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

We introduce a new realistic input model for geometric graphs and nonconvex polyhedra. A geometric graph G is local if (1) the longest edge at every vertex v is only a constant factor longer than the distance from v to its Euclidean nearest neighbor and (2) the lengths of the longest and shortest edges differ by at most a polynomial factor. A polyhedron is local if all its faces are simplices and its edges form a local geometric graph. We show that any boolean combination of any two local polyhedra in IRd, each with n vertices, can be computed in O(n log n) time, using a standard hierarchy of axis-aligned bounding boxes. Using results of de Berg, we also show that any local polyhedron in IRd has a binary space partition tree of size O(n logd-1 n). Finally, we describe efficient algorithms for computing Minkowski sums of local polyhedra in two and three dimensions.

/pdf/local-polyhedra-and-geometric-graphs-17qh401jc5.pdf

Local polyhedra and geometric graphs

Almost exactly 100 years ago, Max Dehn described the first combinatorial algorithm to determine whether two given cycles on a compact surface are homotopic, meaning one cycle can be continuously deformed into the other without leaving the surface. We describe a simple variant of Dehn's algorithm that runs in linear time, with no hidden dependence on the genus of the surface. Specifically, given two closed vertex-edge walks of length at most e in a combinatorial surface of complexity n, our algorithm determines whether the walks are homotopic in O(n + e) time. Our algorithm simplifies and corrects a similar algorithm of Dey and Guha [JCSS 1999] and simplifies the more recent algorithm of Lazarus and Rivaud [FOCS 2012], who identified a subtle flaw in Dey and Guha's results. Our algorithm combines components of these earlier algorithms, classical results in small cancellation theory by Gersten and Short [Inventiones 1990], and simple run-length encoding.

/pdf/transforming-curves-on-surfaces-redux-45bczba0cc.pdf

Transforming curves on surfaces redux

A cycle on a combinatorial surface is tight if it as short as possible in its (free) homotopy class. We describe an algorithm to compute a single tight, noncontractible, essentially simple cycle on a given orientable combinatorial surface in O(n log n) time. The only method previously known for this problem was to compute the globally shortest noncontractible or nonseparating cycle in O(min{g3,n}, n log n) time, where g is the genus of the surface. As a consequence, we can compute the shortest cycle freely homotopic to a chosen boundary cycle in O(n log n) time, a tight octagonal decomposition in O(gnlog n) time, and a shortest contractible cycle enclosing a nonempty set of faces in O(nlog2n) time.

Finding one tight cycle

We develop the first nontrivial lower bounds on the complexity of online hyperplane and halfspace emptiness queries. Our lower bounds apply to a general class of geometric range query data structures called partition graphs. Informally, a partition graph is a directed acyclic graph that describes a recursive decomposition of space. We show that any partition graph that supports hyperplane emptiness queries implicitly defines a halfspace range query data structure in the Fredman/Yao semigroup arithmetic model, with the same asymptotic space and time bounds. Thus, results of Bronnimann, Chazelle, and Pach imply that any partition graph of size s that supports hyperplane emptiness queries in time t satisfies the inequality $st^d = \Omega((n/\log n)^{d - (d-1)/(d+1)})$. Using different techniques, we improve previous lower bounds for Hopcroft's problem---Given a set of points and hyperplanes, does any hyperplane contain a point?---in dimensions four and higher. Using this offline result, we show that for online hyperplane emptiness queries, $\Omega(n^d/{\mbox{ polylog }} n)$ space is required to achieve polylogarithmic query time, and $\Omega(n^{(d-1)/d}/{\mbox{ polylog }} n)$ query time is required if only O(n polylog n) space is available. These two lower bounds are optimal up to polylogarithmic factors. For two-dimensional queries, we obtain an optimal continuous tradeoff $st^2=\Omega(n^2)$ between these two extremes. Finally, using a lifting argument, we show that the same lower bounds hold for both offline and online halfspace emptiness queries in ${\mathbb{R}}^{d(d+3)/2}$.

Space-Time Tradeoffs for Emptiness Queries

This paper addresses the problem of capturing an arbitrary convex object P in the plane with three congruent disc-shaped robots. Given two stationary robots in contact with P, we characterize the set of positions of a third robot that prevent P from escaping to infinity and show that the computation of this so-called capture region reduces to the resolution of a visibility problem. We present two algorithms for solving this problem and computing the capture region when P is a polygon and the robots are points (zero-radius discs). The first algorithm is exact and has polynomial-time complexity. The second one uses simple hidden-surface removal techniques from computer graphics to output an arbitrarily accurate approximation of the capture region; it has been implemented and examples are presented.

Jeff Erickson

Papers

Local polyhedra and geometric graphs

Transforming curves on surfaces redux

Finding one tight cycle

Space-Time Tradeoffs for Emptiness Queries

Capturing a convex object with three discs