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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.,} $h_iinmathcal{A}$ for all $iin{1,2,ldots, n}$). 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 $q_iin mathcal{A}$ 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 $f(n, n)ge n - loglog n$ for all sufficiently large $n$. Second, we show that Knuths 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 $f(n,k)=Omega(nlog k)$.
Suppose that the vertices of a graph $G$ are colored with two colors in an unknown way. The color that occurs on more than half of the vertices is called the majority color (if it exists), and any vertex of this color is called a majority vertex. We
The {em Total Influence} ({em Average Sensitivity) of a discrete function is one of its fundamental measures. We study the problem of approximating the total influence of a monotone Boolean function ifnumplusminus=1 $f: {pm1}^n longrightarrow {pm1}$,
We show that finding a graph realization with the minimum Randic index for a given degree sequence is solvable in polynomial time by formulating the problem as a minimum weight perfect b-matching problem. However, the realization found via this reduc
Given a family of graphs $mathcal{F}$, we prove that the normalized edit distance of any given graph $Gamma$ to being induced $mathcal{F}$-free is estimable with a query complexity that depends only on the bounds of the Frieze--Kannan Regularity Lemma and on a Removal Lemma for $mathcal{F}$.
In this work, we consider a variant of the classical Longest Common Subsequence problem called Doubly-Constrained Longest Common Subsequence (DC-LCS). Given two strings s1 and s2 over an alphabet A, a set C_s of strings, and a function Co from A to N