A chip search problem on binary numbers

Abstract

Suppose that we have an unknown point d in the interval (0, 1) and an unbounded reservoir of chips (pebbles) on both points 0 and 1. In every step, we can either move two pebbles from points x and y to (x + y)/2, or we can ask whether d < x or d > x, but only if there is currently a pebble at x. Our aim is to determine an interval of length 2⁻ⁿ including d, and we are interested in the exact number of necessary moves in the worst case, especially for the first values of n.

First we analyze a natural GREEDY strategy solving this problem in roughly n ²/6 moves which improves our previous n ²/4 result in [5]. On the other hand, n ²/12 is a lower bound. Our analysis allows to compute the exact worst-case number of moves g(n) that GREEDY takes for the first values of n, although a nice general expression for this is missing.

GREEDY sends pebbles only to points being binary approximations of the target d. Moreover, our analysis will show that GREEDY is optimal among all strategies sharing this property. Hence any such strategy needs γ(n ²) moves in the worst case. It might seem that sending pebbles to other points brings no advantages. So it is surprising that, without the mentioned restriction, we can achieve an asymptotic result of O(n ^{1+2/√log n}) moves, by an acceleration technique. Hence GREEDY is not optimal in general, nevertheless it seems to be the best choice for small (i.e. practical?) n, since it is simple, and we have no strategy beating g(n) if n < 20. An open problem is to determine the maximum n for which g(n) moves are optimal.

In a final section we discuss a way to save tests if d is very small, and we briefly consider a variant of the problem where a test at x destroys a pebble lying there.

To our best knowledge, the present problem has not been studied before