1. Combinatorial Explosion
Limitation of computation
Edward Tsang (Copyright)
Wednesday, 01 May 2019

2. Combinations In A Password
Suppose you can only use to make a password
Suppose your password has 4 digits
How many possible combinations? 104
Because there are 10 choices for the first digit, 10 for the second digit, …, etc.
So there are 10×10×10×10 combinations
For a password with n digits, there are 10n possibilities
The number of possibilities grows exponentially as n grows
This is called combinatorial explosion
Edward Tsang (Copyright)
Wednesday, 01 May 2019

3. Combinatorial Explosion Impact
Suppose you enter 1 password per second
A 4 digit password would take 104 seconds
Or approximately 3 hours
5 digits: 28 hours
6 digits: 12 days
7 digits: 116 days
8 digits: 3 years
Entering 10 passwords per second helps
but not very meaningful for very long passwords
Edward Tsang (Copyright)
Wednesday, 01 May 2019

4. Combinatorial Explosion in Chess
A Naïve chess program explores every possible moves, and every possible moves after that.
There are 20 possible moves at start, typically more possibilities in mid-game.
Assuming 20 possibilities per move, the program that considers k moves ahead will look at 20k possibilities
The number of possibilities grows exponentially as k grows
A grand master often looks 10 moves ahead
There are 2010 possibilities, or 10 trillion to evaluates
A program that calculates 1 million possibilities per second will take 118 days to make one move

5. Challenge in Computation
Today’s computers are fast
But they fail to contain combinatorial explosion
Combinatorial Explosion is a core issue in computation
It prevented computers from cracking passwords, beating chess world champions sooner, …
Much research in:
Finding out what problems can be solved efficiently
Developing specialized methods for special problems
By exploiting their properties

6. Limited by Combinatorial Explosion What could one do?
Finding near-solutions
Improving our odds

7. Finding approximate solutions
One may not be able to find the optimal solution, but…
For large operations, a saving of 0.5% means £millions
Lower cost  higher chance to win a bid for construction projects
So it helps to find near-solutions
This is where optimization techniques may help

8. Improving the odds
Given a fixed amount of time, one may not be able to guarantee finding solutions
But being able to find solutions 65% of the time is better than 46%
This is where heuristics may help

9. Summary Today’s computers are fast
But not fast enough for some problems
Due to the nature of the problem
Combinatorial explosion is a major issue in computation