142 research outputs found
Progressive Simplification of Polygonal Curves
Simplifying polygonal curves at different levels of detail is an important
problem with many applications. Existing geometric optimization algorithms are
only capable of minimizing the complexity of a simplified curve for a single
level of detail. We present an -time algorithm that takes a polygonal
curve of n vertices and produces a set of consistent simplifications for m
scales while minimizing the cumulative simplification complexity. This
algorithm is compatible with distance measures such as the Hausdorff, the
Fr\'echet and area-based distances, and enables simplification for continuous
scaling in time. To speed up this algorithm in practice, we present
new techniques for constructing and representing so-called shortcut graphs.
Experimental evaluation of these techniques on trajectory data reveals a
significant improvement of using shortcut graphs for progressive and
non-progressive curve simplification, both in terms of running time and memory
usage.Comment: 20 pages, 20 figure
Computing the Similarity Between Moving Curves
In this paper we study similarity measures for moving curves which can, for
example, model changing coastlines or retreating glacier termini. Points on a
moving curve have two parameters, namely the position along the curve as well
as time. We therefore focus on similarity measures for surfaces, specifically
the Fr\'echet distance between surfaces. While the Fr\'echet distance between
surfaces is not even known to be computable, we show for variants arising in
the context of moving curves that they are polynomial-time solvable or
NP-complete depending on the restrictions imposed on how the moving curves are
matched. We achieve the polynomial-time solutions by a novel approach for
computing a surface in the so-called free-space diagram based on max-flow
min-cut duality
Fast Frechet Distance Between Curves With Long Edges
Computing the Fr\'echet distance between two polygonal curves takes roughly
quadratic time. In this paper, we show that for a special class of curves the
Fr\'echet distance computations become easier. Let and be two polygonal
curves in with and vertices, respectively. We prove four
results for the case when all edges of both curves are long compared to the
Fr\'echet distance between them: (1) a linear-time algorithm for deciding the
Fr\'echet distance between two curves, (2) an algorithm that computes the
Fr\'echet distance in time, (3) a linear-time
-approximation algorithm, and (4) a data structure that supports
-time decision queries, where is the number of vertices of
the query curve and the number of vertices of the preprocessed curve
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
Trajectory Similarity Measurement: An Efficiency Perspective
Trajectories that capture object movement have numerous applications, in
which similarity computation between trajectories often plays a key role.
Traditionally, the similarity between two trajectories is quantified by means
of heuristic measures, e.g., Hausdorff or ERP, that operate directly on the
trajectories. In contrast, recent studies exploit deep learning to map
trajectories to d-dimensional vectors, called embeddings. Then, some distance
measure, e.g., Manhattan or Euclidean, is applied to the embeddings to quantify
trajectory similarity. The resulting similarities are inaccurate: they only
approximate the similarities obtained using the heuristic measures. As distance
computation on embeddings is efficient, focus has been on achieving embeddings
yielding high accuracy.
Adopting an efficiency perspective, we analyze the time complexities of both
the heuristic and the learning-based approaches, finding that the time
complexities of the former approaches are not necessarily higher. Through
extensive experiments on open datasets, we find that, on both CPUs and GPUs,
only a few learning-based approaches can deliver the promised higher
efficiency, when the embeddings can be pre-computed, while heuristic approaches
are more efficient for one-off computations. Among the learning-based
approaches, the self-attention-based ones are the fastest to learn embeddings
that also yield the highest accuracy for similarity queries. These results have
implications for the use of trajectory similarity approaches given different
application requirements
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 Fréchet distance, the Fréchet 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 Fréchet 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 ähnliche Daten verarbeiten. Ein wichtiger Bestandteil dieser Anwendungen ist die Berechnung der Ähnlichkeit von Objekten. Diese Dissertation präsentiert Resultate, genauer gesagt Algorithmen, untere Komplexitätsschranken und Algorithm Engineering der wichtigsten Ähnlichkeitsmaße für Punktmengen und Sequenzen, wie zum Beispiel Fréchetdistanz, Fréchetdistanz unter Translation und Hausdorffdistanz unter Translation. Als eine Erweiterung der bloßen Berechnung von Ähnlichkeit betrachten wir auch das Near Neighbor Problem für die kontinuierliche Fréchetdistanz auf Zeitfolgen und zeigen obere und untere Schranken dafür
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