Playing twenty questions with a procrastinator
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Searching games with errors---fifty years of coping with liars
Theoretical Computer Science
Approximate Location of Relevant Variables under the Crossover Distribution
SAGA '01 Proceedings of the International Symposium on Stochastic Algorithms: Foundations and Applications
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A function $f : \Re \rightarrow \Re$ is {\em $k$-modal} if its $k$th derivative has a unique zero. We study the problem of finding the smallest possible interval containing the unique zero of the $k$th derivative of such a function, assuming that the function is evaluated only at integer points. We present optimal algorithms for the case when $k$ is even and for $k=3$ and near-optimal algorithms when $k \geq 5$ and odd. A novel generalization of Kraft's inequality is used to prove lower bounds on the number of function evaluations required. We show how our algorithms lead to an efficient divide-and-conquer algorithm to determine all turning points or zeros of a $k$-modal function. Unbounded $k$-modal search is introduced and some problems in extending previous approaches for unbounded searching to the $k$-modal case are discussed.