51 research outputs found
Computing a Subtrajectory Cluster from c-packed Trajectories
We present a near-linear time approximation algorithm for the subtrajectory
cluster problem of -packed trajectories. The problem involves finding
subtrajectories within a given trajectory such that their Fr\'echet
distances are at most , and at least one subtrajectory must
be of length~ or longer. A trajectory is -packed if the intersection
of and any ball with radius is at most in length.
Previous results by Gudmundsson and Wong
\cite{GudmundssonWong2022Cubicupperlower} established an lower
bound unless the Strong Exponential Time Hypothesis fails, and they presented
an time algorithm. We circumvent this conditional lower bound
by studying subtrajectory cluster on -packed trajectories, resulting in an
algorithm with an time complexity
Fine-grained complexity and algorithm engineering of geometric similarity measures
Point sets and sequences are fundamental geometric objects that arise in any application that considers movement data, geometric shapes, and many more. A crucial task on these objects is to measure their similarity. Therefore, this thesis presents results on algorithms, complexity lower bounds, and algorithm engineering of the most important point set and sequence similarity measures like the FrĂ©chet distance, the FrĂ©chet distance under translation, and the Hausdorff distance under translation. As an extension to the mere computation of similarity, also the approximate near neighbor problem for the continuous FrĂ©chet distance on time series is considered and matching upper and lower bounds are shown.Punktmengen und Sequenzen sind fundamentale geometrische Objekte, welche in vielen Anwendungen auftauchen, insbesondere in solchen die Bewegungsdaten, geometrische Formen, und Ă€hnliche Daten verarbeiten. Ein wichtiger Bestandteil dieser Anwendungen ist die Berechnung der Ăhnlichkeit von Objekten. Diese Dissertation prĂ€sentiert Resultate, genauer gesagt Algorithmen, untere KomplexitĂ€tsschranken und Algorithm Engineering der wichtigsten ĂhnlichkeitsmaĂe fĂŒr Punktmengen und Sequenzen, wie zum Beispiel FrĂ©chetdistanz, FrĂ©chetdistanz unter Translation und Hausdorffdistanz unter Translation. Als eine Erweiterung der bloĂen Berechnung von Ăhnlichkeit betrachten wir auch das Near Neighbor Problem fĂŒr die kontinuierliche FrĂ©chetdistanz auf Zeitfolgen und zeigen obere und untere Schranken dafĂŒr
Approximating the Packedness of Polygonal Curves
In 2012 Driemel et al. \cite{DBLP:journals/dcg/DriemelHW12} introduced the
concept of -packed curves as a realistic input model. In the case when
is a constant they gave a near linear time -approximation
algorithm for computing the Fr\'echet distance between two -packed polygonal
curves. Since then a number of papers have used the model.
In this paper we consider the problem of computing the smallest for which
a given polygonal curve in is -packed. We present two
approximation algorithms. The first algorithm is a -approximation algorithm
and runs in time. In the case we develop a faster
algorithm that returns a -approximation and runs in
time.
We also implemented the first algorithm and computed the approximate
packedness-value for 16 sets of real-world trajectories. The experiments
indicate that the notion of -packedness is a useful realistic input model
for many curves and trajectories.Comment: A preliminary version to appear in ISAAC 202
Approximability of the Discrete {Fr\'echet} Distance
<p>The Fréchet distance is a popular and widespread distance measure for point sequences and for curves. About two years ago, Agarwal et al. [SIAM J. Comput. 2014] presented a new (mildly) subquadratic algorithm for the discrete version of the problem. This spawned a flurry of activity that has led to several new algorithms and lower bounds.</p><p>In this paper, we study the approximability of the discrete Fréchet distance. Building on a recent result by Bringmann [FOCS 2014], we present a new conditional lower bound showing that strongly subquadratic algorithms for the discrete Fréchet distance are unlikely to exist, even in the one-dimensional case and even if the solution may be approximated up to a factor of 1.399.</p><p>This raises the question of how well we can approximate the Fréchet distance (of two given -dimensional point sequences of length ) in strongly subquadratic time. Previously, no general results were known. We present the first such algorithm by analysing the approximation ratio of a simple, linear-time greedy algorithm to be . Moreover, we design an -approximation algorithm that runs in time , for any . Hence, an -approximation of the Fréchet distance can be computed in strongly subquadratic time, for any \varepsilon > 0.</p
Approximating -center clustering for curves
The Euclidean -center problem is a classical problem that has been
extensively studied in computer science. Given a set of
points in Euclidean space, the problem is to determine a set of
centers (not necessarily part of ) such that the maximum
distance between a point in and its nearest neighbor in
is minimized. In this paper we study the corresponding
-center problem for polygonal curves under the Fr\'echet distance,
that is, given a set of polygonal curves in ,
each of complexity , determine a set of polygonal curves
in , each of complexity , such that the maximum Fr\'echet
distance of a curve in to its closest curve in is
minimized. In this paper, we substantially extend and improve the known
approximation bounds for curves in dimension and higher. We show that, if
is part of the input, then there is no polynomial-time approximation
scheme unless . Our constructions yield different
bounds for one and two-dimensional curves and the discrete and continuous
Fr\'echet distance. In the case of the discrete Fr\'echet distance on
two-dimensional curves, we show hardness of approximation within a factor close
to . This result also holds when , and the -hardness
extends to the case that , i.e., for the problem of computing the
minimum-enclosing ball under the Fr\'echet distance. Finally, we observe that a
careful adaptation of Gonzalez' algorithm in combination with a curve
simplification yields a -approximation in any dimension, provided that an
optimal simplification can be computed exactly. We conclude that our
approximation bounds are close to being tight.Comment: 24 pages; results on minimum-enclosing ball added, additional author
added, general revisio
Tight(er) bounds for similarity measures, smoothed approximation and broadcasting
In this thesis, we prove upper and lower bounds on the complexity of sequence similarity measures, the approximability of geometric problems on realistic inputs, and the performance of randomized broadcasting protocols.
The first part approaches the question why a number of fundamental polynomial-time problems - specifically, Dynamic Time Warping, Longest Common Subsequence (LCS), and the Levenshtein distance - resists decades-long attempts to obtain polynomial improvements over their simple dynamic programming solutions. We prove that any (strongly) subquadratic algorithm for these and related sequence similarity measures would refute the Strong Exponential Time Hypothesis (SETH). Focusing particularly on LCS, we determine a tight running time bound (up to lower order factors and conditional on SETH) when the running time is expressed in terms of all input parameters that have been previously exploited in the extensive literature.
In the second part, we investigate the approximation performance of the popular 2-Opt heuristic for the Traveling Salesperson Problem using the smoothed analysis paradigm. For the Fréchet distance, we design an improved approximation algorithm for the natural input class of c-packed curves, matching a conditional lower bound.
Finally, in the third part we prove tighter performance bounds for processes that disseminate a piece of information, either as quickly as possible (rumor spreading) or as anonymously as possible (cryptogenography).Die vorliegende Dissertation beweist obere und untere Schranken an die KomplexitĂ€t von SequenzĂ€hnlichkeitsmaĂen, an die Approximierbarkeit geometrischer Probleme auf realistischen Eingaben und an die EffektivitĂ€t randomisierter Kommunikationsprotokolle.
Der erste Teil befasst sich mit der Frage, warum fĂŒr eine Vielzahl fundamentaler Probleme im Polynomialzeitbereich - insbesondere fĂŒr das Dynamic-Time-Warping, die lĂ€ngste gemeinsame Teilfolge (LCS) und die Levenshtein-Distanz - seit Jahrzehnten keine Algorithmen gefunden werden konnten, die polynomiell schneller sind als ihre einfachen Lösungen mittels dynamischer Programmierung. Wir zeigen, dass ein (im strengen Sinne) subquadratischer Algorithmus fĂŒr diese und verwandte ĂhnlichkeitsmaĂe die starke Exponentialzeithypothese (SETH) widerlegen wĂŒrde. FĂŒr LCS zeigen wir eine scharfe Schranke an die optimale Laufzeit (unter der SETH und bis auf Faktoren niedrigerer Ordnung) in AbhĂ€ngigkeit aller bisher untersuchten Eingabeparameter.
Im zweiten Teil untersuchen wir die ApproximationsgĂŒte der klassischen 2-Opt-Heuristik fĂŒr das Problem des Handlungsreisenden anhand des Smoothed-Analysis-Paradigmas. Weiterhin entwickeln wir einen verbesserten Approximationsalgorithmus fĂŒr die FrĂ©chet-Distanz auf einer Klasse natĂŒrlicher Eingaben.
Der letzte Teil beweist neue Schranken fĂŒr die EffektivitĂ€t von Prozessen, die Informationen entweder so schnell wie möglich (Rumor-Spreading) oder so anonym wie möglich (Kryptogenografie) verbreiten
- âŠ