Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Query Complexity of Mastermind Variants   We study variants of Mastermind, a popular board game in which the objective
is sequence reconstruction. In this two-player game, the so-called
\textit{codemaker} constructs a hidden sequence $H = (h_1, h_2, \ldots, h_n)$
of colors selected from an alphabet $\mathcal{A} = \{1,2,\ldots, k\}$
(\textit{i.e.,} $h_i\in\mathcal{A}$ for all $i\in\{1,2,\ldots, n\}$). The game
then proceeds in turns, each of which consists of two parts: in turn $t$, the
second player (the \textit{codebreaker}) first submits a query sequence $Q_t =
(q_1, q_2, \ldots, q_n)$ with $q_i\in \mathcal{A}$ for all $i$, and second
receives feedback $\Delta(Q_t, H)$, where $\Delta$ is some agreed-upon function
of distance between two sequences with $n$ components. The game terminates when
$Q_t = H$, and the codebreaker seeks to end the game in as few turns as
possible. Throughout we let $f(n,k)$ denote the smallest integer such that the
codebreaker can determine any $H$ in $f(n,k)$ turns. We prove three main
results: First, when $H$ is known to be a permutation of $\{1,2,\ldots, n\}$,
we prove that $f(n, n)\ge n - \log\log n$ for all sufficiently large $n$.
Second, we show that Knuth's Minimax algorithm identifies any $H$ in at most
$nk$ queries. Third, when feedback is not received until all queries have been
submitted, we show that $f(n,k)=\Omega(n\log k)$.

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The categories of this paper are: physics
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The categories of this paper are: atom-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: q-alg
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The categories of this paper are: patt-sol
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The categories of this paper are: funct-an
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The categories of this paper are: nlin
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The categories of this paper are: hep-ex
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The categories of this paper are: solv-int
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-th
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The categories of this paper are: quant-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: supr-con
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The categories of this paper are: nucl-th