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.

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

The straight skeleton of a polygon is a variant of the medial axis introduced by Aichholzer et al., defined by a shrinking process in which each edge of the polygon moves inward at a fixed rate. We construct the straight skeleton of an n -gon with r reflex vertices in time O(n 1+e + n 8/11+e r 9/11+e ) , for any fixed e >0 , improving the previous best upper bound of O(nr log n) . Our algorithm simulates the sequence of collisions between edges and vertices during the shrinking process, using a technique of Eppstein for maintaining extrema of binary functions to reduce the problem of finding successive interactions to two dynamic range query problems: (1) maintain a changing set of triangles in R3 and answer queries asking which triangle is first hit by a query ray, and (2) maintain a changing set of rays in R3 and answer queries asking for the lowest intersection of any ray with a query triangle. We also exploit a novel characterization of the straight skeleton as a lower envelope of triangles in R3 . The same time bounds apply to constructing non-self-intersecting offset curves with mitered or beveled corners, and similar methods extend to other problems of simulating collisions and other pairwise interactions among sets of moving objects.

/pdf/raising-roofs-crashing-cycles-and-playing-pool-applications-9zlhoo4ix0.pdf

Raising roofs, crashing cycles, and playing pool: applications of a data structure for finding pairwise interactions

We consider the problem of cutting a subset of the edges of a polyhedral manifold surface, possibly with boundary, to obtain a single topological disk, minimizing either the total number of cut edges or their total length. We show that this problem is NP-hard in general, even for manifolds without boundary and for punctured spheres. We also describe an algorithm with running time n O(g+k), where n is the combinatorial complexity, g is the genus, and k is the number of boundary components of the input surface. Finally, we describe a greedy algorithm that outputs a O(log2 g)-approximation of the minimum cut graph in O(g 2 n log n) time.

/pdf/optimally-cutting-a-surface-into-a-disk-2pc3o2bkxj.pdf

Optimally Cutting a Surface into a Disk

We propose three indexing schemes for storing a set S of N points in the plane, each moving along a linear trajectory, so that a query of the following form can be answered quickly: Given a rectangle R and a real value tq, report all K points of S that lie inside R at time tq. We first present an indexing structure that, for any given constant e > 0, uses O(N/B) disk blocks, where B is the block size, and answers a query in O((N/B)1/2+e + K/B) I/Os. It can also report all the points of S that lie inside R during a given time interval. A point can be inserted or deleted, or the trajectory of a point can be changed, in O(log2B N) I/Os. Next, we present a general approach that improves the query time if the queries arrive in chronological order, by allowing the index to evolve over time. We obtain a trade off between the query time and the number of times the index needs to be updated as the points move. We also describe an indexing scheme in which the number of I/Os required to answer a query depends monotonically on the difference between tq and the current time. Finally, we develop an efficient indexing scheme to answer approximate nearest-neighbor queries among moving points.

/pdf/indexing-moving-points-extended-abstract-4ulycgzk5h.pdf

Indexing moving points (extended abstract)

We introduce a new method for finding several types of optimalk-point sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of theO(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, which were based on high-order Voronoi diagrams. Our technique allows us for the first time to maintain minimal sets efficiently as new points are inserted, to generalize our algorithms to higher dimensions, to find minimal convexk-vertex polygons and polytopes, and to improve many previous results. We achieve many of our results via a new algorithm for finding rectilinear nearest neighbors in the plane in timeO(n logn+kn). We also demonstrate a related technique for finding minimum areak-point sets in the plane, based on testing sets of nearest vertical neighbors to each line segment determined by a pair of points. A generalization of this technique also allows us to find minimum volume and boundary measure sets in arbitrary dimensions.

/pdf/iterated-nearest-neighbors-and-finding-minimal-polytypes-3ky1jv9jzg.pdf

Iterated nearest neighbors and finding minimal polytypes

/pdf/iterated-nearest-neighbors-and-finding-minimal-polytopes-2r8q09ok3q.pdf

Jeff Erickson

Papers

Raising roofs, crashing cycles, and playing pool: applications of a data structure for finding pairwise interactions

Optimally Cutting a Surface into a Disk

Indexing moving points (extended abstract)

Iterated nearest neighbors and finding minimal polytypes

Iterated nearest neighbors and finding minimal polytopes