We investigate techniques for analysis and retrieval of object trajectories in two or three dimensional space. Such data usually contain a large amount of noise, that has made previously used metrics fail. Therefore, we formalize non-metric similarity functions based on the longest common subsequence (LCSS), which are very robust to noise and furthermore provide an intuitive notion of similarity between trajectories by giving more weight to similar portions of the sequences. Stretching of sequences in time is allowed, as well as global translation of the sequences in space. Efficient approximate algorithms that compute these similarity measures are also provided. We compare these new methods to the widely used Euclidean and time warping distance functions (for real and synthetic data) and show the superiority of our approach, especially in the strong presence of noise. We prove a weaker version of the triangle inequality and employ it in an indexing structure to answer nearest neighbor queries. Finally, we present experimental results that validate the accuracy and efficiency of our approach.

/pdf/discovering-similar-multidimensional-trajectories-3ofj6nb9yv.pdf

Discovering similar multidimensional trajectories

Convex bodies: the brunn–minkowski theory

Computational geometry

At SIGMOD 2015, an article was presented with the title “DBSCAN Revisited: Mis-Claim, Un-Fixability, and Approximation” that won the conference’s best paper award. In this technical correspondence, we want to point out some inaccuracies in the way DBSCAN was represented, and why the criticism should have been directed at the assumption about the performance of spatial index structures such as R-trees and not at an algorithm that can use such indexes. We will also discuss the relationship of DBSCAN performance and the indexability of the dataset, and discuss some heuristics for choosing appropriate DBSCAN parameters. Some indicators of bad parameters will be proposed to help guide future users of this algorithm in choosing parameters such as to obtain both meaningful results and good performance. In new experiments, we show that the new SIGMOD 2015 methods do not appear to offer practical benefits if the DBSCAN parameters are well chosen and thus they are primarily of theoretical interest. In conclusion, the original DBSCAN algorithm with effective indexes and reasonably chosen parameter values performs competitively compared to the method proposed by Gan and Tao.

DBSCAN Revisited, Revisited: Why and How You Should (Still) Use DBSCAN

We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c > 1 and given any n nodes in ℛ2, a randomized version of the scheme finds a (1 + 1/c)-approximation to the optimum traveling salesman tour in O(n(log n)O(c)) time. When the nodes are in ℛd, the running time increases to O(n(log n)(O(√c))d-1). For every fixed c, d the running time is n · poly(logn), that is nearly linear in n. The algorithmm can be derandomized, but this increases the running time by a factor O(nd). The previous best approximation algorithm for the problem (due to Christofides) achieves a 3/2-aproximation in polynomial time.We also give similar approximation schemes for some other NP-hard Euclidean problems: Minimum Steiner Tree, k-TSP, and k-MST. (The running times of the algorithm for k-TSP and k-MST involve an additional multiplicative factor k.) The previous best approximation algorithms for all these problems achieved a constant-factor approximation. We also give efficient approximation schemes for Euclidean Min-Cost Matching, a problem that can be solved exactly in polynomial time.All our algorithms also work, with almost no modification, when distance is measured using any geometric norm (such as lp for p ≥ 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.

/pdf/polynomial-time-approximation-schemes-for-euclidean-k5kn9e00dl.pdf

Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems

Many questions about homotopy are provably hard or even unsolvable in general. However, in specific settings, it is possible to efficiently test homotopy-equivalence or compute shortest cycles with prescribed homotopy. We focus on computing such "interesting" topological features in three settings. The first two results are about cycles on surfaces; the third is about classes of homotopies in R2 minus a set of obstacles; and the final result is about paths and cycles in Rips complexes. 
First, we examine two problems in the combinatorial surface model. Combinatorial surfaces combine properties of graphs and manifolds, making a rich set of techniques available for analysis and algorithm design. We give algorithms to find the shortest noncontractible and nonseparating cycles in a combinatorial surface in O(g3n log n) time. Our main tool is a data structure that kinetically maintains the shortest path tree as the root of the tree moves around the vertices of a single face. The total running time is O(g2n log n). By maintaining the data structure persistently, we can answer shortest path queries in O(log n) time. 
Next we consider finding the shortest splitting cycle in a combinatorial surface, or simple cycle which is both separating and noncontractible; such cycles divide the topology of the surface as well as the underlying graph. We prove that finding the shortest splitting cycle is NP-Hard. We then give an algorithm that runs gO(g) n log n time, which is polynomial if the surface is fixed. 
We then examine a very different setting, namely similarity between curves in some underlying metric space. If we imagine a homotopy between the curves as a way to morph one curve into the other, we can optimize the morphing so that the maximum distance any point must travel is minimized This is a generalization of the more well known Frechet distance, with the additional requirement that the leash to move continuously in the ambient space. We call this distance the homotopic Frechet distance. We give a polynomial time algorithm to compute the homotopic Frechet distance between two curves in the plane minus a set of polygonal obstacles. We also extend our characterization of optimal morphings to surfaces of nonpositive curvature. 
Finally, we examine a more fundamental homotopy problem in a different setting. A Rips complex is a simplicial complex defined by a set of points from some metric space where every pair of points within distance 1 is connected by an edge, and every (k + 1)-clique in that graph forms a k-simplex. We prove that the projection map which takes each k-simplex in the Rips complex to the convex hull of the original points in the plane induces an isomorphism between the fundamental groups of both spaces. Since the union of these convex hulls is a polygonal region in the plane, possibly with holes, our result implies that the fundamental group of a planar Rips complex is a free group, allowing us to design efficient algorithms to answer homotopy questions in planar Rips complexes.

Computing interesting topological features

We present an algorithm to construct meshes suitable for space-time discontinuous Galerkin finite-element methods. Our method generalizes and improves the `Tent Pitcher' algorithm of Ungor and Sheffer. Given an arbitrary simplicially meshed domain M of any dimension and a time interval [0,T], our algorithm builds a simplicial mesh of the space-time domain Mx[0,T], in constant time per element. Our algorithm avoids the limitations of previous methods by carefully adapting the durations of space-time elements to the local quality and feature size of the underlying space mesh.

Building Space-Time Meshes over Arbitrary Spatial Domains

We propose a hybrid image-space/object-space solution to the classical hidden surface removal problem: Given n disjoint triangles in Real^3 and p sample points (``pixels'') in the xy-plane, determine the first triangle directly behind each pixel. Our algorithm constructs the sampled visibility map of the triangles with respect to the pixels, which is the subset of the trapezoids in a trapezoidal decomposition of the analytic visibility map that contain at least one pixel. The sampled visibility map adapts to local changes in image complexity, and its complexity is bounded both by the number of pixels and by the complexity of the analytic visibility map. Our algorithm runs in time O(n^{1+e} + n^{2/3+e}t^{2/3} + p), where t is the output size and e is any positive constant. This is nearly optimal in the worst case and compares favorably with the best output-sensitive algorithms for both ray casting and analytic hidden surface removal. In the special case where the pixels form a regular grid, a sweepline variant of our algorithm runs in time O(n^{1+e} + n^{2/3+e}t^{2/3} + t log p), which is usually sublinear in the number of pixels.

Finite-resolution hidden surface removal

We consider three classes of geodesic embeddings of graphs on Euclidean flat tori: 
- A torus graph G is equilibrium if it is possible to place positive weights on the edges, such that the weighted edge vectors incident to each vertex of G sum to zero 
- A torus graph G is reciprocal if there is a geodesic embedding of the dual graph G^* on the same flat torus, where each edge of G is orthogonal to the corresponding dual edge in G^* 
- A torus graph G is coherent if it is possible to assign weights to the vertices, so that G is the (intrinsic) weighted Delaunay graph of its vertices The classical Maxwell-Cremona correspondence and the well-known correspondence between convex hulls and weighted Delaunay triangulations imply that the analogous concepts for plane graphs (with convex outer faces) are equivalent Indeed, all three conditions are equivalent to G being the projection of the 1-skeleton of the lower convex hull of points in ℝ³ However, this three-way equivalence does not extend directly to geodesic graphs on flat tori On any flat torus, reciprocal and coherent graphs are equivalent, and every reciprocal graph is equilibrium, but not every equilibrium graph is reciprocal We establish a weaker correspondence: Every equilibrium graph on any flat torus is affinely equivalent to a reciprocal/coherent graph on some flat torus

https://drops.dagstuhl.de/opus/volltexte/2020/12198/pdf/LIPIcs-SoCG-2020-40.pdf/

A Toroidal Maxwell-Cremona-Delaunay Correspondence

We present the first algorithm to morph graphs on the torus. Given two isotopic essentially 3-connected embeddings of the same graph on the Euclidean flat torus, where the edges in both drawings are geodesics, our algorithm computes a continuous deformation from one drawing to the other, such that all edges are geodesics at all times. Previously even the existence of such a morph was not known. Our algorithm runs in $O(n^{1+\omega/2})$ time, where $\omega$ is the matrix multiplication exponent, and the computed morph consists of $O(n)$ parallel linear morphing steps. Existing techniques for morphing planar straight-line graphs do not immediately generalize to graphs on the torus; in particular, Cairns' original 1944 proof and its more recent improvements rely on the fact that every planar graph contains a vertex of degree at most 5. Our proof relies on a subtle geometric analysis of 6-regular triangulations of the torus. We also make heavy use of a natural extension of Tutte's spring embedding theorem to torus graphs.

Jeff Erickson

Papers

Computing interesting topological features

Building Space-Time Meshes over Arbitrary Spatial Domains

Finite-resolution hidden surface removal

A Toroidal Maxwell-Cremona-Delaunay Correspondence

How to Morph Graphs on the Torus