On-line construction of suffix trees

Topological Data Analysis is a recent and fast growing field providing a set of new topological and geometric tools to infer relevant features for possibly complex data. This paper is a brief introduction, through a few selected topics, to basic fundamental and practical aspects of \tda\ for non experts.

An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists

In many Intelligent Transportation System (ITS) applications that crowd-source data from probe vehicles, a crucial step is to accurately map the GPS trajectories to the road network in real time. This process, known as map-matching, often needs to account for noise and sparseness of the data because (1) highly precise GPS traces are rarely available, and (2) dense trajectories are costly for live transmission and storage.

/pdf/online-map-matching-based-on-hidden-markov-model-for-real-5b98nb56fi.pdf

Online map-matching based on Hidden Markov model for real-time traffic sensing applications

The Frechet distance is a well-studied and very popular measure of similarity of two curves. Many variants and extensions have been studied since Alt and Godau introduced this measure to computational geometry in 1991. Their original algorithm to compute the Frechet distance of two polygonal curves with n vertices has a runtime of O(n^2 log n). More than 20 years later, the state of the art algorithms for most variants still take time more than O(n2 / log n), but no matching lower bounds are known, not even under reasonable complexity theoretic assumptions. To obtain a conditional lower bound, in this paper we assume the Strong Exponential Time Hypothesis or, more precisely, that there is no O*((2-adelta;)N) algorithm for CNF-SAT for any delta > 0. Under this assumption we show that the Frechet distance cannot be computed in strongly subquadratic time, i.e., in time O(n2-adelta;) for any delta > 0. This means that finding faster algorithms for the Frechet distance is as hard as finding faster CNF-SAT algorithms, and the existence of a strongly subquadratic algorithm can be considered unlikely. Our result holds for both the continuous and the discrete Frechet distance. We extend the main result in various directions. Based on the same assumption we (1) show non-existence of a strongly subquadratic 1.001-approximation, (2) present tight lower bounds in case the numbers of vertices of the two curves are imbalanced, and (3) examine realistic input assumptions (c-packed curves).

/pdf/why-walking-the-dog-takes-time-frechet-distance-has-no-1lmcwv0nae.pdf

Why Walking the Dog Takes Time: Frechet Distance Has No Strongly Subquadratic Algorithms Unless SETH Fails

The Frechet distance is a well-studied and very popular measure of similarity of two curves. Many variants and extensions have been studied since Alt and Godau introduced this measure to computational geometry in 1991. Their original algorithm to compute the Frechet distance of two polygonal curves with n vertices has a runtime of O(n^2 log n). More than 20 years later, the state of the art algorithms for most variants still take time more than O(n^2 / log n), but no matching lower bounds are known, not even under reasonable complexity theoretic assumptions. 
To obtain a conditional lower bound, in this paper we assume the Strong Exponential Time Hypothesis or, more precisely, that there is no O*((2-delta)^N) algorithm for CNF-SAT for any delta > 0. Under this assumption we show that the Frechet distance cannot be computed in strongly subquadratic time, i.e., in time O(n^{2-delta}) for any delta > 0. This means that finding faster algorithms for the Frechet distance is as hard as finding faster CNF-SAT algorithms, and the existence of a strongly subquadratic algorithm can be considered unlikely. 
Our result holds for both the continuous and the discrete Frechet distance. We extend the main result in various directions. Based on the same assumption we (1) show non-existence of a strongly subquadratic 1.001-approximation, (2) present tight lower bounds in case the numbers of vertices of the two curves are imbalanced, and (3) examine realistic input assumptions (c-packed curves).

Why walking the dog takes time: Frechet distance has no strongly subquadratic algorithms unless SETH fails

We extend recent results using curve simplification for approximating the Frechet distance of realistic curves in near linear time to map matching: the problem of matching a curve in an embedded graph. We show that the theoretical bounds on the running time of the previous result still hold if only one of the curves is simplified during the course of the approximation algorithm. This enables our extension to the case of map matching under the assumption that the graph is p-low density for a constant p. We present experimental evidence for this assumption and implement the extended approximate matching algorithm. We show that it performs well on real world data, such as GPS traces and road networks of urban areas. In particular, it is able to perform matching tasks that took several hours with the exact matching algorithm in under a second.

/pdf/approximate-map-matching-with-respect-to-the-frechet-31zvhe1zhp.pdf

Approximate map matching with respect to the Fréchet distance

We consider the problem of reconstructing a road network from a collection of path traces and provide guarantees on the accuracy of the reconstruction under reasonable assumptions. Our algorithm can be used to process a collection of polygonal paths in the plane so that shared structures (subpaths) among the paths in the collection can be discovered and the collection can be organized to allow efficient path similarity queries against new query paths on the same road network. This is a timely problem, as GPS and other location traces of both people and vehicles are becoming available on a large scale and there is a real need to create appropriate data structures and data bases for such data.

https://epubs.siam.org/doi/pdf/10.1137/1.9781611973075.105

Road network reconstruction for organizing paths

Many real-world data sets can be viewed of as noisy samples of special types of metric spaces called metric graphs.19 Building on the notions of correspondence and Gromov-Hausdorff distance in metric geometry, we describe a model for such data sets as an approximation of an underlying metric graph. We present a novel algorithm that takes as an input such a data set, and outputs a metric graph that is homeomorphic to the underlying metric graph and has bounded distortion of distances. We also implement the algorithm, and evaluate its performance on a variety of real world data sets.

Metric Graph Reconstruction From Noisy Data

Many real-world data sets can be viewed of as noisy samples of special types of metric spaces called metric graphs [16]. Building on the notions of correspondence and Gromov-Hausdorff distance in metric geometry, we describe a model for such data sets as an approximation of an underlying metric graph. We present a novel algorithm that takes as an input such a data set, and outputs the underlying metric graph with guarantees. We also implement the algorithm, and evaluate its performance on a variety of real world data sets.

/pdf/metric-graph-reconstruction-from-noisy-data-24dgu5j09e.pdf

Metric graph reconstruction from noisy data

Motivated by the increasing availability of large collections of noisy GPS traces, we present a new data-driven framework for smoothing trajectory data. The framework, which can be viewed of as a generalization of the classical moving average technique, naturally leads to efficient algorithms for various smoothing objectives. We analyze an algorithm based on this framework and provide connections to previous smoothing techniques. We implement a variation of the algorithm to smooth an entire collection of trajectories and show that it performs well on both synthetic data and massive collections of GPS traces.

/pdf/data-driven-trajectory-smoothing-12k78chha5.pdf

Daniel Chen

Papers

Approximate map matching with respect to the Fréchet distance

Road network reconstruction for organizing paths

Metric Graph Reconstruction From Noisy Data

Metric graph reconstruction from noisy data

Data-driven trajectory smoothing