Which point sets realize a given distance multiset? Interesting cases include the "turnpike problem" where the points lie on a line, the "beltway problem" where the points lie on a loop, and multidimensional versions. We are interested both in the algorithmic problem of determining such point sets for a given collection of distances and the combinatorial problem of finding bounds on the maximum number of different solutions. These problems have applications in genetics and crystallography.

In a previous paper we showed that, for any n ≥ m + 2, most sets of n points in R m are determined (up to rotations, reflections, translations and relabeling of the points) by the distribution of their pairwise distances. But there are some exceptional point configurations which are not reconstructible from the distribution of distances in the above sense. In this paper, we concentrate on the planar case m = 2 and present a reconstructibility test with running time O(n 11). The cases of orientation preserving rigid motions (rotations and translations) and scalings are also discussed.

In computational molecular biology, the aim of restriction mapping is to locate the restriction sites of a given enzyme on a DNA molecule. Double digest and partial digest are two well-studied techniques for restriction mapping. While double digest is NP-complete, there is no known polynomial algorithm for partial digest. Another disadvantage of the above techniques is that there can be multiple solutions for reconstruction. In this

The PARTIAL DIGEST problem well-known for its applications in computational biology and for the intriguingly open status of its computational complexity asks for the coordinates of n points on a line such that the pairwise distances of the points form a given multi-set of () distances. In an effort to model real-life data, we study the computational complexity of a minimization version of PARTIAL DIGEST, in which only a subset of all pairwise distances is given and the rest are lacking due to experimental errors. We show that this variation is NP-hard to solve exactly, thus making the existence of polynomial-time algorithms for this problem extremely unlikely. Our result answers an open question posed by Pevzner (2000). We then study a maximiza- tion version of PARTIAL DIGEST where a superset of all pairwise distances is given, with some additional distances due to inaccurate measurements. We show that this maximization version is NP-hard to approximate to within a factor of [D[ -c for any e 0, where [D[ is the number of input distances, which implies that polynomial-time algorithms cannot even guarantee to find a solution for the problem that comes close to the optimum. Our inapproximabilky result is tight up to low-order terms as we give a trivial approximation algorithm that achieves a matching approximation ratio. Our optimization variations model two different error types that occur in real-life data.

Pattern matching in point sets is a well studied problem with numerous applications. We assume that the point sets may contain outliers (missing or spurious points) and are subject to an unknown translation. We define the distance between any two point sets to be the minimum size of their symmetric difference over all translations of one set relative to the other. We consider the problem in the context of similarity search. We assume that a large database of point sets is to be preprocessed so that given any query point set, the closest matches in the database can be computed efficiently. Our approach is based on showing that there is a randomized algorithm that computes a translation-invariant embedding of any point set of size at most n into the L1 metric, so that with high probability, distances are subject to a distortion that is O(log² n).

Abstract. The turnpike problem is one of the few “natural ” problems that are neither known to be NP-complete nor solvable by efficient algorithms. We seek to study this problem in a more general setting. We consider the generalized problem which tries to resolve set A = {a1,a2, ·· ·,an} from pairwise function values {f(ai,aj)|1 ≤ i, j ≤ n} for a given bivariate function f. WecallthisproblemtheNumber Reconstruction problem. Our results include efficient algorithms when f is monotone and non-trivial bounds on the number of solutions when f is the sum. We also generalize previous backtracking and algebraic algorithms for the turnpike problem such that they work for the family of anti-monotone functions and linear-decomposable functions. Finally, we propose an efficient algorithm for the string reconstruction problem, which is related to an approach to protein reconstruction. 1