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Transcript of MASTERMIND Henning Thomas (joint with Benjamin Doerr, Reto Spöhel and Carola Winzen) TexPoint fonts...
![Page 1: MASTERMIND Henning Thomas (joint with Benjamin Doerr, Reto Spöhel and Carola Winzen) TexPoint fonts used in EMF. Read the TexPoint manual before you delete.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649cef5503460f949bdb5e/html5/thumbnails/1.jpg)
MASTERMIND
Henning Thomas(joint with Benjamin Doerr,Reto Spöhel and Carola Winzen)
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Henning Thomas Mastermind ETH Zurich 2012
Mastermind
Board game invented by Mordechai Meirovitz in 1970
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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
![Page 4: MASTERMIND Henning Thomas (joint with Benjamin Doerr, Reto Spöhel and Carola Winzen) TexPoint fonts used in EMF. Read the TexPoint manual before you delete.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649cef5503460f949bdb5e/html5/thumbnails/4.jpg)
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
![Page 5: MASTERMIND Henning Thomas (joint with Benjamin Doerr, Reto Spöhel and Carola Winzen) TexPoint fonts used in EMF. Read the TexPoint manual before you delete.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649cef5503460f949bdb5e/html5/thumbnails/5.jpg)
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
![Page 6: MASTERMIND Henning Thomas (joint with Benjamin Doerr, Reto Spöhel and Carola Winzen) TexPoint fonts used in EMF. Read the TexPoint manual before you delete.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649cef5503460f949bdb5e/html5/thumbnails/6.jpg)
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?
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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.
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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
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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
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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
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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)
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Henning Thomas Mastermind ETH Zurich 2012
Upper Bound (Goodrich)
Idea: Answer “0” is good since we can eliminate one
color from every slot!
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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
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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
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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.
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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
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Henning Thomas Mastermind ETH Zurich 2012
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Henning Thomas Mastermind ETH Zurich 2012