Optimized WireWorld Algorithm

As usual, this entry was written because it is an uncommon subject that I just happen to be working on.

I've been slowly working on a virtual reality world and decided that it would be nice to add some technology to the simulation. I decided that a custom version of the WireWorld cellular automaton in 3 dimensions would be perfect since it is Turing complete, can run relatively fast, and would be rather simple to integrate with any other in-game technologies.

It had to be customized because it had to be fast (to prevent simulator sickness in VR), but there is very little interest in this automaton and all the code I've seen uses the traditional implementation. This is why I came up with this version (which someone probably has done and I just didn't know about it).

In this blog post, I will be describing an optimized version of the algorithm for WireWorld. In theory this version is a little slower than the classical 2 array version of WireWorld on small simulations, but should slowly become faster than the traditional implementation as the simulation gets bigger.

The rules of the automaton state that when a cell is an electron head, it will degrade into an electron tail. When the cell is an electron tail, it will degrade into a conductor. When the cell is a conductor, it can turn into an electron head only if one or two of its immediate neighbors are electron heads.

The way this is usually implemented is with an array where each element holds the state of the cell. A value of 0 is nothing, 1 is an electron head, 2 is an electron tail, and 3 is a conductor. The simulation goes through and writes the updated states to a second array in order to retain the new states without altering the current frame. Then the second array is copied back onto the first and is displayed and the process starts all over again. This works fine, but requires you to iterate through each array element and it if it has live neighbors and degrade its state until it becomes a conductor. Going through each element for every single frame is generally an O(n^2) time complexity, but is better on a small scale because it would use less memory and less time in general.

The way I will write about here will only go through the cells that actually have a state that isn't 0. Instead of holding the states in an array, the cells are held in a list of data structures that contain the x & y coordinates, the state, the addresses of the immediate neighbors, and a count of the live neighbors. The simulation goes through each cell a first time and if the cell is an electron head, then it will go through the list of neighbors and update the live neighbor count if the neighbor is a conductor. Then each cell will be iterated through a second time to degrade the state and then change its state based on if it had the 1 or 2 neighbors or not. This way all the empty "cells" that would have been in an array are skipped over since they are not in the list.

Of course, each cell in the second implementation takes more time and memory to execute on a small scale, but on a large scale there would be much more empty space that could be skipped and at a certain point the second implementation should be able to run faster than the first implementation. The time complexity for the second implementation should be something like O(n). In practice, this is only better in large scales because even though O(n) is less than O(n^2), it uses more memory and computing power per cell, but doesn't rise in power and memory as fast as O(n^2) would.

Enough of this theory stuff, lets get some actual pseudo code.

define cell data type

    x position

    y position

    state

    dynamic list 'neighbors'

    live neighbor count

initialize dynamic list 'wires'

set erase mode to false

define find function, input x and y coordinates

    loop through each element in 'wires', return index if coordinates match

    return -1 if no cell found at those coordinates

define get neighbors function, input cell

    loop through each coordinate around the cell's position

        if cells exist at those coordinates, add reference to cell's neighbor list

define add cell function, input x and y coordinates

    add new blank cell to wires list

    run get neighbors function on this new cell

    iterate through the neighbor list and add this cell to those neighbor's neighbor list

define delete cell function, input cell

    loop through each cell in cell's neighbor list

        remove the reference to this cell from the neighbor's neighbor list

    remove reference to this cell from the wires list

infinite loop

    clear screen

    if enter is pressed, toggle erase mode

    get mouse position and mouse buttons

    draw mouse cursor onto screen 

    if mouse click

        get index of cell at mouse coordinates with find function

        if left click and not erase mode and no cell existing at mouse coords

            call add cell function with mouse coordinates

        if left click and erase mode and cell existing at mouse coords

            call delete cell function with cell from index in wires list

        if right click and cell existing at mouse coordinates

            set cell at index to have stage of 1 (electron head)

    loop through cells in wires list

        if cell state is electron head

            loop through neighbors that are conductors

                increment live neighbor count of that neighbor cell

    loop through cells in wires list

        if cell is not 3 (conductor)

            increment cell state

        if cell's live neighbor count is 1 or 2

            change state to 2 (electron head )

        clear cell's live neighbor count to 0

        set color equal to black for 0 (void), blue for 1 (electron head), red for 2 (electron tail), yellow for 3 (conductor)

        draw cell on screen with coordinates and color

The Python source for this looks like this:
(Left click adds a conductor, right click activates it, pressing enter turns on delete mode)

#!/usr/bin/env python
import pygame
from pygame.locals import *
pygame.display.init()
pygame.font.init()
screen = pygame.display.set_mode((640,480))
pygame.display.set_caption("")
font = pygame.font.Font(None,20)
clock = pygame.time.Clock()
           # 0   1   2      3          4
wires = [] #[xp, yp, state, neighbors, live]
# 0 black void, 1 blue electron head, 2 red electron tail, 3 yellow wire
erase = False

def findWire(xp,yp):
 for w in xrange(len(wires)):
  if wires[w][0]==xp and wires[w][1]==yp: return w
 return -1
def updateNeighbors(ww):
 for x in xrange(-1,2):
  for y in xrange(-1,2):
   wp = findWire(ww[0]+x,ww[1]+y)
   if (x,y) != (0,0) and wp != -1:
    ww[3].append(wires[wp])
def insertWire(wx,wy):
 wires.append([wx,wy,3,[],0])
 updateNeighbors(wires[-1])
 for ww in wires[-1][3]:
  ww[3].append(wires[-1])
def deleteWire(ww):
 for w in ww[3]: #remove references from neighbors
  for wg in xrange(len(w[3])):
   if ww[0]==w[3][wg][0] and ww[1]==w[3][wg][1]:
    w[3].pop(wg)
    break
 wires.pop(findWire(ww[0], ww[1])) #remove reference from wires

while True:
 for event in pygame.event.get():
  if event.type == QUIT: exit()
  elif event.type == KEYDOWN:
   if event.key == K_RETURN: erase = not erase
   elif event.key == K_ESCAPE: exit()
 screen.fill((0,0,0))
 
 mx,my = pygame.mouse.get_pos()
 b = pygame.mouse.get_pressed()
 if erase: c = (100,0,0)
 else: c = (50,50,50)
 pygame.draw.rect(screen,c,(mx/5*5,my/5*5,5,5))
 mx /= 5
 my /= 5
 w = findWire(mx,my)
 if b[0]:
  if erase and w != -1: deleteWire(wires[w])
  elif not erase and w == -1: insertWire(mx,my)
 elif b[2] and w != -1:
  wires[w][2] = 1

 for w in wires:
  if w[2] == 1: #inc neighbor counters
   for v in w[3]:
    if v[2] == 3: v[4] += 1

 for w in wires:
  if w[2] != 3: w[2] += 1
  if w[4] == 1 or w[4] == 2: w[2] = 1
  w[4] = 0
  c = [(0,0,0),(0,0,255),(255,0,0),(255,255,0)][w[2]]
  #c = [(0,0,0),(0,255,255),(0,100,0),(40,40,40)][w[2]]
  pygame.draw.rect(screen,c,(w[0]*5,w[1]*5,5,5))
 
 clock.tick()
 screen.blit(font.render("%.2f"%clock.get_fps(),-1,(255,255,255)),(0,0))
 
 pygame.display.flip()

And as always, if you have some suggestions or ways to optimize this algorithm further, please let me know and I'll be more than happy to update this post with the new information.