| Tight Bounds for Blind Search on the Integers (2008) | |||||||||||||||
Abstract | |||||||||||||||
| We analyze a simple random process in which a token is moved in the interval $A=\{0,\dots,n\$: Fix a probability distribution $\mu$ over $\{1,\dots,n\$. Initially, the token is placed in a random position in $A$. In round $t$, a random value $d$ is chosen according to $\mu$. If the token is in position $a\geq d$, then it is moved to position $a-d$. Otherwise it stays put. Let $T$ be the number of rounds until the token reaches position 0. We show tight bounds for the expectation of $T$ for the optimal distribution $\mu$. More precisely, we show that $\min_\mu\{E_\mu(T)\=\Theta\left((\log n)^2\right)$. For the proof, a novel potential function argument is introduced. The research is motivated by the problem of approximating the minimum of a continuous function over $[0,1]$ with a ``blind'' optimization strategy. | |||||||||||||||
Details der Publikation | |||||||||||||||
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