G5AIAI%20Introduction%20to%20AI

1
G5AIAIIntroduction 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
G5AIAIIntroduction to AI

• Graham Kendall

End Combinatorial Explosion