Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Mittagsseminar Talk Information

Date and Time: Tuesday, March 19, 2013, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Benjamin Doerr (MPI Saarbruecken)

Mastermind With Many Colors

We analyze the general version of the classic guessing game Mastermind with $n$~positions and $k$~colors. Since the case $k \le n^{1-\eps}$, $\eps>0$ constant, is well understood, we concentrate on larger numbers of colors. For the most prominent case $k = n$, our results imply that Codebreaker can find the secret code with $O(n \log \log n)$ guesses. This bound is valid also when only black answer-pegs are used. It improves the $O(n \log n)$ bound first proven by Chv\'atal (Combinatorica 3 (1983), 325--329). We also show that if both black and white answer-pegs are used, then the $O(n \log\log n)$ bound holds for up to $n^2 \log\log n$ colors. These bounds are almost tight as the known lower bound of $\Omega(n)$ shows. Unlike for $k \le n^{1-\eps}$, simply guessing at random until the secret code is determined is not sufficient. In fact, we show that an optimal non-adaptive strategy (deterministic or randomized) needs $\Theta(n \log n)$ guesses.

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