Cellular Automata Method for Generating Random Cave-Like Levels

From RogueBasin
Revision as of 22:43, 21 May 2012 by RylandAlmanza (Talk | contribs)

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It is an old and fairly well documented trick to use cellular automata to generate cave-like structures. The basic idea is to fill the first map randomly, then repeatedly create new maps using the 4-5 rule: a tile becomes a wall if it was a wall and 4 or more of its nine neighbors were walls, or if it was not a wall and 5 or more neighbors were. Put more succinctly, a tile is a wall if the 3x3 region centered on it contained at least 5 walls. Each iteration makes each tile more like its neighbors, and the amount of overall "noise" is gradually reduced:

original:           iteration 1:        2:                  3:                  4:
 #       ### ##     ####  ##########    ####  ##########    ####  ##########    ####  ##########
# ##  ## ## #       # #      ##### #    ###      #######    ###      #######    ###      #######
 # # ##   #### #    # #   #   #### #    ##        ######    ##        ######    ##        ######
 # #  #  # ## #     ###  #     ### #    ##         #####    ##         #####    ##         #####
### # #     # #     #    #     #####    ##         #####    ##         #####    ##        ######
    # # ## #####    ##     ###  ####    #       ## #####    ##      ########    ##      ########
##     ####  #      #      #### ####    ##     ####  ###    ##     #### ####    ##     #########
## ## # ## #  ##    ###    ####    #    ###    ###   ###    ###    ####  ###    ###    #########
# ##  ###  #  #     ##### ###    ###    ####  #####  ###    ####  ##### ####    ####   #########
#  # # #  # ###     # ### ##########    ###### #########    #####  #########    ####   #########
  ## ## #### # #    #####    #######    ####    ########    ####    ########    ####    ########
####    # # # #     # #      #######    ##        ######    ##       #######    ##       #######
       #   ## ##    #          #####    #         ######    #         ######    #         ######
   # #    # ####               #####    #         ######    #         ######    #         ######
# # #  ## ######    #        #######    #        #######    ##       #######    ##       #######
#  #  # ####  #     ################    ################    ################    ################

If 45% of the original random map contains walls and the process is repeated 5 times, the output might look like the following:

############################################################
###....####################################.....############
##......######################..#########.........##########
##......#####################....#######...........####.####
##......###################.........................##...###
##......##################..........................###...##
#........##############.............................###...##
#........#############...............................#....##
##.......##############..................................###
##.......###..############..............................####
##.......##....############.............................####
#..............############...###........................###
#...............###########..#####...............##.......##
#................#################...............##.......##
##.....#####..........###########....#..........###.......##
##....#######...........########....###.........####......##
##....#######............######....####........#####......##
##....#######.............####....#####.......#####......###
#......######..............###....####........####......####
#.......######.............###...####.........###.......####
#........#####.............###..####.....................###
##........####..............#...####.....................###
#####......##...................####.....................###
######...........................##.....................####
######..................................................####
######.........###.....................####.............####
######......#########.................######............####
#######....#############.......##############.....###..#####
##############################################..############
############################################################ 

The problem is that the results are inconsistent. The algorithm prone to generating disconnected maps:

############################################################
#####################################################..#####
#####.....##################...###############............##
#####......###########.####....########.#####..............#
#####.......#########..###.....###############.............#
####.............#####.###....###################.........##
###...............########...####################........###
##.................#######...####################........###
##.......##.........#####.....##################.........###
##......####.........###.......################...........##
##.....########.................#######..######...........##
##...###########................######...#######..........##
#########..######..............######....########........###
########....######..#####......#####.....##########......###
#######......#############.....#####.....###########.....###
#######.......############......###.......###########.....##
#######.......###########.......###.......###########.....##
######.......####..######.......####.......#########......##
#####.......####....#####.......####..........######.....###
####........####......####......####...........#####.....###
####.........###.......###......####...##......######.....##
####...##.....###.......#......###########.....#######.....#
#####.####.....#####...........############....########....#
##########.....######..........############....#########..##
#########.......#####...........##########.....#############
#########.......####...............#####........############
##########......####................###...........##########
###########....#####.....######.....####...........#########
################################...##########.....##########
############################################################

It also sometimes generates maps which consist of one huge open space:

############################################################
############################################################
#################.####.######..######........###############
########...#####...##...####....####..........######...#####
###.####...####....###..####....####..........#####.....####
##...###....##.....###...##.....###............###......####
###.####...........###..........###.....###.........########
########...........###...........#.......#.........#########
#######.............#..............................#########
######.....##..........................................#####
###.......###...........................................####
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##......................................................####
##.....................................................#####
##.....................................................#####
#.....................................##...............#####
#.....................................##................####
#...........................###......###................####
##...........###............###.....#####...............####
###...........#..............##.....######............######
######.......................##......######........#########
##########..................##.........#####......##########
###########................###..........####......##########
############................#...........####......##########
###.....#####.........##...............#####.......####..###
###.....#####.......#####.............#####..............###
###.....####.......#####..............#####.....##.......###
####...#####.......#####.....#####...######....####.....####
#############.##########....################..##############
############################################################

We can fix the disjoint segments problem in one of three ways. Either throw away maps that have disjoint segments in them, connect the segments after the fact, or fill in all but the biggest segment. We can't just retry when we get a disjoint map, because if the map is big then, statistically, that will be almost 100% of the time. Filling in all but the biggest segment will tend to produce a small area in a map that was supposed to be big. Connecting up the regions works, but it tends to look unnatural, as in the example from above, now connected:

############################################################
#####################################################..#####
#####.....##################...###############............##
#####......###########.####....########....................#
#####.......#########..###.....###############.............#
####.............####..###....###################.........##
###...............###.####...####################........###
##.................##.####...####################........###
##.......##.........#.###.....##################.........###
##......####..........##.......################...........##
##.....########.................#######..######...........##
##...###########................######...#######..........##
#########..######..............######....########........###
########............#####......#####.....##########......###
#######......#############...............###########.....###
#######.......############......###.......###########.....##
#######.......###########.......###.......###########.....##
######.......####..######.......####.......#########......##
#####.......####....#####.......####.....................###
####........####......####......####...........#####.....###
####.........###.......###......####...##......######.....##
####...##.....###.......#......###########.....#######.....#
#####.####.....#####...........############....########....#
##########.....######..........############....#########..##
#########.......#####...........##########.....#############
#########.......####...............#####........############
##########......####................###...........##########
###########....#####.....######.....####...........#########
################################...##########.....##########
############################################################

The solution to both problems, as it turns out, is to revisit the original cellular automata rules. Recall that the original rule was

  • There is a wall initially at P with 45% probability
  • In the next generation, there is a wall at spot P if the number of tiles around P which are walls is at least 5

Or, in more compact notation:

  • Winit(p) = rand[0,100) < 45
  • R(p) = the number of tiles within 1 step of p which are walls
  • W?(p) = R(p) ? 5

Now, one of the problems was that we tend to get big, open areas. So why not try filling those areas in? Change the rule to

  • W?(p) = R(p) ? 5 or p is in the middle of an open space

Or more formally,

  • Rn(p) = the number of tiles within n steps of p which are walls
  • W?(p) = R1(p)?5 || R2(p)=0

So how does it look?

Winit(p) = rand[0,100) < 45
Repeat 5: W?(p) = R1(p) ? 5 || R2(p) ? 1
############################################################
##....######################################################
#.......###..#####..####....###########################..###
#...#.........###.............##########..############....##
##...###...#..###..............########....######..###....##
###...######...#..##...........######.....######....###..###
####..######......##..##........####...#..######....########
####..###.#.......#...##...#.....##...###..######...########
#####.........##...........##........####.....##.....#######
#####.........##...........#.......##.....#.............####
####...###...##................#...##......###...........###
####...###..##...###....##.....##..##..##..###....##.....###
#########..###..####...###.............###..##..####.....###
########...##..#####...##...............#...#...####....####
#######........######......###...##....................#####
#######..##.....######....##########...................#####
######..####.......####...#########...................######
####....####..##....##.....#######...##..............#######
###.....###..#####......#...####....##................######
##..##..###..###........##.........#....#......##......#####
#..####..##...##.................###...##.....####......####
#.....#..###..#..##..........#..###..###.....#####......####
##.......####.....#...##........##..###....#######......####
######....###.......................###...#######....#######
#######......................##.....###...#######...########
########.................#######....####...#####....########
########..............###########..######...........########
#########....####....######################........#########
###################.########################################
############################################################

This is more interesting - it doesn't have any big open areas, it has a decent layout. It's almost fully connected. Still, it has some new problems: there are isolated single-tile walls in places, and in general it's not very smooth. But with a little tweaking:

Winit(p) = rand[0,100) < 40
Repeat 4: W?(p) = R1(p) ? 5 || R2(p) ? 2
Repeat 3: W?(p) = R1(p) ? 5
############################################################
###...###########..#############################.....#######
##..........####....################..#########.........####
##...........##.....####..#########.......####..######...###
##.......#..........###....###.................########..###
##......###........###........................#########..###
##.......##.........#........................##########...##
##.......###...........##.............###....#########.....#
##.......######.......####...........#####....#####........#
###.....#########....#####...........######...#####........#
###########################...........#####...#######.....##
#############...###########.............##....########....##
############.........#######...................#######....##
###########...........########......###............##....###
###..#####.............#########...##########............###
##....###...............######################..........####
###..........................######..#########..........####
####..........................###.....#######...........####
####.................##................##................###
####...###..........####...............#..................##
###...#####.........####..............##......##...........#
##....########......#####............####....####..........#
##....#########.....#####............####....####..........#
##.....######.......#####.............##.....####...##.....#
##......##..........#####....................####..####....#
###.................####.........###........############..##
###............##..######.###...############################
####..........##############################################
######..####################################################
############################################################

Notice that the initial fill percentage is a little lower, the cutoffs are different, and we switch rules after a few generations. This is more like the desired result. So, with these parameters, I give you some more samples, at various sizes.

##############################
##..###....###.....###########
#...........................##
#.................##.........#
##..###..........#######.....#
########.........#######.....#
############......####......##
#############...............##
###..########...............##
##....#######...###..#......##
##.....######...#######.....##
##......#####...######......##
##...#...###....#####.......##
##..##.........######.......##
##..........##########......##
###........#####..####.....###
###........####....###.....###
###....##...##.....###.....###
###...####........####......##
###...#####......#####......##
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##.....###.....####.........##
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###.........................##
####........................##
####.......................###
####..........##...........###
#####...###..####.........####
####################.....#####
##############################
############################################################
#########......###...##........####...####.....#############
####............................##.....##............#######
###................###.....##..........................#####
###...............#####...####..........................####
###...............#####....####............#............####
####...###.........####.......##.......................#####
##########..........###........##.....................######
##########...........#..........##...................#######
#########.......................#####..............#########
######.........................#######...#......############
#####..........................############....#############
####...........................###########......######..####
###..........##.................#########................###
##.......#######........#..........######...###.........####
##......########.......##............###...######.....######
###.....#######...............#####........########..#######
###......#####...##...........######........################
###......#####..####...........#####.........###############
#######..#####..####............###...........#######....###
########..###...#####......###.................#####......##
########.......######......####.................###.......##
########.......######.......##....##..................##..##
#######.......######....##.......####................####..#
######.......#######....###......####..........###..#####..#
#####........######.....######....##..........##########...#
######........###........######...............########.....#
##########...............#######.............########.....##
#############...#######..###########...#####################
############################################################
################################################################
#######..#####............####..####...###....###########...####
###......####..................................#########.....###
##.......###.......................................####......###
##......####.............##.........................##......####
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##.....################..######.........#.............####...###
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###.....#####..##...##....####.........###.............###....##
####....####..............###..........###...................###
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###.....####...........#####.........................##.....####
##.......##............#####....................##..####...#####
##.....................######................###########...#####
###..##........##......###############.....############....#####
########......####....#################..###############..######
#########...####################################################
################################################################ 
############################################################
#########......###...##........####...####.....#############
####............................##.....##............#######
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###...............#####....####............#............####
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##########..........###........##.....................######
##########...........#..........##...................#######
#########.......................#####..............#########
######.........................#######...#......############
#####..........................############....#############
####...........................###########......######..####
###..........##.................#########................###
##.......#######........#..........######...###.........####
##......########.......##............###...######.....######
###.....#######...............#####........########..#######
###......#####...##...........######........################
###......#####..####...........#####.........###############
#######..#####..####............###...........#######....###
########..###...#####......###.................#####......##
########.......######......####.................###.......##
########.......######.......##....##..................##..##
#######.......######....##.......####.........##.....####..#
#####........#######....###......####........#####..#####..#
####........#######.....######...#####.......############..#
####.......######..........####..#########..#############..#
###........#####............###..########################.##
##.........####.............###..################.######..##
#....###...####......####....#...######..#######...#####..##
#.....#.....##......######.......#####....#####....#####..##
#...................######........####....###.....#####...##
#....................#####........#####..###......##......##
#....................######........########................#
##......#............########.......#######................#
##......##...........#########......#######...............##
###.....#............####...##.....#######..##...........###
###..........#......####...........######..####..........###
##..........####....####...........#####..######.........###
##...........####..#####............##....######.........###
##............###..######......#...........####..........###
##............###..#######.....##........................###
##.......###...#....#######....#..........................##
###......###.........######.........................##.....#
###.......#..........######........#####...........####....#
###............###...######......########.........#####....#
###...........#####..######.....#########.........####.....#
###...........#####.#######.....########...........###.....#
###...........####..########...#########......##...###....##
###...........####...##################......####..###....##
###...........####......##############.......####..###....##
###...........####..........##########........##...###....##
###............####..........#########.............####..###
###...........#####...........#######..............#########
###.....##########............######.......##......#########
##.....###########.....###.....####.......####......########
##.....############....###......##.......#####........######
###...##############..#####.............#######.......######
################################...##..#####################
############################################################

There's still no guarantee of connectedness, though. However, it's now consistently almost-connected, so that you can reasonably just drop the isolated chunks.

Finally, here is the C program I used to try out different parameters. Before putting this into an actual game, handling of disconnected regions is needed.

 #include <stdio.h>
 #include <stdlib.h>
 #include <time.h>

 #define TILE_FLOOR 0
 #define TILE_WALL 1

 typedef struct {
 	int r1_cutoff, r2_cutoff;
 	int reps;
 } generation_params; 
 
 int **grid;
 int **grid2; 
 
 int fillprob = 40;
 int r1_cutoff = 5, r2_cutoff = 2;
 int size_x = 64, size_y = 20;
 generation_params *params;  

 generation_params *params_set;
 int generations;

 int randpick(void)
 {
 	if(rand()%100 < fillprob)
 		return TILE_WALL;
 	else
 		return TILE_FLOOR;
 }

 void initmap(void)
 {
	int xi, yi;
	
	grid  = (int**)malloc(sizeof(int*) * size_y);
	grid2 = (int**)malloc(sizeof(int*) * size_y);
	
	for(yi=0; yi<size_y; yi++)
	{
		grid [yi] = (int*)malloc(sizeof(int) * size_x);
		grid2[yi] = (int*)malloc(sizeof(int) * size_x);
	}
	
	for(yi=1; yi<size_y-1; yi++)
	for(xi=1; xi<size_x-1; xi++)
		grid[yi][xi] = randpick();
	
	for(yi=0; yi<size_y; yi++)
	for(xi=0; xi<size_x; xi++)
		grid2[yi][xi] = TILE_WALL;
	
	for(yi=0; yi<size_y; yi++)
		grid[yi][0] = grid[yi][size_x-1] = TILE_WALL;
	for(xi=0; xi<size_x; xi++)
		grid[0][xi] = grid[size_y-1][xi] = TILE_WALL;
 }

 void generation(void)
 {
	int xi, yi, ii, jj;
	
	for(yi=1; yi<size_y-1; yi++)
	for(xi=1; xi<size_x-1; xi++)
 	{
 		int adjcount_r1 = 0,
 		    adjcount_r2 = 0;
 		
 		for(ii=-1; ii<=1; ii++)
		for(jj=-1; jj<=1; jj++)
 		{
 			if(grid[yi+ii][xi+jj] != TILE_FLOOR)
 				adjcount_r1++;
 		}
 		for(ii=yi-2; ii<=yi+2; ii++)
 		for(jj=xi-2; jj<=xi+2; jj++)
 		{
 			if(abs(ii-yi)==2 && abs(jj-xi)==2)
 				continue;
 			if(ii<0 || jj<0 || ii>=size_y || jj>=size_x)
 				continue;
 			if(grid[ii][jj] != TILE_FLOOR)
 				adjcount_r2++;
 		}
 		if(adjcount_r1 >= params->r1_cutoff || adjcount_r2 <= params->r2_cutoff)
 			grid2[yi][xi] = TILE_WALL;
 		else
 			grid2[yi][xi] = TILE_FLOOR;
 	}
 	for(yi=1; yi<size_y-1; yi++)
 	for(xi=1; xi<size_x-1; xi++)
 		grid[yi][xi] = grid2[yi][xi];
 } 

 void printfunc(void)
 {
 	int ii;
 	
 	printf("W[0](p) = rand[0,100) < %i\n", fillprob);
 	
 	for(ii=0; ii<generations; ii++)
 	{
 		printf("Repeat %i: W'(p) = R[1](p) >= %i",
 			params_set[ii].reps, params_set[ii].r1_cutoff);
 		
 		if(params_set[ii].r2_cutoff >= 0)
 			printf(" || R[2](p) <= %i\n", params_set[ii].r2_cutoff);
 		else
 			putchar('\n');
 	}
 }
 
 void printmap(void)
 {
 	int xi, yi;
 	
 	for(yi=0; yi<size_y; yi++)
 	{
 		for(xi=0; xi<size_x; xi++)
 		{
 			switch(grid[yi][xi]) {
 				case TILE_WALL:  putchar('#'); break;
 				case TILE_FLOOR: putchar('.'); break;
 			}
 		}
 		putchar('\n');
 	}
 }

 int main(int argc, char **argv)
 {
 	int ii, jj;
 	
 	if(argc < 7) {
 		printf("Usage: %s xsize ysize fill (r1 r2 count)+\n", argv[0]);
 		return 1;
 	}
 	size_x     = atoi(argv[1]);
 	size_y     = atoi(argv[2]);
 	fillprob   = atoi(argv[3]);
 	
 	generations = (argc-4)/3;
 	
 	params = params_set = (generation_params*)malloc( sizeof(generation_params) * generations );
 	
 	for(ii=4; ii+2<argc; ii+=3)
 	{
 		params->r1_cutoff  = atoi(argv[ii]);
 		params->r2_cutoff  = atoi(argv[ii+1]);
 		params->reps = atoi(argv[ii+2]);
 		params++;
 	}
 	
 	srand(time(NULL));
 	
 	initmap();
 	
 	for(ii=0; ii<generations; ii++)
 	{
 		params = &params_set[ii];
 		for(jj=0; jj<params->reps; jj++)
 			generation();
 	}
 	printfunc();
 	printmap();
 	return 0;
 }

(Original article by Jim Babcock)

Tips

I used this method on my game (Arcan Myth RL but only in developing version) but I limited the repetition cycle to 3: I like the result very much and I don't have the problem about big isolated cave. However very good method. Thank you :-) --Provolik


This algorithm creates good looking caves, but the problem is isolated caves. The way I solved this is by picking a random, open point on the map and flood filling. Any open point outside the flood filled portion gets turned back into a wall. I then check to see if the flood filled portion of the map is more than some threshold percent of the map. If the map is ~45% open it usually looks alright. If the percent is below this, I just start over.

This turns out to work quite well. On large maps, almost always you will select the largest open connected area, and you won't have to restart the regeneration very often, if at all. On very small maps, this doesn't happen as often, but its also very quick to regenerate because the map is small. It ends up that this method just does the right thing to rectify the non-connected maps naturally!

-Jlund3

External Links

Animated GIF of the cellular automata cave process with 12 smoothing iterations (Generated using haskell): cavegen.gif

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