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MASTERMIND
Henning Thomas(joint with Benjamin Doerr,Reto Spöhel and Carola Winzen)
Henning Thomas Mastermind ETH Zurich 2012
Mastermind
Board game invented by Mordechai Meirovitz in 1970
Henning Thomas Mastermind ETH Zurich 2012
Mastermind
The “Codemaker” generates a secret color combination of length 4 with 6 colors,
The “Codebreaker” queries such color combinations,
The answer by Codemaker is , depicted by black pegs , depicted by
white pegs
The goal of Codemaker is to identify m with as few queries as possible.secret
query answer
Henning Thomas Mastermind ETH Zurich 2012
Mastermind with n slots and k colors
The “Codemaker” generates a secret color combination of length n with k colors,
The “Codebreaker” queries such color combinations,
The answer by Codemaker is , depicted by black pegs , depicted by
white pegs
The goal of Codemaker is to identify m with as few queries as possible.secret
query answer
Henning Thomas Mastermind ETH Zurich 2012
Mastermind with n slots and k colors
The “Codemaker” generates a secret color combination of length n with k colors,
The “Codebreaker” queries such color combinations,
The answer by Codemaker is , depicted by black pegs , depicted by
white pegs
The goal of Codemaker is to identify m with as few queries as possible.secret
query answer
This talk:Black PegMastermind
Henning Thomas Mastermind ETH Zurich 2012
Mastermind with n slots and k colors
The “Codemaker” generates a secret color combination of length n with k colors,
The “Codebreaker” queries such color combinations,
The answer by Codemaker is , depicted by black pegs , depicted by
white pegs
The goal of Codemaker is to identify m with as few queries as possible.secret
query answer
This talk:Black PegMastermind
What is the minimum number t = t(k,n) of queries such that there exists a
deterministic strategy to identify every secret color combination?
Henning Thomas Mastermind ETH Zurich 2012
Some Known Results & Our Results
[Knuth ’76], In the original board game (4 slots, 6 colors) 5 queries are optimal.
Henning Thomas Mastermind ETH Zurich 2012
Some Known Results & Our Results
[Knuth ’76], In the original board game (4 slots, 6 colors) 5 queries are optimal.
[Erdős, Rényi, ’63], Analysis of non-adaptive strategies for 0-1-Mastermind
In this talk: [Chvátal, ’83], Asymptotically optimal strategy for
using random queries [Goodrich, ’09], Improvement of Chvátals results
by a factor of 2 using deterministic strategyOur Result: Improved bound for k=n by combining Chvátal
and Goodrich
Henning Thomas Mastermind ETH Zurich 2012
Lower Bound
Information theoretic argument:
...
start
query 1
query 2
1 leaf
n leaves
n2 leaves
query t nt leaves
0 n
Henning Thomas Mastermind ETH Zurich 2012
Upper Bound (Chvátal)
Idea: Ask Random Queries. Intuition:
The number of black pegs of a query is Bin(n, 1/k) distributed.
Hence, we ‚learn‘ roughly bits per query. We need to learn n log k bits. t satisfies
0 n
Henning Thomas Mastermind ETH Zurich 2012
Comparison Lower Bound vs Chvátal
The optimal number of queries t satisfies
Problem for k=n: Non-Adaptive: Learning does not improve during
the game. For k=n we expect 1 black peg per query. We learn a constant number of bits. This yields
good ifk=o(n)
Henning Thomas Mastermind ETH Zurich 2012
Upper Bound (Goodrich)
Idea: Answer “0” is good since we can eliminate one
color from every slot!
Henning Thomas Mastermind ETH Zurich 2012
Upper Bound (Goodrich)
Implementation: Divide and Conquer1. Ask monochromatic queries for every color.
Obtain Xi = # appearances of color i.
2. Ask
3. Calculate Li = # appearnace of color i in left halfRi = # appearnace of color i in right
half
11 ... 1 22 ... 211 ... 1 33 ... 3
11 ... 1 kk ... k
b2
b3
bk
11 ... 122 ... 2
kk ... k
Henning Thomas Mastermind ETH Zurich 2012
Upper Bound (Goodrich)
Implementation: Divide and Conquer1. Ask monochromatic queries for every color.
Obtain Xi = # appearances of color i.
2. Ask
3. Calculate Li = # appearnace of color i in left halfRi = # appearnace of color i in right half
4.Recurse in the left and right half (without step 1)
5.Runtime for k=n:
11 ... 1 22 ... 211 ... 1 33 ... 3
11 ... 1 kk ... k
b2
b3
bk
11 ... 122 ... 2
kk ... k
Henning Thomas Mastermind ETH Zurich 2012
Comparison Lower Bound vs Goodrich
For k=n Goodrich yields
Problem: When Goodrich runs for a while, the blocks
eventually become too small that we cannot learn as many bits as we would like to.
Henning Thomas Mastermind ETH Zurich 2012
Combining Chvátal and Goodrich
Goodrich is good at eliminating colors. Chvátal is good for k << n.
Idea: 2 phases.
(i)Goodrich(ii)Chvátal
Henning Thomas Mastermind ETH Zurich 2012
Henning Thomas Mastermind ETH Zurich 2012