Point placement on the line by distance data Original Research Article

Given partial distance information in a set of n points on the real line, we want to figure out the positions of these points, subject to translation and reflection. This type of problem is motivated by DNA mapping. We show the following results: If we can ask arbitrary distance queries for pairs of points then 2n−3 adaptive queries will be optimal. (In contrast, all (n2) queries must be asked in the nonadaptive case.) Surprisingly, if the learner knows in advance that the n points have distinct locations, 85n nonadaptive queries, or alternatively 32n queries in 2 rounds will be sufficient. This might be further improved, as we only have the lower bounds 43n and n, respectively. The subject is related to some rigidity concept for graphs. In another version of the problem, the graph of distance measures is already given, that means, we cannot choose our distance queries at our own discretion. Here, we give a simple efficient algorithm which produces a representation of all linear layouts if the given graph is chordal.