Tight Bounds for Blind Search on the Integers

Abstract

We analyze a simple random process in which a token is moved in the interval $A=\{0,...,n\$:Fixaprobabilitydistribution$μ$over$\{1,...,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 $a\ge d$, 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 ${\min }_{\mu }\{E_{\mu }(T)\bar{\Theta }((\log n{)}^2)$. 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.