When building a large map RTS game, my team are experiencing some performance issues regarding pathfinding.
A* is obviously inefficient due to not only janky path finding, but processing costs for large groups of units moving at the same time.
After research, the obvious solution would be to use FlowField pathfinding, the industry standard for RTS games as it stands.
The issue we are now having after creating the base algorithm is that the map is quite large requiring a grid of around 766 x 485. This creates a noticeable processing freeze or lag when computing the flowfield for the units to follow.
Has anybody experienced this before or have any solutions on how to make the flowfields more efficient? I have tried the following:
Adding flowfields to a list when it is created and referencing later (Works once it has been created, but obviously lags on creation.)
Processing flowfields before the game is started and referencing the list (Due to the sheer amount of cells, this simply doesn't work.)
Creating a grid based upon the distance between the furthest selected unit and the destination point (Works for short distances, not if moving from one end of the map to the other).
I was thinking about maybe splitting up the map into multiple flowfields, but I'm trying to work out how I would make them move from field to field.
Any advice on this?
Thanks in advance!
Maybe this is a bit late answer. Since you have mentioned that this is a large RTS game, then the computation should not be limited to one CPU core. There are a few advice for you to use flowfield more efficiently.
Use multithreads to compute new flow fields for each unit moving command
Group units, so that all units in same command group share the same flowfield
Partition the flowfield grids, so you only have to update any partition that had modification in path (new building/ moving units)
Pre baked flowfields grid slot cost:you prebake basic costs of the grids(based on environments or other static values that won't change during the game).
Divide, e.g. you have 766 x 485 map, set it as 800 * 500, divide it into 100 * 80 * 50 partitions as stated in advice 3.
You have a grid of 10 * 10 = 100 slots, create a directed graph (https://en.wikipedia.org/wiki/Graph_theory) using the a initial flowfield map (without considering any game units), and use A* algorihtm to search the flowfield grid before the game begins, so that you know all the connections between partitions.
For each new flowfield, Build flowfield only with partitions marked by a simple A* search in the graph. And then use alternative route if one node of the route given by A* is totally blocked from reaching the next one (mark the node as blocked and do A* again in this graph)
6.Cache, save flowfield result from step.5 for further usage (same unit spawning from home and going to the enemy base. Same partition routes. Invalidate cache if there is any change in the path, and only invalidate the cache of the changed partition first, check if this partition still connects to other sides, then only minor change will be made within the partition only)
Runtime late updating the units' command. If the map is large enough. Move the units immediately to the next partition without using the flowfield, (use A* first to search the 10*10 graph to get the next partition). And during this time of movement, in the background, build the flowfield using previous step 1-6. (in fact you only need few milliseconds to do the calculation if optimized properly, then the units changes their route accordingly. Most of the time there is no difference and player won't notice a thing. In the worst case, where we finally have to search all patitions to get the only possible route, it is only the first time there will be any delay, and the cache will minimise the time since it is the only way and the cache will be used repeatitively)
Re-do the build process above every once per few seconds for each command group (in the background), just in case anything changes in the middle of the way.
I could get this working with much larger random map (2000*2000) with no fps drop at all.
Hope this helps anyone in the future.
Related
I need to display a crypto currency price graph based similar to what is done on CoinMarketCap: https://coinmarketcap.com/currencies/bitcoin/
There could be gigabytes of data for one currency pair over a long period of time, so sending all the data to the client is not an option.
After doing some research I ended up using a Douglas-Peucker Line Approximation Algorithm: https://www.codeproject.com/Articles/18936/A-C-Implementation-of-Douglas-Peucker-Line-Appro It allows to reduce the amount of dots that is sent to the client but there's one problem: every time there's new data I have to go through all the data on the server and as I'd like to update data on the client in real time, it takes a lot of resources.
So I'm thinking about a some kind of progressive algorithm where, let's say, if I need to display data for the last month I can split data into 5 minute intervals, preprocess only the last interval and when it's completed, remove the first one. I'm debating either customising the Douglas-Peucker algorithm (but I'm not sure if it fits this scenario) or finding an algorithm designed for this purpose (any hint would be highly appreciated)
Constantly re-computing the entire reduction points when the new data arrives would change your graph continuously. The graph will lack consistency. The graph seen by one user would be different from the graph seen by another user and the graph would change when the user refreshes the page(this shouldn't happen!), and even in case of server/application shutdown, your data needs to be consistent to what it was before.
This is how I would approach:
Your reduced points should be as they are. For suppose, you are getting data for every second and you have computed reduced points for a 5-minute interval graph, save those data points in a limiting queue. Now gather all seconds data for next 5-minutes and perform the reduction operation on these 600 data points and add the final reduced point to your limiting queue.
I would make the Queue synchronous and the main thread would return the data points in the queue whenever there is an API call. And the worker thread would compute the reduction point on the 5-minute data once the data for the entire 5-minute interval is available.
I'd use tree.
A sub-node contains the "precision" and "average" values.
"precision" means the date-range. For example: 1 minute, 10 minutes, 1 day, 1 month, etc. This also means a level in the tree.
"average" is the value that best represents the price for a range. You can use a simple average, a linear regression, or whatever you decide as "best".
So if you need 600 points (say you get the window size), you can find the precision by prec=total_date_range/600, and some rounding to your existing ranges.
Now you have the 'prec' you just need to retrieve the nodes for that 'prec` level.
Being gigabytes of data, I'd slice them into std::vector objects. The tree would store ids to these vectors for the lowest nodes. The rest of nodes could also be implemented by indices to vectors.
Updating with new data only requires to update a branch (or even creating a new one), starting from root, but with not so many sub-nodes.
I'm trying to implement a MCTS algorithm for the AI of a small game. The game is a rpg-simulation. The AI should decides what moves to play in battle. It's a turn base battle (FF6-7 style). There is no movement involved.
I won't go into details but we can safely assume that we know with certainty what move will chose the player in any given situation when it is its turn to play.
Games end-up when one party has no unit alive (4v4). It can take any number of turn (may also never end). There is a lot of RNG element in the damage computation & skill processing (attacks can hit/miss, crit or not, there is a lots of procs going on that can "proc" or not, buffs can have % value to happens ect...).
Units have around 6 skills each to give an idea of the branching factor.
I've build-up a preliminary version of the MCTS that gives poor results for now. I'm having trouble with a few things :
One of my main issue is how to handle the non-deterministic states of my moves. I've read a few papers about this but I'm still in the dark.
Some suggest determinizing the game information and run a MCTS tree on that, repeat the process N times to cover a broad range of possible game states and use that information to take your final decision. In the end, it does multiply by a huge factor our computing time since we have to compute N times a MCTS tree instead of one. I cannot rely on that since over the course of a fight I've got thousands of RNG element : 2^1000 MCTS tree to compute where i already struggle with one is not an option :)
I had the idea of adding X children for the same move but it does not seems to be leading to a good answer either. It smooth the RNG curve a bit but can shift it in the opposite direction if the value of X is too big/small compared to the percentage of a particular RNG. And since I got multiple RNG par move (hit change, crit chance, percentage to proc something etc...) I cannot find a decent value of X that satisfies every cases. More of a badband-aid than anythign else.
Likewise adding 1 node per RNG tuple {hit or miss ,crit or not,proc1 or not,proc2 or not,etc...} for each move should cover every possible situations but has some heavy drawbacks : with 5 RNG mecanisms only that means 2^5 node to consider for each move, it is way too much to compute. If we manage to create them all, we could assign them a probability ( linked to the probability of each RNG element in the node's tuple) and use that probability during our selection phase. This should work overall but be really hard on the cpu :/
I also cannot "merge" them in one single node since I've got no way of averaging the player/monsters stat's value accuractely based on two different game state and averaging the move's result during the move processing itself is doable but requieres a lot of simplifcation that are a pain to code and will hurt our accuracy really fast anyway.
Do you have any ideas how to approach this problem ?
Some other aspects of the algorithm are eluding me:
I cannot do a full playout untill a end state because A) It would take a lot of my computing time and B) Some battle may never ends (by design). I've got 2 solutions (that i can mix)
- Do a random playout for X turns
- Use an evaluation function to try and score the situation.
Even if I consider only health point to evaluate I'm failing to find a good evaluation function to return a reliable value for a given situation (between 1-4 units for the player and the same for the monsters ; I know their hp current/max value). What bothers me is that the fights can vary greatly in length / disparity of powers. That means that sometimes a 0.01% change in Hp matters (for a long game vs a boss for example) and sometimes it is just insignificant (when the player farm a low lvl zone compared to him).
The disparity of power and Hp variance between fights means that my Biais parameter in the UCB selection process is hard to fix. i'm currently using something very low, like 0.03. Anything > 0.1 and the exploration factor is so high that my tree is constructed depth by depth :/
For now I'm also using a biaised way to choose move during my simulation phase : it select the move that the player would choose in the situation and random ones for the AI, leading to a simulation biaised in favor of the player. I've tried using a pure random one for both, but it seems to give worse results. Do you think having a biaised simulation phase works against the purpose of the alogorithm? I'm inclined to think it would just give a pessimistic view to the AI and would not impact the end result too much. Maybe I'm wrong thought.
Any help is welcome :)
I think this question is way too broad for StackOverflow, but I'll give you some thoughts:
Using stochastic or probability in tree searches is usually called expectimax searches. You can find a good summary and pseudo-code for Expectimax Approximation with Monte-Carlo Tree Search in chapter 4, but I would recommend using a normal minimax tree search with the expectimax extension. There are a few modifications like Star1, Star2 and Star2.5 for a better runtime (similiar to alpha-beta pruning).
It boils down to not only having decision nodes, but also chance nodes. The probability of each possible outcome should be known and the expected value of each node is multiplied with its probability to know its real expected value.
2^5 nodes per move is high, but not impossibly high, especially for low number of moves and a shallow search. Even a 1-3 depth search shoulld give you some results. In my tetris AI, there are ~30 different possible moves to consider and I calculate the result of three following pieces (for each possible) to select my move. This is done in 2 seconds. I'm sure you have much more time for calculation since you're waiting for user input.
If you know what move the player is obvious, shouldn't it also obvious for your AI?
You don't need to consider a single value (hp), you can have several factors that are weighted different to calculate the expected value. If I come back to my tetris AI, there are 7 factors (bumpiness, highest piece, number of holes, ...) that are calculated, weighted and added together. To get the weights, you could use different methods, I used a genetic algorithm to find the combination of weights that resulted in most lines cleared.
Short Version:
I'm looking for a technique to keep nearly-sorted data in nearly-sorted order over time, despite the values changing slightly.
Here's the scenario:
In the world of 3D graphics, it is often beneficial to order your objects from front-to-back before drawing. As your scene changes or your view of the scene changes, this data may require re-sorting, however it will usually be very close to the sorted order (i.e. it won't change very much between frames). It's also not critical that the data be exactly in sorted order. The worst thing that will happen is that a polygon will be rendered and then completely hidden. It's a small performance hit, but not the end of the world.
With this in mind, is it possible to sort the data once ahead of time and then apply a minimal patch to the data once per frame to ensure that the data stays mostly sorted? In this scenario, the data would be considered mostly sorted if most of the objects were in ascending order. That is, 1 object that is 10 steps away from it's proper location is much better (10x better) than 10 objects that are 1 step away from their proper location.
It's also worth noting that the data could continue to be patched on a semi regular basis, as the data is typically rendered 30 times per second (or so). As long as the calculation was efficient, it could continue to be done over time until the changes stop and the list was completely sorted.
Existing Idea:
My knee jerk reaction to this problem is:
Apply an n log n sort to the data when it is loaded, and on large changes (which I can track pretty easily).
When the data starts changing slowly (e.g. when the scene is rotated), apply a single (linear) pass of some sort on the data to swap backwards neighbors and try to maintain sort order (I think this is basically shell sort - maybe there is a better algorithm to use for this single pass).
Keep doing a single pass of the partial sort each frame until the changes stop and the data is completely sorted
Go back to step 2 and wait for more changes.
There are a variety of sorts that run in O(n) time if the input is mostly sorted, and O(n log n) if the data is not sorted. It sounds like you can use that pretty easily. Timsort is one such sort and, I believe, is the default sort now in both python and java. Smoothsort is another one that is fairly easy to implement yourself.
From your description it sounds like the sort order changes without you changing the data itself. E.g. you change the camera, so the sort order should change, even though you have not modified any polygons.
If so, you can't detect sort order changes directly when they happen. If you could, I would create buckets for the list of polygons, and resort buckets when 'enough' polygons in that bucket have been touched.
But I'm betting your system doesn't work that way. The sort is determined by the view port. In that case polygons at the front of the sort matter much more than ones at the end.
So I'd segment the poly list into fifths or something like that. Front to back, so that the first fifth is the part closest to the camera. I'd completely sort the first segment every frame. I'd divide the second segment into sub segments - say 5 again - and sort each sub segment every frame, such that every 5 frames the second fifth is completely sorted. segment the third through 5th segments into 15 sub segments and do those every 5 frames each such that the rest get sorted completely every 75 frames. At 60 fps you'd have the display list completely resorted a little more than once per second.
The nice thing about prioritizing the front of the list, is
1. Polys at the front are going to tend to be larger on the screen, and will fail depth test more often. Bad orders at the end of the list will more often than not just not matter.
2. the front of the list is more susceptible sort changes due to camera changes.
Also chose those segment ranges with a little overlap, so that polygons can migrate to their correct segment in 2 sorts.
#OP: Thinking about it a little more. You are probably more concerned with having the sorting cost stay bounded - instead of exploding with scene complexity. Especially since a very complex scene should - surprisingly - be less susceptible to bad sorts ( because generally the polys get smaller ).
You could define a fixed amount of sorting you are willing to do per frame. Use say 50% of the budget for as much of the front of the list as you can afford, 25% of the budget to sort the next region and 25% to spend equally on the rest.
Say you budget 1000 polys sorted per frame, and you have 10000 polys in the scene. Sort the first 500 polys every frame. Sort 250 polys every tenth frame for the next region. So 501-750 on frame 1, 751-1000 on frame 2 etc. And then divide the rest of the list into 250 frame segments and sort them round robin for however many frames you need to.
This keeps the sorting cost fixed s the scene gets more and less complex, and it is easy to tune, you just adjust the sorting budget to what you can afford.
I'll suggest a solution that borrows from a number of others here. Of course we start with a full sort of the objects on initialisation.
What I would do is always perform, say, 10 linear-time runs over your objects for every frame (with early termination if you find out that your objects are already completely sorted). Each run can be, say, one pass of bubble sort with a shell sort-style gap over the whole array: for all i from 0 to n-gap-1, compare A[i] and A[i+gap], and exchange them if they are not sorted. You can use a fixed sequence of gaps, or maybe better, let it vary between frames; either way, if you do sufficiently many frames where the objects do not change, you'll have a fully sorted sequence. You could even mix different types of sub-algorithms to do your runs, as long as each iteration improves the 'sortedness'.
You can add Rafael Baptista's idea of prioritizing the front of the scene easily by doing one extra run on the front segment, or choosing to divide the gap by two for the front half, or something like that.
It doesn't work out as neatly as the problem you've supposed because all you have to do is turn the camera 90 degrees and the basis for being sorted is on a different axis entirely. (X and Y axis are independent, for example -- looking down the X axis will cause the sort order to not rely on the X axis, and looking down the Y axis will cause the sort order to not rely on the Y axis.) Even a 5 degree turn can cause far away "close" (as far as Z-order is concerned) things to be suddenly "far".
Let's be honest -- generating the draw calls for the objects is normally going to take much more time than sorting them, especially if you have an optimized sorting algorithm for your scenario and your game is of modern visual complexity.
Sorting can be practically O(n), especially with histogram-based algorithms or radix-style algorithms. (Yes, radix sort applies to integers, so you'd have to scale your world coordinates to integers, but normally that's more than good enough unless you have a gigantic world.)
That being said, since you're already doing O(n) ops for everything you're drawing, resorting per frame isn't going to be a huge problem, especially with both high and low level optimization.
Another common way of addressing this issue is with a scene graph, but for your purposes it ends up essentially being a re-sort per frame. However, you can build frustum culling, shadow culling, and level of detail calculations into the scene graph traversal.
If you're looking for approximations, instead of doing a z-distance sort do a true distance sort and update the sort order more often for close by objects and less often for further objects (depending on distance the camera has traveled). This can work because if you're further away from an object, moving doesn't cause the angle to the viewer to change as often which, in turn, means the old sorting data is more likely to be valid. I'm not a fan of this because I like algorithms which allow my game to teleport across the map without any issues. (Mind you, streaming assets from disk becomes the real issue for teleporting.)
Shell sort is good for lists with few unique values and some scenarios that "need short code and do not use the call stack".
In your case, you need something called Adaptive sort, which means algorithms "takes advantage of existing order in its input".
If your space is tight, you can just use Straight Insertion Sort, which is adaptive and in place.
Otherwise you can try Timsort and Smoothsort as #RunningWild suggested, they are both adaptive sort algorithms.
In order to save bandwith and so as to not to have generate pictures/graphs ourselves I plan on using Google's charting API:
http://code.google.com/apis/chart/
which works by simply issuing a (potentially long) GET (or a POST) and then Google generate and serve the graph themselves.
As of now I've got graphs made of about two thousands entries and I'd like to trim this down to some arbitrary number of entries (e.g. by keeping only 50% of the original entries, or 10% of the original entries).
How can I decide which entries I should keep so as to have my new graph the closest to the original graph?
Is this some kind of curve-fitting problem?
Note that I know that I can do POST to Google's chart API with up to 16K of data and this may be enough for my needs, but I'm still curious
The flot-downsample plugin for the Flot JavaScript graphing library could do what you are looking for, up to a point.
The purpose is to try retain the visual characteristics of the original line using considerably fewer data points.
The research behind this algorithm is documented in the author's thesis.
Note that it doesn't work for any kind of series, and won't give meaningful results when you want a downsampling factor beyond 10, in my experience.
The problem is that it cuts the series in windows of equal sizes then keep one point per window. Since you may have denser data in some windows than others the result is not necessarily optimal. But it's efficient (runs in linear time).
What you are looking to do is known as downsampling or decimation. Essentially you filter the data and then drop N - 1 out of every N samples (decimation or down-sampling by factor of N). A crude filter is just taking a local moving average. E.g. if you want to decimate by a factor of N = 10 then replace every 10 points by the average of those 10 points.
Note that with the above scheme you may lose some high frequency data from your plot (since you are effectively low pass filtering the data) - if it's important to see short term variability then an alternative approach is to plot every N points as a single vertical bar which represents the range (i.e. min..max) of those N points.
Graph (time series data) summarization is a very hard problem. It's like deciding, in a text, what is the "relevant" part to keep in an automatic summarization of it. I suggest you use one of the most respected libraries for finding "patterns of interest" in time series data by Eamonn Keogh
To experiment, I've (long ago) implemented Conway's Game of Life (and I'm aware of this related question!).
My implementation worked by keeping 2 arrays of booleans, representing the 'last state', and the 'state being updated' (the 2 arrays being swapped at each iteration). While this is reasonably fast, I've often wondered about how to optimize this.
One idea, for example, would be to precompute at iteration N the zones that could be modified at iteration (N+1) (so that if a cell does not belong to such a zone, it won't even be considered for modification at iteration (N+1)). I'm aware that this is very vague, and I never took time to go into the details...
Do you have any ideas (or experience!) of how to go about optimizing (for speed) Game of Life iterations?
I am going to quote my answer from the other question, because the chapters I mention have some very interesting and fine-tuned solutions. Some of the implementation details are in c and/or assembly, yes, but for the most part the algorithms can work in any language:
Chapters 17 and 18 of
Michael Abrash's Graphics
Programmer's Black Book are one of
the most interesting reads I have ever
had. It is a lesson in thinking
outside the box. The whole book is
great really, but the final optimized
solutions to the Game of Life are
incredible bits of programming.
There are some super-fast implementations that (from memory) represent cells of 8 or more adjacent squares as bit patterns and use that as an index into a large array of precalculated values to determine in a single machine instruction if a cell is live or dead.
Check out here:
http://dotat.at/prog/life/life.html
Also XLife:
http://linux.maruhn.com/sec/xlife.html
You should look into Hashlife, the ultimate optimization. It uses the quadtree approach that skinp mentioned.
As mentioned in Arbash's Black Book, one of the most simple and straight forward ways to get a huge speedup is to keep a change list.
Instead of iterating through the entire cell grid each time, keep a copy of all the cells that you change.
This will narrow down the work you have to do on each iteration.
The algorithm itself is inherently parallelizable. Using the same double-buffered method in an unoptimized CUDA kernel, I'm getting around 25ms per generation in a 4096x4096 wrapped world.
what is the most efficient algo mainly depends on the initial state.
if the majority of cells is dead, you could save a lot of CPU time by skipping empty parts and not calculating stuff cell by cell.
im my opinion it can make sense to check for completely dead spaces first, when your initial state is something like "random, but with chance for life lower than 5%."
i would just divide the matrix up into halves and start checking the bigger ones first.
so if you have a field of 10,000 * 10,000, you´d first accumulate the states of the upper left quarter of 5,000 * 5,000.
and if the sum of states is zero in the first quarter, you can ignore this first quarter completely now and check the upper right 5,000 * 5,000 for life next.
if its sum of states is >0, you will now divide up the second quarter into 4 pieces again - and repeat this check for life for each of these subspaces.
you could go down to subframes of 8*8 or 10*10 (not sure what makes the most sense here) now.
whenever you find life, you mark these subspaces as "has life".
only spaces which "have life" need to be divided into smaller subspaces - the empty ones can be skipped.
when you are finished assigning the "has life" attribute to all possible subspaces, you end up with a list of subspaces which you now simply extend by +1 to each direction - with empty cells - and perform the regular (or modified) game of life rules to them.
you might think that dividn up a 10,000*10,000 spae into subspaces of 8*8 is a lot os tasks - but accumulating their states values is in fact much, much less computing work than performing the GoL algo to each cell plus their 8 neighbours plus comparing the number and storing the new state for the net iteration somewhere...
but like i said above, for a random init state with 30% population this wont make much sense, as there will be not many completely dead 8*8 subspaces to find (leave alone dead 256*256 subpaces)
and of course, the way of perfect optimisation will last but not least depend on your language.
-110
Two ideas:
(1) Many configurations are mostly empty space. Keep a linked list (not necessarily in order, that would take more time) of the live cells, and during an update, only update around the live cells (this is similar to your vague suggestion, OysterD :)
(2) Keep an extra array which stores the # of live cells in each row of 3 positions (left-center-right). Now when you compute the new dead/live value of a cell, you need only 4 read operations (top/bottom rows and the center-side positions), and 4 write operations (update the 3 affected row summary values, and the dead/live value of the new cell). This is a slight improvement from 8 reads and 1 write, assuming writes are no slower than reads. I'm guessing you might be able to be more clever with such configurations and arrive at an even better improvement along these lines.
If you don't want anything too complex, then you can use a grid to slice it up, and if that part of the grid is empty, don't try to simulate it (please view Tyler's answer). However, you could do a few optimizations:
Set different grid sizes depending on the amount of live cells, so if there's not a lot of live cells, that likely means they are in a tiny place.
When you randomize it, don't use the grid code until the user changes the data: I've personally tested randomizing it, and even after a long amount of time, it still fills most of the board (unless for a sufficiently small grid, at which point it won't help that much anymore)
If you are showing it to the screen, don't use rectangles for pixel size 1 and 2: instead set the pixels of the output. Any higher pixel size and I find it's okay to use the native rectangle-filling code. Also, preset the background so you don't have to fill the rectangles for the dead cells (not live, because live cells disappear pretty quickly)
Don't exactly know how this can be done, but I remember some of my friends had to represent this game's grid with a Quadtree for a assignment. I'm guess it's real good for optimizing the space of the grid since you basically only represent the occupied cells. I don't know about execution speed though.
It's a two dimensional automaton, so you can probably look up optimization techniques. Your notion seems to be about compressing the number of cells you need to check at each step. Since you only ever need to check cells that are occupied or adjacent to an occupied cell, perhaps you could keep a buffer of all such cells, updating it at each step as you process each cell.
If your field is initially empty, this will be much faster. You probably can find some balance point at which maintaining the buffer is more costly than processing all the cells.
There are table-driven solutions for this that resolve multiple cells in each table lookup. A google query should give you some examples.
I implemented this in C#:
All cells have a location, a neighbor count, a state, and access to the rule.
Put all the live cells in array B in array A.
Have all the cells in array A add 1 to the neighbor count of their
neighbors.
Have all the cells in array A put themselves and their neighbors in array B.
All the cells in Array B Update according to the rule and their state.
All the cells in Array B set their neighbors to 0.
Pros:
Ignores cells that don't need to be updated
Cons:
4 arrays: a 2d array for the grid, an array for the live cells, and an array
for the active cells.
Can't process rule B0.
Processes cells one by one.
Cells aren't just booleans
Possible improvements:
Cells also have an "Updated" value, they are updated only if they haven't
updated in the current tick, removing the need of array B as mentioned above
Instead of array B being the ones with live neighbors, array B could be the
cells without, and those check for rule B0.