MASTERMIND Henning Thomas (joint with Benjamin Doerr, Reto Spöhel and Carola Winzen) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA

Henning ThomasMastermindETH Zurich 2012 Mastermind  Board game invented by Mordechai Meirovitz in 1970

Henning ThomasMastermindETH 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 ThomasMastermindETH 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 ThomasMastermindETH 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 Peg Mastermind

Henning ThomasMastermindETH 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 Peg Mastermind 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 ThomasMastermindETH Zurich 2012 Some Known Results & Our Results  [Knuth ’76], In the original board game (4 slots, 6 colors) 5 queries are optimal.

Henning ThomasMastermindETH 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 strategy Our Result:  Improved bound for k=n by combining Chvátal and Goodrich

Henning ThomasMastermindETH Zurich 2012 Lower Bound  Information theoretic argument:... start query 1 query 2 1 leaf n leaves n 2 leaves query t n t leaves 0n

Henning ThomasMastermindETH 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 0n

Henning ThomasMastermindETH 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 if k=o(n)

Henning ThomasMastermindETH Zurich 2012 Upper Bound (Goodrich)  Idea:  Answer “0” is good since we can eliminate one color from every slot!

Henning ThomasMastermindETH Zurich 2012 Upper Bound (Goodrich)  Implementation: Divide and Conquer 1.Ask monochromatic queries for every color. Obtain X i = # appearances of color i. 2.Ask 3.Calculate L i = # appearnace of color i in left half R i = # appearnace of color i in right half kk... k b2b2 b3b3 bkbk kk... k

Henning ThomasMastermindETH Zurich 2012 Upper Bound (Goodrich)  Implementation: Divide and Conquer 1.Ask monochromatic queries for every color. Obtain X i = # appearances of color i. 2.Ask 3.Calculate L i = # appearnace of color i in left half R i = # appearnace of color i in right half 4.Recurse in the left and right half (without step 1)  Runtime for k=n: kk... k b2b2 b3b3 bkbk kk... k

Henning ThomasMastermindETH 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 ThomasMastermindETH Zurich 2012 Combining Chvátal and Goodrich  Goodrich is good at eliminating colors.  Chvátal is good for k << n. Idea: 2 phases.  Goodrich  Chvátal

Henning ThomasMastermindETH Zurich 2012

Henning ThomasMastermindETH Zurich 2012