Tight Bounds for Blind Search on the Integers
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.
241-252
Regular Paper
Martin
Dietzfelbinger
Martin Dietzfelbinger
Jonathan E.
Rowe
Jonathan E. Rowe
Ingo
Wegener
Ingo Wegener
Philipp
Woelfel
Philipp Woelfel
10.4230/LIPIcs.STACS.2008.1348
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