Generalized jewels and the point placement problem

The point placement problem is to determine the positions of a linear set of points, P = {p1, p2, p3, . . . , pn}, uniquely, up to translation and reflection, from the fewest possible distance queries between pairs of points. Each distance query corresponds to an edge in a graph, called point placement graph (ppg), whose vertex set is P . The uniqueness requirement of the placement translates to line rigidity of the ppg. In this paper we show how to construct in 2 rounds a line rigid ppg of size 10n/7+O(1) from small rigid components called 5:5 jewels, which are an extension of the 4:4 jewel of [2]. Though this result is slightly worse than the 4n/3 + O( √ n) upper bound, reported in [1], this is more than offset by the potential for generalization of this construction.