
}
sub move ( $dir, @cur, $block ) {
my ( $x, $y ) = @cur;
given $dir {
when Up { @perc[--$y; $x] = $block }
when Down { @perc[++$y; $x] = $block }
when Left { @perc[$y; --$x] = $block }
when Right { @perc[$y; ++$x] = $block }
}
[$x, $y]
}
}
{{out}}
. . 1 . 2 . . 3 . . . 4 . . .
2 2 . . 2 2 2 . 5 5 . 4 4 4 4
2 2 2 2 2 . 2 . 5 . . 4 . . 4
2 . 2 . 2 2 . . . . 4 4 4 . 4
. . . . . 2 . . . . 4 4 4 . .
6 6 6 6 . . 7 7 . . 4 . 4 4 .
6 . 6 . . . . . 4 4 4 4 4 . .
6 6 6 . . . 8 8 . 4 4 4 . . 4
6 . . . 9 . . . . . . 4 4 . 4
. 10 . 11 . . 12 12 . 4 . . 4 4 4
11 . 11 11 11 11 . 12 . 4 . 4 4 4 .
11 11 11 11 11 11 11 . . 4 4 . . 4 .
11 11 11 11 . 11 . . . 4 4 4 4 4 .
11 11 11 . 11 11 11 . . 4 4 4 4 . 13
. 11 11 . 11 11 . . . . 4 4 . 14 .
𝘱 = 0.5, 𝘕 = 15, 𝘒 = 0.062222
𝘱 = 0.5, trials = 5, 𝘕 = 10, 𝘒 = 0.114
𝘱 = 0.5, trials = 5, 𝘕 = 30, 𝘒 = 0.082444
𝘱 = 0.5, trials = 5, 𝘕 = 100, 𝘒 = 0.06862
𝘱 = 0.5, trials = 5, 𝘕 = 300, 𝘒 = 0.066889
𝘱 = 0.5, trials = 5, 𝘕 = 1000, 𝘒 = 0.0659358
Phix
{{trans|C}}
sequence grid
integer w, ww
procedure make_grid(atom p)
ww = w*w
grid = repeat(0,ww)
for i=1 to ww do
grid[i] = -(rnd()<p)
end for
end procedure
constant alpha = "+.ABCDEFGHIJKLMNOPQRSTUVWXYZ"&
"abcdefghijklmnopqrstuvwxyz"
procedure show_cluster()
for i=1 to ww do
integer gi = grid[i]+2
grid[i] = iff(gi<=length(alpha)?alpha[gi]:'?')
end for
puts(1,join_by(grid,w,w,""))
end procedure
procedure recur(integer x, v)
if x>=1 and x<=ww and grid[x]==-1 then
grid[x] = v
recur(x-w, v)
recur(x-1, v)
recur(x+1, v)
recur(x+w, v)
end if
end procedure
function count_clusters()
integer cls = 0
for i=1 to ww do
if grid[i]=-1 then
cls += 1
recur(i, cls)
end if
end for
return cls
end function
function tests(int n, atom p)
atom k = 0
for i=1 to n do
make_grid(p)
k += count_clusters()/ww
end for
return k / n
end function
procedure main()
w = 15
make_grid(0.5)
printf(1,"width=15, p=0.5, %d clusters:\n", count_clusters())
show_cluster()
printf(1,"\np=0.5, iter=5:\n")
w = 4
while w<=4096 do
printf(1,"%5d %9.6f\n", {w, tests(5,0.5)})
w *= 4
end while
end procedure
main()
{{out}}
width=15, p=0.5, 18 clusters:
..EE.FF...OO..P
.......KK.OOO.P
A..B..I....OO.P
.BBB.G.LL....PP
BBB...J....J.P.
..BB..JJJJJJ.P.
.BBBB.JJJ.J.J..
BBB....JJJ.JJ.Q
BB.D.D.J.JJJ.QQ
..DDDD.JJ.JJJ.Q
.DDDD.JJ..JJ...
C.DDD.J.N.JJ...
C.D...J..JJ....
.....H...J.....
.......M..O...R
p=0.5, iter=5:
4 0.137500
16 0.080469
64 0.068164
256 0.066809
1024 0.066018
4096 0.065777
Python
from __future__ import division from random import random import string from math import fsum n_range, p, t = (2**n2 for n2 in range(4, 14, 2)), 0.5, 5 N = M = 15 NOT_CLUSTERED = 1 # filled but not clustered cell cell2char = ' #' + string.ascii_letters def newgrid(n, p): return [[int(random() < p) for x in range(n)] for y in range(n)] def pgrid(cell): for n in range(N): print( '%i) ' % (n % 10) + ' '.join(cell2char[cell[n][m]] for m in range(M))) def cluster_density(n, p): cc = clustercount(newgrid(n, p)) return cc / n / n def clustercount(cell): walk_index = 1 for n in range(N): for m in range(M): if cell[n][m] == NOT_CLUSTERED: walk_index += 1 walk_maze(m, n, cell, walk_index) return walk_index - 1 def walk_maze(m, n, cell, indx): # fill cell cell[n][m] = indx # down if n < N - 1 and cell[n+1][m] == NOT_CLUSTERED: walk_maze(m, n+1, cell, indx) # right if m < M - 1 and cell[n][m + 1] == NOT_CLUSTERED: walk_maze(m+1, n, cell, indx) # left if m and cell[n][m - 1] == NOT_CLUSTERED: walk_maze(m-1, n, cell, indx) # up if n and cell[n-1][m] == NOT_CLUSTERED: walk_maze(m, n-1, cell, indx) if __name__ == '__main__': cell = newgrid(n=N, p=0.5) print('Found %i clusters in this %i by %i grid\n' % (clustercount(cell), N, N)) pgrid(cell) print('') for n in n_range: N = M = n sim = fsum(cluster_density(n, p) for i in range(t)) / t print('t=%3i p=%4.2f n=%5i sim=%7.5f' % (t, p, n, sim))
{{out}}
Found 20 clusters in this 15 by 15 grid
0) a a b c d d d d
1) a a e f g g d
2) e f f f d
3) h h e f i i d d
4) e j d d d d
5) k k k k l d d
6) k k k k k k l m n
7) k k k k k l o p p
8) k k k k l l l q
9) k k k k k l q q q
0) k k k k l q q q
1) k k k k k r r
2) k k k r r r s s
3) k k k k r r r r r s s
4) k k t r r r s s
t= 5 p=0.50 n= 16 sim=0.08984
t= 5 p=0.50 n= 64 sim=0.07310
t= 5 p=0.50 n= 256 sim=0.06706
t= 5 p=0.50 n= 1024 sim=0.06612
t= 5 p=0.50 n= 4096 sim=0.06587
As n increases, the sim result gets closer to 0.065770...
Racket
#lang racket
(require srfi/14) ; character sets
; much faster than safe fixnum functions
(require
racket/require ; for fancy require clause below
(filtered-in
(lambda (name) (regexp-replace #rx"unsafe-" name ""))
racket/unsafe/ops)
; these aren't in racket/unsafe/ops
(only-in racket/fixnum for/fxvector in-fxvector fxvector-copy))
; ...(but less safe). if in doubt use this rather than the one above
; (require racket/fixnum)
(define t (make-parameter 5))
(define (build-random-grid p M N)
(define p-num (numerator p))
(define p-den (denominator p))
(for/fxvector #:length (fx* M N) ((_ (in-range (* M N))))
(if (< (random p-den) p-num) 1 0)))
(define letters
(sort (char-set->list (char-set-intersection
char-set:letter
; char-set:ascii
)) char<?))
(define n-letters (length letters))
(define cell->char
(match-lambda
(0 #\space) (1 #\.)
(c (list-ref letters (modulo (- c 2) n-letters)))))
(define (draw-percol-grid M N . gs)
(for ((r N))
(for ((g gs))
(define row-str
(list->string
(for/list ((idx (in-range (* r M) (* (+ r 1) M))))
(cell->char (fxvector-ref g idx)))))
(printf "|~a| " row-str))
(newline)))
(define (count-clusters! M N g)
(define (gather-cluster! k c)
(when (fx= 1 (fxvector-ref g k))
(define k-r (fxquotient k M))
(define k-c (fxremainder k M))
(fxvector-set! g k c)
(define-syntax-rule (gather-surrounds range? k+)
(let ((idx k+))
(when (and range? (fx= 1 (fxvector-ref g idx)))
(gather-cluster! idx c))))
(gather-surrounds (fx> k-r 0) (fx- k M))
(gather-surrounds (fx> k-c 0) (fx- k 1))
(gather-surrounds (fx< k-c (fx- M 1)) (fx+ k 1))
(gather-surrounds (fx< k-r (fx- N 1)) (fx+ k M))))
(define-values (rv _c)
(for/fold ((rv 0) (c 2))
((pos (in-range (fx* M N)))
#:when (fx= 1 (fxvector-ref g pos)))
(gather-cluster! pos c)
(values (fx+ rv 1) (fx+ c 1))))
rv)
(define (display-sample-clustering p)
(printf "Percolation cluster sample: p=~a~%" p)
(define g (build-random-grid p 15 15))
(define g+ (fxvector-copy g))
(define g-count (count-clusters! 15 15 g+))
(draw-percol-grid 15 15 g g+)
(printf "~a clusters~%" g-count))
(define (experiment p n t)
(printf "Experiment: ~a ~a ~a\t" p n t) (flush-output)
(define sum-Cn
(for/sum ((run (in-range t)))
(printf "[~a" run) (flush-output)
(define g (build-random-grid p n n))
(printf "*") (flush-output)
(define Cn (count-clusters! n n g))
(printf "]") (flush-output)
Cn))
(printf "\tmean K(p) = ~a~%" (real->decimal-string (/ sum-Cn t (sqr n)) 6)))
(module+ main
(t 10)
(for ((n (in-list '(4000 1000 750 500 400 300 200 100 15))))
(experiment 1/2 n (t)))
(display-sample-clustering 1/2))
(module+ test
(define grd (build-random-grid 1/2 1000 1000))
(/ (for/sum ((g (in-fxvector grd)) #:when (zero? g)) 1) (fxvector-length grd))
(display-sample-clustering 1/2))
{{out}}
Run from DrRacket, which runs the test and main modules. From the command line, you'll
want two commands: racket percolation_m_c_d.rkt and raco test percolation_m_c_d.rkt
for the same result.
Experiment: 1/2 4000 10 [0*][1*][2*][3*][4*][5*][6*][7*][8*][9*] mean K(p) = 0.065860
Experiment: 1/2 1000 10 [0*][1*][2*][3*][4*][5*][6*][7*][8*][9*] mean K(p) = 0.066130
Experiment: 1/2 750 10 [0*][1*][2*][3*][4*][5*][6*][7*][8*][9*] mean K(p) = 0.066195
Experiment: 1/2 500 10 [0*][1*][2*][3*][4*][5*][6*][7*][8*][9*] mean K(p) = 0.066522
Experiment: 1/2 400 10 [0*][1*][2*][3*][4*][5*][6*][7*][8*][9*] mean K(p) = 0.066778
Experiment: 1/2 300 10 [0*][1*][2*][3*][4*][5*][6*][7*][8*][9*] mean K(p) = 0.066813
Experiment: 1/2 200 10 [0*][1*][2*][3*][4*][5*][6*][7*][8*][9*] mean K(p) = 0.067908
Experiment: 1/2 100 10 [0*][1*][2*][3*][4*][5*][6*][7*][8*][9*] mean K(p) = 0.069980
Experiment: 1/2 15 10 [0*][1*][2*][3*][4*][5*][6*][7*][8*][9*] mean K(p) = 0.089778
Percolation cluster sample: p=1/2
|. ... . . | |A BBB A A |
|... .. .... | |AAA AA AAAA |
|. . .... ... | |A A AAAA AAA |
|. . . .........| |A A C AAAAAAAAA|
| ... .. ....| | AAA AA AAAA|
|.. ......... ..| |AA AAAAAAAAA AA|
| . ... | | A AAA |
|. .. .. | |D AA AA |
| .. ... . .. | | AA AAA E AA |
|. .. .. . . | |F AA AA A A |
|. ........ . ..| |F AAAAAAAA A AA|
|.. . .... ... | |FF A AAAA AAA |
| . . . .... | | F G A AAAA |
|.... .. .. . .| |FFFF HH AA A A|
| . .. .....| | F HH AAAAA|
8 clusters
Tcl
Note that the queue (variables q and k) used to remember where to find cells when flood-filling the cluster is maintained as a list ''segment''; the front of the list is not trimmed for performance reasons. (This would matter with very long queues, in which case the queue could be shortened occasionally; ''frequent'' trimming is still slower though, because Tcl backs its “list” datatype with arrays and not linked lists.)
{{works with|Tcl|8.6}}
package require Tcl 8.6 proc determineClusters {w h p} { # Construct the grid set grid [lrepeat $h [lrepeat $w 0]] for {set i 0} {$i < $h} {incr i} { for {set j 0} {$j < $w} {incr j} { lset grid $i $j [expr {rand() < $p ? -1 : 0}] } } # Find (and count) the clusters set cl 0 for {set i 0} {$i < $h} {incr i} { for {set j 0} {$j < $w} {incr j} { if {[lindex $grid $i $j] == -1} { incr cl for {set q [list $i $j];set k 0} {$k<[llength $q]} {incr k} { set y [lindex $q $k] set x [lindex $q [incr k]] if {[lindex $grid $y $x] != -1} continue lset grid $y $x $cl foreach dx {1 0 -1 0} dy {0 1 0 -1} { set nx [expr {$x+$dx}] set ny [expr {$y+$dy}] if { $nx >= 0 && $ny >= 0 && $nx < $w && $ny < $h && [lindex $grid $ny $nx] == -1 } then { lappend q $ny $nx } } } } } } return [list $cl $grid] } # Print a sample 15x15 grid lassign [determineClusters 15 15 0.5] n g puts "15x15 grid, p=0.5, with $n clusters" puts "+[string repeat - 15]+" foreach r $g {puts |[join [lmap x $r {format %c [expr {$x==0?32:64+$x}]}] ""]|} puts "+[string repeat - 15]+" # Determine the densities as the grid size increases puts "p=0.5, iter=5" foreach n {5 30 180 1080 6480} { set tot 0 for {set i 0} {$i < 5} {incr i} { lassign [determineClusters $n $n 0.5] nC incr tot $nC } puts "n=$n, K(p)=[expr {$tot/5.0/$n**2}]" }
{{out}}
15x15 grid, p=0.5, with 21 clusters
+---------------+
| A B CCCCC|
| D A BBB C |
|E B F CCCC|
| B B F CC C|
|BBB B BB CCC|
|B BBBBBB CCCCC|
| B B G C C|
|H II G G J |
|HH II G GG K|
|HH II GGG GG K|
| I G GGGG |
|LL GGG GG M N|
| L G G O P |
|LLLL Q R |
|L L S T UUU|
+---------------+
p=0.5, iter=5
n=5, K(p)=0.184
n=30, K(p)=0.07155555555555557
n=180, K(p)=0.06880246913580246
n=1080, K(p)=0.0661267146776406
n=6480, K(p)=0.06582889898643499
zkl
{{trans|C}}
const X=-1; // the sentinal that marks an untouched cell
var C,N,NN,P;
fcn createC(n,p){
N,P=n,p; NN=N*N;
C=NN.pump(List.createLong(NN),0); // vector of ints
foreach n in (NN){ C[n]=X*(Float.random(1)<=P) } // X is the sentinal
}
fcn showCluster{
alpha:="-ABCDEFGHIJKLMNOPQRSTUVWXYZ" "abcdefghijklmnopqrstuvwxyz";
foreach n in ([0..NN,N]){ C[n,N].pump(String,alpha.get).println() }
}
fcn countClusters{
clusters:=0;
foreach n in (NN){
if(X!=C[n]) continue;
fcn(n,v){
if((0<=n<NN) and C[n]==X){
C[n]=v;
self.fcn(n-N,v); self.fcn(n-1,v); self.fcn(n+1,v); self.fcn(n+N,v);
}
}(n,clusters+=1);
}
clusters
}
fcn tests(N,n,p){
k:=0.0;
foreach z in (n){ createC(N,p); k+=countClusters().toFloat()/NN; }
k/n
}
createC(15,0.5);
println("width=%d, p=%.1f, %d clusters:".fmt(N,P,countClusters()));
showCluster();
println("p=0.5, 5 iterations:");
w:=4; do(6){ println("%5d %9.6f".fmt(w,tests(w, 5, 0.5))); w*=4; }
{{out}}
width=15, p=0.5, 16 clusters:
-AAA-BB-BBB---C
------BBBB--D--
E---F---BB--DD-
EE----G-BB---DD
--H-I--J--J--DD
-K--I--JJ-J--D-
-K--I--JJJJ-L--
KK-III-------MM
-K-I--I--NN-I--
I-IIIII-NNN-III
I-II--I-N-N-II-
III-III--NNN-II
I-II-II-O---I--
I-I-IIII-PP-III
I-II--I---P--II
p=0.5, 5 iterations:
4 0.062500
16 0.070312
64 0.067627
256 0.067078
1024 0.065834
4096 0.065771