Randomized vs. deterministic distance query strategies for point location on the line

Abstract

Suppose that n points are located at n mutually distinct but unknown positions on the line, and we can measure their pairwise distances. How many measurements are needed to determine their relative positions uniquely? The problem is motivated by DNA mapping techniques based on pairwise distance measures. It is also interesting by itself for its own and surprisingly deep. Continuing our earlier work on this problem, we give a simple randomized two-round strategy that needs, with high probability, only ( 1 + o ( 1 ) ) n measurements. We show that deterministic strategies cannot manage the task in two rounds with ( 1 + o ( 1 ) ) n measurements in the worst case. We improve an earlier deterministic bound to roughly 4 n / 3 measurements.