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
Mastermind • Board game invented by Mordechai Meirovitz in 1970
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
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
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. This talk: Black Peg Mastermind secret query answer
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. This talk: Black Peg Mastermind Whatistheminimumnumber t = t(k,n) ofqueries such thatthereexists a deterministicstrategytoidentifyeverysecretcolorcombination? secret query answer
Some Known Results & Our Results • [Knuth ’76], In the original board game (4 slots, 6 colors) 5 queries are optimal.
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
Lower Bound • Information theoretic argument: start 1 leaf 0 n ... query 1 n leaves query 2 n2 leaves query t nt leaves
Upper Bound (Chvátal) 0 n • 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
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)
Upper Bound (Goodrich) • Idea: • Answer “0”is good since we can eliminate one color from every slot!
Upper Bound (Goodrich) • Implementation:DivideandConquer • Askmonochromaticqueriesforeverycolor.ObtainXi = # appearancesofcolor i. • Ask • CalculateLi = # appearnaceofcolor i in lefthalfRi= # appearnaceofcolori in righthalf 11 ... 1 22 ... 2 kk ... k b2 11 ... 1 22 ... 2 b3 11 ... 1 33 ... 3 bk 11 ... 1 kk ... k
Upper Bound (Goodrich) • Implementation:DivideandConquer • Askmonochromaticqueriesforeverycolor.ObtainXi = # appearancesofcolor i. • Ask • CalculateLi = # appearnaceofcolor i in lefthalfRi= # appearnaceofcolori in righthalf • Recursein theleftandright half (withoutstep 1) • Runtimefork=n: 11 ... 1 22 ... 2 kk ... k b2 11 ... 1 22 ... 2 b3 11 ... 1 33 ... 3 bk 11 ... 1 kk ... k
Comparison Lower Bound vs Goodrich • For k=n Goodrich yields • Problem: • When Goodrich runs for a while, the blockseventually become too small that we cannot learn as many bits as we would like to.
Combining Chvátal and Goodrich • Goodrich is good at eliminating colors. • Chvátal is good for k << n. Idea: 2 phases. • Goodrich • Chvátal
