http://www.thelowlyprogrammer.com/2011/05/game-of-life-part-2-hashlife.html
HashLife is a simple yet interesting algorithm. Invented in 1984 by Bill Gosper (of Gosper glider gun fame), it exploits repeated patterns to dramatically cut down the work required to support large patterns over vast numbers of iterations
Together these properties allow for ridiculous speedups. Hashing and sharing nodes drastically reduces the space requirements, with exponentially more sharing the further down the tree you go. There are only 16 possible leaf nodes, after all! From this, calculating the future core for a node requires exponentially less time than a naïve implementation would. It can be done by recursively calculating the inner core of smaller nodes, where the better caching comes into play, and then combining them together into a new node. You might be wondering if the gains from caching are lost to the increasing difficulty of determining which nodes are equal, but with a couple careful invariants we actually get that for free. First, nodes must be immutable - this one’s pretty straightforward. Second, nodes must be unique at all times. This forces us to build the tree from the bottom up, but then checking if a new node duplicates an existing one is simply a matter of checking if there are any existing nodes that point to the same set of quadrants in the same order, a problem that hash tables trivially solve.
http://pzemtsov.github.io/2015/04/24/game-of-life-hash-tables-and-hash-codes.html
There are two major classes of Life programs. The matrix-based ones, such as those in my old experiments, store a part of the universe (or the entire universe considered being a toroidal array, as in the examples above) as a matrix. The entire matrix is re-calculated on each step. These programs are very simple but have two disadvantages. Firstly, they are limited by the size of the matrix, and Life structures are known to be able to grow indefinitely (for example, by spawning gliders). This means such programs can only produce approximate results, such as the result we saw for the R-pentamino above. Secondly, most Life colonies contain much fewer live cells than there’re squares in the matrix, which suggests a way of improving speed – to keep track of the occupied cells in some data structure and only process those.
This is the second class of Life programs. Very sophisticated data structures have been used for Life simulation, some with extraordinary performance. For instance, the HashLife approach makes use of identifying repeating patterns both in space and time, with great results.
http://www.drdobbs.com/jvm/an-algorithm-for-compressing-space-and-t/184406478
- combines quadtrees and memoization to yield astronomical speedup to the Game of Life
The Game of Life (Figure 1) is a solitaire game played on an infinite two-dimensional grid. Each cell in the grid is either alive (1, denoted by a black circle) or dead. Every living cell that has two or three living neighbors (of the eight immediately adjacent) remains alive, otherwise it dies. Any dead cell surrounded by exactly three living neighbors is a birth cell and is born into the next generation. All the rules are applied to all the cells at the same instant.
These rules lead to life forms that are stable and oscillate—gliders and spaceships that transport themselves across the universe, life forms that grow without bound or completely die away. Seed patterns with as few as nine initial living cells can mutate for many thousands of generations, with constantly moving areas of chaotic activity spewing forth gliders among a large stable menagerie.
http://pzemtsov.github.io/2015/04/24/game-of-life-hash-tables-and-hash-codes.html
https://discuss.leetcode.com/topic/37410/the-conway-game-of-life-using-hashtable-in-java-fast-solution-which-only-check-alive-cell-and-its-neighbors
HashLife is a simple yet interesting algorithm. Invented in 1984 by Bill Gosper (of Gosper glider gun fame), it exploits repeated patterns to dramatically cut down the work required to support large patterns over vast numbers of iterations
At it’s heart, HashLife is built around the concept of a quadtree. If you’re unfamiliar with it, a quadtree takes a square region and breaks it into four quadrants, each a quarter the size of the original. Each quadrant is further broken down into quadrants of its own, and on down. At the bottom, in squares of some minimum size like 2x2, actual points are stored.
This structure is usually used to make spatial queries like “what points intersect this bounding box” efficient, but in this case two other properties are taken advantage of. First, nodes at any level are uniquely defined by the points within their region, which means duplicated regions can be backed by the same node in memory. For the Game of Life, where there are repeated patterns and empty regions galore, this can drastically reduce the space required. Second, in the Game of Life a square of (n)x(n) points fully dictates the inner (n-2)x(n-2) core one generation forward, the inner (n/2)x(n/2) core n/4 generations forward, irrespective of what cells are adjacent to it. So the future core of a node can be calculated once and will apply at any future point in time, anywhere in the tree.
Together these properties allow for ridiculous speedups. Hashing and sharing nodes drastically reduces the space requirements, with exponentially more sharing the further down the tree you go. There are only 16 possible leaf nodes, after all! From this, calculating the future core for a node requires exponentially less time than a naïve implementation would. It can be done by recursively calculating the inner core of smaller nodes, where the better caching comes into play, and then combining them together into a new node. You might be wondering if the gains from caching are lost to the increasing difficulty of determining which nodes are equal, but with a couple careful invariants we actually get that for free. First, nodes must be immutable - this one’s pretty straightforward. Second, nodes must be unique at all times. This forces us to build the tree from the bottom up, but then checking if a new node duplicates an existing one is simply a matter of checking if there are any existing nodes that point to the same set of quadrants in the same order, a problem that hash tables trivially solve.There are two major classes of Life programs. The matrix-based ones, such as those in my old experiments, store a part of the universe (or the entire universe considered being a toroidal array, as in the examples above) as a matrix. The entire matrix is re-calculated on each step. These programs are very simple but have two disadvantages. Firstly, they are limited by the size of the matrix, and Life structures are known to be able to grow indefinitely (for example, by spawning gliders). This means such programs can only produce approximate results, such as the result we saw for the R-pentamino above. Secondly, most Life colonies contain much fewer live cells than there’re squares in the matrix, which suggests a way of improving speed – to keep track of the occupied cells in some data structure and only process those.
This is the second class of Life programs. Very sophisticated data structures have been used for Life simulation, some with extraordinary performance. For instance, the HashLife approach makes use of identifying repeating patterns both in space and time, with great results.
http://www.drdobbs.com/jvm/an-algorithm-for-compressing-space-and-t/184406478
- combines quadtrees and memoization to yield astronomical speedup to the Game of Life
The Game of Life (Figure 1) is a solitaire game played on an infinite two-dimensional grid. Each cell in the grid is either alive (1, denoted by a black circle) or dead. Every living cell that has two or three living neighbors (of the eight immediately adjacent) remains alive, otherwise it dies. Any dead cell surrounded by exactly three living neighbors is a birth cell and is born into the next generation. All the rules are applied to all the cells at the same instant.
These rules lead to life forms that are stable and oscillate—gliders and spaceships that transport themselves across the universe, life forms that grow without bound or completely die away. Seed patterns with as few as nine initial living cells can mutate for many thousands of generations, with constantly moving areas of chaotic activity spewing forth gliders among a large stable menagerie.
http://pzemtsov.github.io/2015/04/24/game-of-life-hash-tables-and-hash-codes.html
https://discuss.leetcode.com/topic/37410/the-conway-game-of-life-using-hashtable-in-java-fast-solution-which-only-check-alive-cell-and-its-neighbors
