Tight Bounds for Blind Search on the Integers

Abstract

We analyze a simple random process in which a token is moved in the interval $A=\{0,...,n\$: Fix a probability distribution$\mu$over$\{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$a\geq d$, 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$\min_\mu\{E_\mu(T)\=\Theta((\log n)^2)$. 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