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Graham Kendall
GXK@CS.NOTT.AC.UK
www.cs.nott.ac.uk/~gxk
+44 (0) 115 846 6514
G5AIAIIntroduction to AI
Graham KendallCombinatorial Explosion
G5G5AIAIAIAI History of AI 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)
G5G5AIAIAIAI History of AI History of AI
Combinatorial Explosion
Travelling Salesman Problem
0
500000
1000000
1500000
2000000
1 2 3 4 5 6 7 8 9 10
Cities
Ro
ute
s
G5G5AIAIAIAI History of AI History of AI
Combinatorial Explosion
Cities Routes
1 12 13 34 125 606 3607 25208 201609 18144010 181440011 19958400
G5G5AIAIAIAI History of AI History of AI
Combinatorial ExplosionA 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 100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 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)
G5G5AIAIAIAI History of AI History of AI
Combinatorial Explosion - Towers of Hanoi
G5G5AIAIAIAI History of AI History of AI
Combinatorial Explosion - Towers of Hanoi
G5G5AIAIAIAI History of AI History of AI
Combinatorial Explosion - Towers of Hanoi
G5G5AIAIAIAI History of AI History of AI
Combinatorial Explosion - Towers of Hanoi
G5G5AIAIAIAI History of AI History of AI
Combinatorial Explosion - Towers of Hanoi
G5G5AIAIAIAI History of AI History of AI
Combinatorial Explosion - Towers of Hanoi
G5G5AIAIAIAI History of AI History of AI
Combinatorial Explosion - Towers of Hanoi
G5G5AIAIAIAI History of AI History of AI
Combinatorial Explosion - Towers of Hanoi
G5G5AIAIAIAI History of AI 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!)
G5G5AIAIAIAI History of AI 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
G5G5AIAIAIAI History of AI 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
2N-1
• Since N appears as the exponent we have an exponential function
G5G5AIAIAIAI History of AI History of AI
Combinatorial Explosion - Towers of Hanoi
Pegs 2N-1
3 74 155 326 63… …10 1023
G5G5AIAIAIAI History of AI 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?
G5G5AIAIAIAI History of AI 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!!!!
G5G5AIAIAIAI History of AI History of AI
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
G5G5AIAIAIAI History of AI History of AI
Combinatorial Explosion
11E+141E+281E+421E+561E+701E+841E+981E+1121E+1261E+1401E+1541E+1681E+1821E+1961E+2101E+2241E+2381E+2521E+2661E+280
2 4 8 16 32 64 128 256 512 1024 2048
5N
N^3
N^5
N^10
1.2^N
2^N
N^N
Microseconds in a Day
Microseconds since Big Bang
G5G5AIAIAIAI History of AI History of AI
Combinatorial Explosion
10 20 50 100 200
N2
N5
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
2N
NN
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
G5AIAIIntroduction to AI
Graham KendallEnd Combinatorial Explosion
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