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 prove an Ω(ndr/2e) lower bound for the following problem: For some fixed linear equation in r variables, given n real numbers, do any r of them satisfy the equation? Our lower bound holds in a restricted linear decision tree model, in which each decision is based on the sign of an arbitrary linear combination of r or fewer inputs. In this model, our lower bound is as large as possible. Previously, this lower bound was known only for a few special cases and only in more specialized models of computation. Our lower bound follows from an adversary argument. We show that for any algorithm, there is a input that contains Ω(ndr/2e) “critical” r-tuples, which have the following important property. None of the critical tuples satisfies the equation; however, if the algorithm does not directly test each critical tuple, then the adversary can modify the input, in a way that is undetectable to the algorithm, so that some untested tuple does satisfy the equation. A key step in the proof is the introduction of formal infinitesimals into the adversary input. A theorem of Tarski implies that if we can construct a single input containing infinitesimals that is hard for every algorithm, then for every decision tree algorithm there exists a corresponding real-valued input which is hard for that algorithm. An extended abstract of this paper can be found in [Eri95].

/pdf/lower-bounds-for-linear-satisfiability-problems-3f0125hd3d.pdf

Lower bounds for linear satisfiability problems

Global Minimum Cuts in Surface-Embedded Graphs.

Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called homotopy moves. We prove that simplifying a planar closed curve with n self-crossings requires $$\Theta (n^{3/2})$$
 homotopy moves in the worst case. Our algorithm improves the best previous upper bound $$O(n^2)$$
 , which is already implicit in the classical work of Steinitz; the matching lower bound follows from the construction of closed curves with large defect, a topological invariant of generic closed curves introduced by Aicardi and Arnold. Our lower bound also implies that $$\Omega (n^{3/2})$$
 facial electrical transformations are required to reduce any plane graph with treewidth $$\Omega (\sqrt{n})$$
 to a single vertex, matching known upper bounds for rectangular and cylindrical grid graphs. More generally, we prove that transforming one immersion of k circles with at most n self-crossings into another requires $$\Theta (n^{3/2} + nk + k^2)$$
 homotopy moves in the worst case. Finally, we prove that transforming one non-contractible closed curve to another on any orientable surface requires $$\Omega (n^2)$$
 homotopy moves in the worst case; this lower bound is tight if the curve is homotopic to a simple closed curve.

/pdf/untangling-planar-curves-1en326a4af.pdf

Untangling Planar Curves

The (Vietoris-)Rips complex of a discrete point-set P is an abstract simplicial complex in which a subset of P defines a simplex if and only if the diameter of that subset is at most 1. We describe an efficient algorithm to determine whether a given cycle in a planar Rips complex is contractible. Our algorithm requires O(m log n) time to preprocess a set of n points in the plane in which m pairs have distance at most 1; after preprocessing, deciding whether a cycle of k Rips edges is contractible requires O(k) time. We also describe an algorithm to compute the shortest non-contractible cycle in a planar Rips complex in O(n2log n + mn) time.

/pdf/testing-contractibility-in-planar-rips-complexes-4s2v4p6pkr.pdf

Testing contractibility in planar rips complexes

We develop lower bounds on the number of primitive operations required to solve several fundamental problems in computational geometry. For example, given a set of points in the plane, are any three colinear? Given a set of points and lines, does any point lie on a line? These and similar questions arise as subproblems or special cases of a large number of more complicated geometric problems, including point location, range searching, motion planning, collision detection, ray shooting, and hidden surface removal.
Previously these problems were studied only in general models of computation, but known techniques for these models are too weak to prove useful results. Our approach is to consider, for each problem, a more specialized model of computation that is still rich enough to describe all known algorithms for that problem. Thus, our results formally demonstrate inherent limitations of current algorithmic techniques. Our lower bounds dramatically improve previously known results and in most cases match known upper bounds, at least up to polylogarithmic factors.
In the first part of the thesis, we develop lower bounds for several degeneracy-detection problems, using adversary arguments. For example, we show that detecting colinear triples of points requires $\Omega(\eta\sp2$) sidedness queries in the worst case. Our lower bound follows from the construction of a set of points in general position with several "collapsible" triangles, any one of which can be made degenerate without changing the orientation of any other triangle. Using similar techniques, we prove lower bounds for deciding, given a set of points in $\IR\sp{\rm d}$, whether any d + 1 points lie on a hyperplane, whether any d + 2 points lie on a sphere, or whether the convex hull of the point is simplicial.
In the second part, we consider offline range searching problems, which are usually solved using geometric divide-and-conquer techniques. To study these problems, we introduce the class of partitioning algorithms. We prove that any partitioning algorithm requires $\Omega(\eta\sp{4/3}$) time to detect point-line incidences in the worst case. Using similar techniques, we prove an $\Omega(\eta\sp{4/3}$) lower bound for deciding if a set of points lies entirely above a set of hyperplanes in dimensions five and higher.

Jeff Erickson

Papers

Lower bounds for linear satisfiability problems

Global Minimum Cuts in Surface-Embedded Graphs.

Untangling Planar Curves

Testing contractibility in planar rips complexes

Lower bounds for fundamental geometric problems