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,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 $ageq d$, 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)=Thetaleft((log n)^2 ight). 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