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, . . . , n}: Fix a probability distribution 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 μ. If the token is in position a ≥ 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 μ. More precisely, we show that minμ{Eμ(T )} = Θ ` (log n) ́ . 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.