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Title: G5AIAI%20Introduction%20to%20AI


1
G5AIAI Introduction to AI
  • Graham Kendall

Combinatorial Explosion
Graham Kendall GXK_at_CS.NOTT.AC.UK www.cs.nott.ac.uk
/gxk 44 (0) 115 846 6514
2
The Travelling Salesman Problem
  • A salesperson has to visit a number of cities
  • (S)He can start at any city and must finish at
    that same city
  • The salesperson must visit each city only once
  • The number of possible routes is (n!)/2 (where n
    is the number of cities)

3
Combinatorial Explosion
4
Combinatorial Explosion
5
Combinatorial Explosion
A 10 city TSP has 181,000 possible solutions A 20
city TSP has 10,000,000,000,000,000 possible
solutions A 50 City TSP has 100,000,000,000,000,00
0,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000 possible solutions
There are 1,000,000,000,000,000,000,000 litres of
water on the planet
Mchalewicz, Z, Evolutionary Algorithms for
Constrained Optimization Problems, CEC 2000
(Tutorial)
6
Combinatorial Explosion - Towers of Hanoi
7
Combinatorial Explosion - Towers of Hanoi
8
Combinatorial Explosion - Towers of Hanoi
9
Combinatorial Explosion - Towers of Hanoi
10
Combinatorial Explosion - Towers of Hanoi
11
Combinatorial Explosion - Towers of Hanoi
12
Combinatorial Explosion - Towers of Hanoi
13
Combinatorial Explosion - Towers of Hanoi
14
Combinatorial Explosion - Towers of Hanoi
  • How many moves does it take to move four rings?
  • You might like to try writing a towers of hanoi
    program (and you may well have to in one of your
    courses!)

15
Combinatorial Explosion - Towers of Hanoi
  • If you are interested in an algorithm here is a
    very simple one
  • Assume the pegs are arranged in a circle
  • 1. Do the following until 1.2 cannot be done
  • 1.1 Move the smallest ring to the peg residing
    next to it, in clockwise order
  • 1.2 Make the only legal move that does not
    involve the smallest ring
  • 2. Stop
  • P. Buneman and L.Levy (1980). The Towers of Hanoi
    Problem, Information Processing Letters, 10, 243-4

16
Combinatorial Explosion - Towers of Hanoi
  • A time analysis of the problem shows that the
    lower bound for the number of moves is
  • 2N-1
  • Since N appears as the exponent we have an
    exponential function

17
Combinatorial Explosion - Towers of Hanoi
18
Combinatorial Explosion - Towers of Hanoi
  • The original problem was stated that a group of
    tibetan monks had to move 64 gold rings which
    were placed on diamond pegs.
  • When they finished this task the world would end.
  • Assume they could move one ring every second (or
    more realistically every five seconds).
  • How long till the end of the world?

19
Combinatorial Explosion - Towers of Hanoi
  • gt 500,000 years!!!!! Or 3 Trillion years
  • Using a computer we could do many more moves than
    one a second so go and try implementing the 64
    rings towers of hanoi problem.
  • If you are still alive at the end, try 1,000
    rings!!!!

20
Combinatorial Explosion - Optimization
  • Optimize f(x1, x2,, x100)
  • where f is complex and xi is 0 or 1
  • The size of the search space is 2100 ? 1030
  • An exhaustive search is not an option
  • At 1000 evaluations per second
  • Start the algorithm at the time the universe was
    created
  • As of now we would have considered 1 of all
    possible solutions

21
Combinatorial Explosion
22
Combinatorial Explosion
Running on a computer capable of 1 million
instructions/second
Ref Harel, D. 2000. Computer Ltd. What they
really cant do, Oxford University Press
23
G5AIAI Introduction to AI
  • Graham Kendall

End Combinatorial Explosion
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