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 present an O(n log n)-time algorithm that determines whether a given planar n-gon is weakly simple. This improves upon an O(n^2 log n)-time algorithm by [Chang, Erickson, and Xu, SODA, 2015]. Weakly simple polygons are required as input for several geometric algorithms. As such, how to recognize simple or weakly simple polygons is a fundamental question.

https://drops.dagstuhl.de/opus/volltexte/2016/5900/pdf/LIPIcs-SoCG-2016-8.pdf/

Recognizing weakly simple polygons

We show that in the worst case, Ω(ndd/2e−1 +n logn) sidedness queries are required to determine whether the convex hull of n points in Rd is simplicial or to determine the number of convex hull facets. This lower bound matches known upper bounds in any odd dimension. Our result follows from a straightforward adversary argument. A key step in the proof is the construction of a quasi-simplicial n-vertex polytope with Ω(ndd/2e−1) degenerate facets. While it has been known for several years that d-dimensional convex hulls can have Ω(nbd/2c) facets, the previously best lower bound for these problems is only Ω(n logn). Using similar techniques, we also obtain simple and correct proofs of Erickson and Seidel’s lower bounds for detecting affine degeneracies in arbitrary dimensions and circular degeneracies in the plane. As a related result, we show that detecting simplicial convex hulls in Rd is dd/2esum-hard in the sense of Gajentaan and Overmars.

https://epubs.siam.org/doi/pdf/10.1137/S0097539797315410

New lower bounds for convex hull problems in odd dimensions

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, non-contractible, 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 non-contractible or non-separating 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 and a tight octagonal decomposition in O(gn log n) time

/pdf/finding-one-tight-cycle-4xfv67cux6.pdf

Finding one tight cycle

This report documents the program and the outcomes of Dagstuhl Seminar 16221
“Algorithms for Optimization Problems in Planar Graphs”. The seminar was held from May 29 to June 3, 2016. This report contains abstracts for the recent developments in planar graph algorithms discussed during the seminar as well as summaries of open problems in this area of research.

Algorithms for Optimization Problems in Planar Graphs (Dagstuhl Seminar 16221)

We present a near-optimal polynomial-time approximation algorithm for the asymmetric traveling salesman problem for graphs of bounded orientable or non-orientable genus. Given any algorithm that achieves an approximation ratio of f(n) on arbitrary n-vertex graphs as a black box, our algorithm achieves an approximation factor of O(f(g)) on graphs with genus g. In particular, the O(log n/loglog n)-approximation algorithm for general graphs by Asadpour et al. [SODA 2010] immediately implies an O(log g/loglog g)-approximation algorithm for genus-g graphs. Moreover, recent results on approximating the genus of graphs imply that our O(log g/loglog g)-approximation algorithm can be applied to bounded-degree graphs even if no genus-g embedding of the graph is given. Our result improves and generalizes the o(√ g log g)-approximation algorithm of Oveis Gharan and Saberi [SODA 2011], which applies only to graphs with orientable genus g and requires a genus-g embedding as part of the input, even for bounded-degree graphs. Finally, our techniques yield a O(1)-approximation algorithm for ATSP on graphs of genus g with running time 2O(g) · nO(1).

Jeff Erickson

Papers

Recognizing weakly simple polygons

New lower bounds for convex hull problems in odd dimensions

Finding one tight cycle

Algorithms for Optimization Problems in Planar Graphs (Dagstuhl Seminar 16221)

A near-optimal approximation algorithm for Asymmetric TSP on embedded graphs