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.

The paper [10] of J. Solymosi and Cs. Toth implicitly raised the following arithmetic problem. Consider n pairwise disjoint s element sets and form all s 2 n sums of pairs of elements of the same set. What is the minimum number of distinct sums one can get this way? This paper proves a lower bound on the number of distinct sums. As an application we improve the Solymosi-Toth bound on an old Erd}os problem: the number of distinct distances n points determine in the plane. Our bound also nds applications in other related results in discrete geometry. Our bounds are proven through an involved calculation of entropies of several random variables.

Improving an old result of Clarkson et al., we show that the number of distinct distances determined by a set P of n points in three-dimensional space is Ω(n77/141−ε) = Ω(n0.546), for any ε> 0. Moreover, there always exists a point p ∈ P from which there are at least so many distinct distances to the remaining elements of P. The same result holds for points on the three-dimensional sphere. As a consequence, we obtain analogous results in higher dimensions. 1

This paper considers a variety of geometric pattern recognition problems on input sets of size n using a coarse grained multicomputer model consisting of p processors with 0(n p) local memory each (i.e., 0(n p) memory cells of 3(log n) bits apiece), where the processors are connected to an arbitrary interconnection network. It introduces efficient scalable parallel algorithms for a number of geometric problems including the rectangle finding problem, the maximal equally spaced collinear points problem, and the point set pattern matching problem. All of the algorithms presented are scalable in that they are applicable and efficient over a very wide range of ratios of problem size to number of processors. In addition to the practicality imparted by scalability, these algorithms are easy to implement in that all required communications can be achieved by a small number of calls to standard global routing operations.

It is proved that, for any " > 0 and n > n 0 ("), every set of n points in the plane has at most n 5e 1 + triples that induce isosceles triangles. (Here e denotes the base of the natural logarithm, so the exponent is roughly 2:136.) This easily implies the best currently known lower bound, n 5e 1 , for the smallest number of distinct distances determined by n points in the plane, due to Solymosi{C. Toth and Tardos.

We prove that a well-distributed subset of R 2 can have a distance set ∆ with #( ∆ ∩ [0, N]) ≤ CN 3/2−ɛ only if the distance is induced by a polygon K. Furthermore, if the above estimate holds with ɛ = 1/2, then K can have only finitely many sides.

The study of geometrically defined graphs, and of the reverse question, the construction of geometric representations of graphs, leads to unexpected connections between geometry and graph theory. We survey the surprisingly large variety of graph properties related to geometric representations, construction methods for geometric representations, and their applications in proofs and algorithms.

Point pattern matching is an important problem in computational geometry, with applications in areas like computer vision, object recognition, molecular modelling, and image registration. Traditionally, it has been studied in an exact formulation, where the input point sets are given with arbitrary precision. This leads to algorithms that typically have running times of the order of high degree polynomials, and require robust calculations of intersection points of high degree surfaces. We study approximate point pattern matching, with the goal of developing algorithms that are more efficient and more practical than exact algorithms. Our work is motivated by the observation that in practice, data sets that form instances of pattern matching problems are noisy, and so approximate formulations are more appropriate. We present new and efficient algorithms for approximate point pattern matching in two and three dimensions, based on approximate combinatorial distance bounds on sets of points, and via the use of methods from combinatorial pattern matching. We also present an average case analysis and a detailed empirical study of our methods.

Recent papers concerned with the Point Set Pattern Matching Problem (PSPM) ( nding all congruent copies of a pattern in a sample set) in Euclidean 3-space, R³, have given algorithms with running times that have decreased as known output bounds for the problem have decreased. In this paper, a recent result of [2] is used to show that the volume of the output is O(kn

This paper considers a variety of geometric problems on input sets of size n using a coarse grained multicomputer model consisting of p processors with ( n n p) local memory each (i.e., ( p) memory cells of (log n) bits apiece), where the processors are connected to an arbitrary interconnection network. It introduces efficient scaleable parallel algorithms for a number of geometric problems including the rectangle finding problem, a variety of lower envelope problems, the maximal equally-spaced collinear points problem, and the point set pattern matching problem. All of the algorithms presented are scaleable in that they are applicable and efficient over a very wide range of ratios of problem size to number of processors. In addition to the practicality imparted by scaleability, these algorithms are easy to implement in that all required communications can be achieved by a small number of calls to standard global routing operations.