377 research outputs found
On the complexity of range searching among curves
Modern tracking technology has made the collection of large numbers of
densely sampled trajectories of moving objects widely available. We consider a
fundamental problem encountered when analysing such data: Given polygonal
curves in , preprocess into a data structure that answers
queries with a query curve and radius for the curves of that
have \Frechet distance at most to .
We initiate a comprehensive analysis of the space/query-time trade-off for
this data structuring problem. Our lower bounds imply that any data structure
in the pointer model model that achieves query time, where is
the output size, has to use roughly space in
the worst case, even if queries are mere points (for the discrete \Frechet
distance) or line segments (for the continuous \Frechet distance). More
importantly, we show that more complex queries and input curves lead to
additional logarithmic factors in the lower bound. Roughly speaking, the number
of logarithmic factors added is linear in the number of edges added to the
query and input curve complexity. This means that the space/query time
trade-off worsens by an exponential factor of input and query complexity. This
behaviour addresses an open question in the range searching literature: whether
it is possible to avoid the additional logarithmic factors in the space and
query time of a multilevel partition tree. We answer this question negatively.
On the positive side, we show we can build data structures for the \Frechet
distance by using semialgebraic range searching. Our solution for the discrete
\Frechet distance is in line with the lower bound, as the number of levels in
the data structure is , where denotes the maximal number of vertices
of a curve. For the continuous \Frechet distance, the number of levels
increases to
Probabilistic embeddings of the Fr\'echet distance
The Fr\'echet distance is a popular distance measure for curves which
naturally lends itself to fundamental computational tasks, such as clustering,
nearest-neighbor searching, and spherical range searching in the corresponding
metric space. However, its inherent complexity poses considerable computational
challenges in practice. To address this problem we study distortion of the
probabilistic embedding that results from projecting the curves to a randomly
chosen line. Such an embedding could be used in combination with, e.g.
locality-sensitive hashing. We show that in the worst case and under reasonable
assumptions, the discrete Fr\'echet distance between two polygonal curves of
complexity in , where , degrades
by a factor linear in with constant probability. We show upper and lower
bounds on the distortion. We also evaluate our findings empirically on a
benchmark data set. The preliminary experimental results stand in stark
contrast with our lower bounds. They indicate that highly distorted projections
happen very rarely in practice, and only for strongly conditioned input curves.
Keywords: Fr\'echet distance, metric embeddings, random projectionsComment: 27 pages, 11 figure
Locality-Sensitive Hashing of Curves
We study data structures for storing a set of polygonal curves in
such that, given a query curve, we can efficiently retrieve similar curves from
the set, where similarity is measured using the discrete Fr\'echet distance or
the dynamic time warping distance. To this end we devise the first
locality-sensitive hashing schemes for these distance measures. A major
challenge is posed by the fact that these distance measures internally optimize
the alignment between the curves. We give solutions for different types of
alignments including constrained and unconstrained versions. For unconstrained
alignments, we improve over a result by Indyk from 2002 for short curves. Let
be the number of input curves and let be the maximum complexity of a
curve in the input. In the particular case where , for some fixed , our solutions imply an approximate near-neighbor
data structure for the discrete Fr\'echet distance that uses space in
and achieves query time in and
constant approximation factor. Furthermore, our solutions provide a trade-off
between approximation quality and computational performance: for any parameter
, we can give a data structure that uses space in , answers queries in time and achieves
approximation factor in .Comment: Proc. of 33rd International Symposium on Computational Geometry
(SoCG), 201
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