Cellular Automata: Life with Simple Rules

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Cellular AutomataLife with Simple Rules

• Sharks and Fish
• Predator/Prey Relationships
• Bill Madden, Nancy Ricca and Jonathan Rizzo
• Graduate Students, Computer Science Department
• Research Project using Departments 20-CPU
  Cluster

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Overview

• Cellular automata can be used to model complex
  systems using simple rules.
• Key features
• divide problem space into cells
• each cell can be in one of several finite states
• cells are affected by neighbors according to
  rules
• all cells are affected simultaneously in a
  generation
• rules are reapplied over many generations

Adapted from Wilkinson,B and M. Allen (1999)
Parallel Programming 2nd Edition, NJ, Pearson
Prentice Hall, p189
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Overview

• This project models a predator/prey relationship
• Begins with a randomly distributed population of
  fish, sharks, and empty cells in a 1000x2000 cell
  grid (2 million cells)
• Initially,
• 50 of the cells are occupied by fish
• 25 are occupied by sharks
• 25 are empty

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Heres the number 2 million

• Fish red sharks yellow empty black

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Rules

• A dozen or so rules describe life in each cell
• birth, longevity and death of a fish or shark
• breeding of fish and sharks
• over- and under-population
• fish/shark interaction
• Important what happens in each cell is
  determined only by rules that apply locally, yet
  which often yield long-term large-scale patterns.

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Do a LOT of computation!

• Apply a dozen rules to each cell
• Do this for 2 million cells in the grid
• Do this for 20,000 generations
• Well over a trillion calculations per run!
• Do this as quickly as you can

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Do a LOT of computation!

• We used a 20-CPU cluster in the Computer Science
  Department (Galaxy)
• Gal is the smaller of two clusters run by the
  Department (larger one has 64 CPUs)
• 15x faster than a single PC
• Longest runs still took about 45 minutes
• GO PARALLEL !!!

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Rules in detail Initial Conditions

• Initially cells contain fish, sharks or are empty
• Empty cells 0 (black pixel)
• Fish 1 (red pixel)
• Sharks 1 (yellow pixel)

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Rules in detail Breeding Rule

• Breeding rule if the current cell is empty
• If there are gt 4 neighbors of one species, and
  gt 3 of them are of breeding age,
• Fish breeding age gt 2,
• Shark breeding age gt3,
• and there are lt4 of the other species
• then create a species of that type
• 1 baby fish (age 1 at birth)
• -1 baby shark (age -1 at birth)

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Breeding Rule Before
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Breeding Rule After
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Rules in Detail Fish Rules

• If the current cell contains a fish
• Fish live for 10 generations
• If gt5 neighbors are sharks, fish dies (shark
  food)
• If all 8 neighbors are fish, fish dies
  (overpopulation)
• If a fish does not die, increment age

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Rules in Detail Shark Rules

• If the current cell contains a shark
• Sharks live for 20 generations
• If gt6 neighbors are sharks and fish neighbors
  0, the shark dies (starvation)
• A shark has a 1/32 (.031) chance of dying due to
  random causes
• If a shark does not die, increment age

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Shark Random Death Before
I Sure Hope that the random number chosen is
gt.031
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Shark Random Death After
YES IT IS!!! I LIVE ?
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Programming logic

• Use 2-dimensional array to represent grid
• At any one (x, y) position, value is
• Positive integer (fish present)
• Negative integer (shark present)
• Zero (empty cell)
• Absolute value of cell is age

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Sample Code (C) Breeding
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Parallelism

• A single CPU has to do it all
• Applies rules to first cell in array
• Repeats rules for each successive cell in array
• After 2 millionth cell is processed, array is
  updated
• One generation has passed
• Repeat this process for many generations
• Every 100 generations or so, convert array to
  red, yellow and black pixels and send results to
  screen

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Parallelism

• How to split the work among 20 CPUs
• 1 CPU acts as Master (has copy of whole array)
• 18 CPUs act as Slaves (handle parts of the array)
• 1 CPU takes care of screen updates
• Problem communication issue concerning cells
  along array boundaries among slaves

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Send Right Boundary Values
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Receive Left Boundary Values
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Send Left Boundary Values
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Receive Right Boundary Values
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Send Right Boundary Values
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Receive Left Boundary Values
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Send Left Boundary Values
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Receive Right Boundary Values
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At intervals, update the master CPU
has copy of entire array
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MPI Classes

• Gal (the 20-CPU cluster we used) has special
  enhancements to C, called MPI Classes, which
  handle such boundary communication issues
• Other than that, each slave processes its portion
  of the array much as a single CPU would process
  the whole array except, of course, they each have
  only 1/18th of the problem to solve!

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Results

• Next several screens show behavior over a span of
  10,000 generations (about 25 minutes on Gal)

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Generation 0
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Generation 100
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Generation 500
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Generation 1,000
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Generation 2,000
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Generation 4,000
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Generation 8,000
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Generation 10,500
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Long-term trends

• Borders tended to harden along vertical,
  horizontal and diagonal lines
• Borders of empty cells form between like species
• Clumps of fish tend to coalesce and form convex
  shapes or communities

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Variations of Initial Conditions

• Still using randomly distributed populations
• Medium-sized population. Fish/sharks
  occupy 1/16th of total grid Fish 62,703
  Sharks 31,301
• Very small population. Fish/sharks
  occupy 1/800th of total grid Initial
  population Fish 1,298 Sharks 609

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Medium-sized population (1/16 of grid)
Generation 100
2000
1000
4000
8000
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Very Small Populations

• Random placement of very small populations can
  favor one species over another
• Fish favored sharks die out
• Sharks favored sharks predominate, but fish
  survive in stable small numbers

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Very Small Populations
Gen. 100
4000
6000
1500
8000
10,000
12,000
14,000
Ultimate welfare of sharks depends on initial
random placement of fish and sharks
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Very small populations

• Fish can live in stable isolated communities as
  small as 20-30
• A community of less than 200 sharks tends not to
  be viable

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Parallel Performance

• Parallel processing suggests speedup of close to
  18 is possible
• Our results 15x speedup
• Reason communication overhead among slaves and
  between slaves and master
• Communication overhead becomes much worse if
  results are communicated more often

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Speed-up
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Communication Overhead
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Acknowledgements

• Bill Madden, Nancy Ricca and Jonathan Rizzo,
  Graduate Students, Computer Science Department,
  Montclair State University
• Dr. Roman Zaritski, Professor, Computer Science
  Department, Montclair State University
• Galaxy 20-CPU cluster in Computer Science
  Department, Montclair State University

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Parallel Processing Laboratory

• This presentation is available on-line at
• http//roman.montclair.edu/Research/Parallel/
• The web site above also hosts a number of other
  interesting parallel computing projects and is
  the on-line home of the Computer Science
  Departments Parallel Processing Laboratory.
• GO PARALLEL !!!