Benjamin Raichel

The frechet distance revisited and extended

Given two simplicial complexes in Rd, and start and end vertices in each complex, we show how to compute curves (in each complex) between these vertices, such that the Frechet distance between these curves is minimized. As a polygonal curve is a complex, this generalizes the regular notion of weak Frechet distance between curves. We also generalize the algorithm to handle an input of k simplicial complexes. Using this new algorithm we can solve a slew of new problems, from computing a mean curve for a given collection of curves, to various motion planning problems. Additionally, we show that for the mean curve problem, when the k input curves are c-packed, one can (1+epsilon)-approximate the mean curve in near linear time, for fixed k and epsilon. Additionally, we present an algorithm for computing the strong Frechet distance between two curves, which is simpler than previous algorithms, and avoids using parametric search.

The fréchet distance revisited and extended

Given two simplicial complexes in Rd and start and end vertices in each complex, we show how to compute curves (in each complex) between these vertices, such that the weak Fréchet distance between these curves is minimized. As a polygonal curve is a complex, this generalizes the regular notion of weak Fréchet distance between curves. We also generalize the algorithm to handle an input of k simplicial complexes. Using this new algorithm, we can solve a slew of new problems, from computing a mean curve for a given collection of curves to various motion planning problems. Additionally, we show that for the mean curve problem, when the k input curves are c-packed, one can (1+ϵ-approximate the mean curve in near-linear time, for fixed k and ϵ. Additionally, we present an algorithm for computing the strong Fréchet distance between two curves, which is simpler than previous algorithms and avoids using parametric search.

Net and prune: a linear time algorithm for euclidean distance problems

We provide a general framework for getting linear time constant factor approximations (and in many cases FPTASs) to a copious amount of well known and well studied problems in Computational Geometry, such as k-center clustering and furthest nearest neighbor. The new approach is robust to variations in the input problem, and yet it is simple, elegant and practical. In particular, many of these well studied problems which fit easily into our framework, either previously had no linear time approximation algorithm, or required rather involved algorithms and analysis. A short list of the problems we consider include furthest nearest neighbor, k-center clustering, smallest disk enclosing k points, k-th largest distance, k-th smallest m-nearest neighbor distance, k-th heaviest edge in the MST and other spanning forest type problems, problems involving upward closed set systems, and more. Finally, we show how to extend our framework such that the linear running time bound holds with high probability.

Geometric packing under non-uniform constraints

We study the problem of discrete geometric packing. Here, given weighted regions (say in the plane) and points (with capacities), one has to pick a maximum weight subset of the regions such that no point is covered more than its capacity. We provide a general framework and an algorithm for approximating the optimal solution for packing in hypergraphs arising out of such geometric settings. Using this framework we get a flotilla of results on this problem (and also on its dual, where one wants to pick a maximum weight subset of the points when the regions have capacities). For example, for the case of fat triangles of similar size, we show an (1)-approximation and prove that no PTAS is possible. See [ehr-gpnuc-11] for the full version of the paper.

On the Complexity of Randomly Weighted Voronoi Diagrams

In this paper, we provide an O(n polylog n) bound on the expected complexity of the randomly weighted Voronoi diagram of a set of n sites in the plane, where the sites can be either points, interior-disjoint convex sets, or other more general objects. Here the randomness is on the weight of the sites, not their location. This compares favorably with the worst case complexity of these diagrams, which is quadratic. As a consequence we get an alternative proof to that of Agarwal et al. [AHKS13] of the near linear complexity of the union of randomly expanded disjoint segments or convex sets (with an improved bound on the latter). The technique we develop is elegant and should be applicable to other problems.

From Proximity to Utility: A Voronoi Partition of Pareto Optima

We present an extension of Voronoi diagrams where when considering which site a client is going to use, in addition to the site distances, other site attributes are also considered (for example, prices or weights). A cell in this diagram is then the locus of all clients that consider the same set of sites to be relevant. In particular, the precise site a client might use from this candidate set depends on parameters that might change between usages, and the candidate set lists all of the relevant sites. The resulting diagram is significantly more expressive than Voronoi diagrams, but naturally has the drawback that its complexity, even in the plane, might be quite high. Nevertheless, we show that if the attributes of the sites are drawn from the same distribution (note that the locations are fixed), then the expected complexity of the candidate diagram is near linear. To this end, we derive several new technical results, which are of independent interest. In particular, we provide a high-probability, asymptotically optimal bound on the number of Pareto optima points in a point set uniformly sampled from the d-dimensional hypercube. To do so we revisit the classical backward analysis technique, both simplifying and improving relevant results in order to achieve the high-probability bounds.

Reality Distortion: Exact and Approximate Algorithms for Embedding into the Line

We describe algorithms for the problem of minimum distortion embeddings of finite metric spaces into the real line (or a finite subset of the line). The time complexities of our algorithms are parametrized by the values of the minimum distortion, δ, and the spread, Δ, of the point set we are embedding. We consider the problem of finding the minimum distortion bijection between two finite subsets of IR. This problem was known to have an exact polynomial time solution when δ is below a specific small constant, and hard to approximate within a factor of δ^1-E, when δ is polynomially large. Let D be the largest adjacent pair distance, a value potentially much smaller than Δ. Then we provide a δ^O(δ²^log²^D)nO⁽¹⁾ time exact algorithm for this problem, which in particular yields a quasipolynomial running time for constant δ, and polynomial D. For the more general problem of embedding any finite metric space (X, dX) into a finite subset of the line, Y , we provide a Δ^O(δ²⁾(mn)^O(1) time O(1)-approximation algorithm (where X = n and Y = m), which runs in polynomial time provided δ is a constant and Δ is polynomial. This in turn allows us to get a Δ^O(δ²⁾(n)O⁽¹⁾ time O(1)-approximation algorithm for embedding (X, dX) into the continuous real line.

A treehouse with custom windows: minimum distortion embeddings into bounded treewidth graphs

We describe a (1 + e)-approximation algorithm for finding the minimum distortion embedding of an n-point metric space X into the shortest path metric space of a weighted graph G with m vertices. The running time of our algorithm is mO(1) · nO(ω) · (Δopt Δ)ω · (1/e)λ + 2 · λ · (O (Δopt))2λ parametrized by the values of the minimum distortion, Δopt, the spread, Δ, of the points of X, the treewidth, ω, of G, and the doubling dimension, λ, of G. In particular, our result implies a PTAS provided an X with polynomial spread, and the doubling dimension of G, the treewidth of G, and Δopt, are all constant. For example, if X has a polynomial spread and Δopt is a constant, we obtain PTASs for embedding X into the following spaces: the line, a cycle, a tree of bounded doubling dimension, and a k-outer planar graph of bounded doubling dimension (for a constant k).

Space Exploration via Proximity Search

We investigate what computational tasks can be performed on a point set in R^d, if we are only given black-box access to it via nearest-neighbor search. This is a reasonable assumption if the underlying point set is either provided implicitly, or it is stored in a data structure that can answer such queries. In particular, we show the following: (A) One can compute an approximate bi-criteria k-center clustering of the point set, and more generally compute a greedy permutation of the point set. (B) One can decide if a query point is (approximately) inside the convex-hull of the point set. We also investigate the problem of clustering the given point set, such that meaningful proximity queries can be carried out on the centers of the clusters, instead of the whole point set.

Computing the Fréchet Gap Distance.

Measuring the similarity of two polygonal curves is a fundamental computational task. Among alternatives, the Frechet distance is one of the most well studied similarity measures. Informally, the Frechet distance is described as the minimum leash length required for a man on one of the curves to walk a dog on the other curve continuously from the starting to the ending points. In this paper we study a variant called the Frechet gap distance. In the man and dog analogy, the Frechet gap distance minimizes the difference of the longest and smallest leash lengths used over the entire walk. This measure in some ways better captures our intuitive notions of curve similarity, for example giving distance zero to translated copies of the same curve. The Frechet gap distance was originally introduced by Filtser and Katz (2015) in the context of the discrete Frechet distance. Here we study the continuous version, which presents a number of additional challenges not present in discrete case. In particular, the continuous nature makes bounding and searching over the critical events a rather difficult task. For this problem we give an O(n^5 log(n)) time exact algorithm and a more efficient O(n^2 log(n) + (n^2/epsilon) log(1/epsilon)) time (1+epsilon)-approximation algorithm, where n is the total number of vertices of the input curves. Note that for (small enough) constant epsilon and ignoring logarithmic factors, our approximation has quadratic running time, matching the lower bound, assuming SETH (Bringmann 2014), for approximating the standard Frechet distance for general curves.