2. Graham Kendall GXK@CS.NOTT.AC.UK www.cs.nott.ac.uk/~gxk +44 (0) 115 846 6514 G5AIAI Introduction to AI Graham Kendall Combinatorial Explosion

3. G5AIAI History of AI 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)

4. G5AIAI History of AI Combinatorial Explosion

5. G5AIAI History of AI Combinatorial Explosion

6. G5AIAI History of AI 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 100,000,000,000,000,000,000,000,000,000,00 0,000,000,000,000,000,000,000,000,000,000 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)

7. G5AIAI History of AI Combinatorial Explosion - Towers of Hanoi

8. G5AIAI History of AI Combinatorial Explosion - Towers of Hanoi

9. G5AIAI History of AI Combinatorial Explosion - Towers of Hanoi

10. G5AIAI History of AI Combinatorial Explosion - Towers of Hanoi

11. G5AIAI History of AI Combinatorial Explosion - Towers of Hanoi

12. G5AIAI History of AI Combinatorial Explosion - Towers of Hanoi

13. G5AIAI History of AI Combinatorial Explosion - Towers of Hanoi

14. G5AIAI History of AI Combinatorial Explosion - Towers of Hanoi

15. G5AIAI History of AI 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!)

16. G5AIAI History of AI 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

17. G5AIAI History of AI Combinatorial Explosion - Towers of Hanoi A time analysis of the problem shows that the lower bound for the number of moves is 2 N -1 Since N appears as the exponent we have an exponential function

18. G5AIAI History of AI Combinatorial Explosion - Towers of Hanoi

19. G5AIAI History of AI 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?

20. G5AIAI History of AI Combinatorial Explosion - Towers of Hanoi > 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!!!!

21. G5AIAI History of AI Combinatorial Explosion - Optimization Optimize f(x 1, x 2,…, x 100 ) where f is complex and x i is 0 or 1 The size of the search space is 2 100  10 30 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

22. G5AIAI History of AI Combinatorial Explosion Microseconds in a Day Microseconds since Big Bang

23. G5AIAI History of AI Combinatorial Explosion 102050100200 N2N2 N5N5 1/10,000 second 1/2500 second 1/400 second 1/100 second 1/25 second 1/10 second 3.2 seconds 5.2 minutes 2.8 hours 3.7 days 2N2N N 1/1000 second 1 second 35.7 years > 400 trillion centuries 45 digit no. of centuries 2.8 hours 3.3 trillion years 70 digit no. of centuries 185 digit no. of centuries 445 digit no. of centuries Running on a computer capable of 1 million instructions/second Ref : Harel, D. 2000. Computer Ltd. : What they really can’t do, Oxford University Press

24. G5AIAI Introduction to AI Graham Kendall End Combinatorial Explosion