The Point Placement Problem on a Line – Improved Bounds for Pairwise Distance Queries

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Abstract

In this paper, we study the adaptive version of the point placement problem on a line, which is motivated by a DNA mapping problem. To identify the relative positions of n distinct points on a straight line, we are allowed to ask queries of pairwise distances of the points in rounds. The problem is to find the number of queries required to determine a unique solution for the positions of the points up to translation and reflection. We improved the bounds for several cases. We show that \(4n/3 + O(\sqrt{n})\) queries are sufficient for the case of two rounds while the best known result was 3n/2 queries. For unlimited number of rounds, the best result was 4n/3 queries. We obtain a much better result of using \(5n/4 + O(\sqrt{n})\) queries with three rounds only. We also improved the lower bound of 30n/29 to 17n/16 for the case of two rounds.