Nature's Algorithms by hcj

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									Nature’s Algorithms
     David C. Uhrig
     Tiffany Sharrard

      CS 477R – Fall 2007
       Dr. George Bebis
What are “Nature’s Algorithms”?
   Combinatorial Optimization Problems
     No easy way to examine all combinations
     Use algorithms based on scientific principles, nature,
      and genetics
   This does not give optimal solutions, but is
    more practical than an exhaustive search
    Example: Traveling Salesman
   The Traveling Salesman problem is one of the
    most famous combinatorial optimization
    problems:
       Given a set of cities (nodes), find the shortest path
        required to visit all cities exactly once.
    Example: Traveling Salesman
   There are (n-1)!/2 possible tours for n cities.
     2520 tours for 8 cities
     653,837,184,000 tours for 16 cities
           Would require a computer nearly 8 days to compute them
            all! (for 1/1,000,000 sec per tour generation)
   Not practical to compute all tours…
    Example: Location Allocation
   Given a set of facilities, each of which can serve
    a certain number of nodes on a graph, place the
    facilities in such a way that the distance between
    each node and the facility is minimized.
     Police stations and population densities
     Schools and student populations
Example: Location Allocation
         Issues with Combinatorial
          Optimization Problems
   Large set of discrete solutions
   Number of solutions grows exponentially
     In the Travelling Salesman problem, the solution
      size grows by n*(n-1)! for each city that is added.
     In the Location Allocation problem, the solution
      size is affected by both the number of nodes but also
      the connections between nodes and the number of
      (and size of) facilities.
    Evolution and the Genetic Algorithm

   Mimics the process of evolution in developing a
    solution.
   Begins with a large set of solutions to the
    problem in a candidate pool.
   Each of these algorithms is tested for „fitness‟
    Evolution and the Genetic Algorithm

   After the fitnesses are calculated, a new set of
    solutions is randomly picked from the „most fit‟
    algorithms.
       Mimics Natural Selection
   Next, two algorithms are paired together and
    swap pieces of each solutions
   Then each algorithm changes (mutates) slightly
Evolution and the Genetic Algorithm
   This pattern of evaluation, reproduction,
    swapping, and mutating mimics the way DNA is
    processed during sexual reproduction.
   This process continues until a satisfactory
    solution is much better than the rest of the
    population of solutions.
Metallurgy and the Simulated Annealing
               Problem
   In metallurgy, when molten metal cools too
    quickly it can lose certain properties, such as
    strength or magnetism.
   Annealing is the process of slowly cooling,
    which can help retain or amplify desired traits.
   In Computer Science, Simulated Annealing
    Algorithms are usually an improvement on
    “Greedy Local Search” algorithms.
 Metallurgy and the Simulated Annealing
                Problem
1.       Start with a proposed solution.
2.       Change the solution slightly
     •     i.e. Swap two cities in the traveling salesman
           problem.
3.       Check to see if the new solution is better than
         the original. If not, undo the change.
4.       Repeat steps 2 and 3 until it is apparent that
         any small change makes the solution worse.
Metallurgy and the Simulated Annealing
               Problem
   A greedy local search
    will only find Locally
    Optimal Solutions.




   This also occurs with Genetic Algorithms when
    mutations stop producing positive changes.
Metallurgy and the Simulated Annealing
               Problem
   Try to improve on Greedy Local Searches by
    taking occasional risks by accepting a worse
    solution.
       Compute ΔC, the difference between the cost of the
        current solution and the proposed solution. Then:
                 for Random(r), 0 < r < 1, if r < e -ΔC/T
        Where T is the metaphoric “temperature” that is
        iteratively decreased until an acceptable temperature
        is reached.
Metallurgy and the Simulated Annealing
               Problem
   Theoretically, the temperature should be
    lowered proportionally to 1/log(n), but this rate
    is too slow.
   Realistically, the temperature is lowered at a
    linear or exponential rate.
   Even with this decreased cooling time, simulated
    annealing has proven to be very effective in
    finding a balance between exploring every
    solution and avoiding locally optimal results.
                 Economics
   The idea of “commodities” and “consumers”
    competing for them
   Prices are determined by competition
   Consumers buy what they can afford
   Prices are influenced by consumers
           Computer Networks
   Consumer = Process
   Competing commodities: processor time, access
    to network printer, etc
                    Example
   Two terminals want to share two database files
   The files will run slower if they are accessed
    from the other location than from the local
    location
   Decision should be made based on how much
    of each file should be stored at each terminal
                      Solution
   Give each file a starting price
   Give each terminal a fixed amount of money
   With constraints of the amount of money, terminals ask
    for a certain portion of each file
   Price changes based on the proportion of how much of
    the files were requested
   Done until equilibrium is reached
     Simulated N-Body Algorithm
   Study of moving particles in physics
   Particles are allowed to move around the graph
   To minimize potential energy in the system,
    particles tend to spread out
       A “boundary” is formed to keep the particles from
        being pushed to the outer edge
Aggregation and Decomposition
   Every particle is grouped into a larger particle
    with combined mass
   Smallest particle is removed
   Small and the largest particle moved to where
    the potential energy is the lowest
   Take next smallest particle and do the same
   Result: reasonably good configuration
                  Particle Physics
   Simulation used to solve traveling salesman
    problems
       Use of charged particles connected by electric bands
               How Does It Work?
   Number of Cities = Number of negatively- charged
    particles = Number of positively-charged particles
   Positive particles are placed in a circle
   Solution:
       Particles are set free
       When particle hits a city-stops
       When all have stopped- elastic string traces a path through
        each city

								
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