By Zhang G.
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Additional info for A 17/10-approximation algorithm for k -bounded space on-line variable-sized bin packing
Equivalently, no general rule exists to determine the length of the shortest searching strategy for arbitrary cardinality of the search space and number of lies. Nonetheless, for any fixed number of lies, e, we can provide asymptotic conditions on the states of knowledge which allow a perfect winning strategy. 5. x0 ; : : : ; xe / Ä 2n and xe Ke ne , then the state is n-winning. Proof (Sketch). x0 ; x1 ; : : : ; xe /, with ch. / D n. For i D 0; 1; : : : ; e 1, let ai be chosen as b x2i c or d x2i e on an alternate basis.
Intuitively, e yields a lower bound for the length of the shortest winning strategy for , considering both Volume and Translation Bound. In addition, e i stands for an upper bound on the length of the shortest winning strategy for i . / provided that there exists a winning strategy for with e questions. This is proved by the following corollary. x0 ; x1 ; : : : ; xe / be a state. Let . 2. 0; : : : ; 0; x0 ; : : : ; xe i / with e i questions. /. i . /D The following lemma states that in most cases the optimal winning strategy for (with respect to the lower bound e ) implicitly behaves like an optimal winning strategy for the states i .
The source’s answer (one bit) is then sent to the receiver via the noisy channel. The receiver gets a (possibly distorted) answer, and then adaptively asks the next question Q2 . At each step t adaptively asks the question Qt Â M knowing the answers to the preceding questions. Since the receiver’s questions range over all possible subsets of a space of cardinality M , one might be led to think that any such question requires M bits to be sent over the noiseless channel. It turns out, however, that only one feedback bit suffices for each bit transmitted by the source.