Metric Spaces with Expensive Distances

Abstract

In algorithms for finite metric spaces, it is common to assume that the distance between two points can be computed in constant time, and complexity bounds are expressed only in terms of the number of points of the metric space. We introduce a different model, where we assume that the computation of a single distance is an expensive operation and consequently, the goal is to minimize the number of such distance queries. This model is motivated by metric spaces that appear in the context of topological data analysis. We consider two standard operations on metric spaces, namely the construction of a [Formula: see text]-spanner and the computation of an approximate nearest neighbor for a given query point. In both cases, we partially explore the metric space through distance queries and infer lower and upper bounds for yet unexplored distances through triangle inequality. For spanners, we evaluate several exploration strategies through extensive experimental evaluation. For approximate nearest neighbors, we prove that our strategy returns an approximate nearest neighbor after a logarithmic number of distance queries.

Approximating Expensive Distance Metrics

“…We implemented an 1 + ε approximation algorithm [5] for use with expensive distance metrics. Our implementation can be found on our GitHub repo.…”

“…We want a way to approximate these distances so we don't have to do the long distance computation so many times. In this paper, we implement an existing approximation algorithm [5] in order to test the runtime improvements compared to the loss of accuracy from the approximation. Our project is based on work by Kerber and Nigmetov [5], a paper that presents an algorithm that computes an approximate distance metric on a fnite metrics space.We implemented the approximation algorithm and tested it on two "expensive" metric spaces: Hausdorf distance [4] on point sets and continuous Frechet distance [8] on paths in ℝ.…”

Light Euclidean Spanners with Steiner Points

“…In particular, it is known that for n-point sets drawn uniformly at random from the unit square, the expected spread is Θ(n), and the expected spread in the unit d-dimensional hypercube is n 2/d for any d = O(1). Researchers have studied random distributions of point sets, in part to explain the success of solving various geometric optimization problems in practice [9,53,30], and there are many results on spanners for random point sets [24,7,37,10,38,14,61,69,15,21,8,56]. Of course, the family of bounded spread point sets is much wider than that of random point sets.…”

Pattern Matching in Doubling Spaces

“…In this paper, we will consider spaces of constant doubling dimension, so we assume that δ = O(1). We will need the following packing lemma: Lemma 1 ( [22]). If a metric space (S, d S ) has doubling dimension δ, then |S| (4Φ(S)) δ .…”