16 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
A New Lower Bound for Semigroup Orthogonal Range Searching
We report the first improvement in the space-time trade-off of lower bounds
for the orthogonal range searching problem in the semigroup model, since
Chazelle's result from 1990. This is one of the very fundamental problems in
range searching with a long history. Previously, Andrew Yao's influential
result had shown that the problem is already non-trivial in one
dimension~\cite{Yao-1Dlb}: using units of space, the query time must
be where is the
inverse Ackermann's function, a very slowly growing function.
In dimensions, Bernard Chazelle~\cite{Chazelle.LB.II} proved that the
query time must be where .
Chazelle's lower bound is known to be tight for when space consumption is
`high' i.e., . We have two main results.
The first is a lower bound that shows Chazelle's lower bound was not tight for
`low space': we prove that we must have . Our lower bound does not close the gap to the existing data
structures, however, our second result is that our analysis is tight. Thus, we
believe the gap is in fact natural since lower bounds are proven for idempotent
semigroups while the data structures are built for general semigroups and thus
they cannot assume (and use) the properties of an idempotent semigroup. As a
result, we believe to close the gap one must study lower bounds for
non-idempotent semigroups or building data structures for idempotent
semigroups. We develope significantly new ideas for both of our results that
could be useful in pursuing either of these directions
A New Lower Bound for Semigroup Orthogonal Range Searching
We report the first improvement in the space-time trade-off of lower bounds for the orthogonal range searching problem in the semigroup model, since Chazelle\u27s result from 1990. This is one of the very fundamental problems in range searching with a long history. Previously, Andrew Yao\u27s influential result had shown that the problem is already non-trivial in one dimension [Yao, 1982]: using m units of space, the query time Q(n) must be Omega(alpha(m,n) + n/(m-n+1)) where alpha(*,*) is the inverse Ackermann\u27s function, a very slowly growing function. In d dimensions, Bernard Chazelle [Chazelle, 1990] proved that the query time must be Q(n) = Omega((log_beta n)^{d-1}) where beta = 2m/n. Chazelle\u27s lower bound is known to be tight for when space consumption is "high" i.e., m = Omega(n log^{d+epsilon}n).
We have two main results. The first is a lower bound that shows Chazelle\u27s lower bound was not tight for "low space": we prove that we must have m Q(n) = Omega(n (log n log log n)^{d-1}). Our lower bound does not close the gap to the existing data structures, however, our second result is that our analysis is tight. Thus, we believe the gap is in fact natural since lower bounds are proven for idempotent semigroups while the data structures are built for general semigroups and thus they cannot assume (and use) the properties of an idempotent semigroup. As a result, we believe to close the gap one must study lower bounds for non-idempotent semigroups or building data structures for idempotent semigroups. We develope significantly new ideas for both of our results that could be useful in pursuing either of these directions
Solving Fr\'echet Distance Problems by Algebraic Geometric Methods
We study several polygonal curve problems under the Fr\'{e}chet distance via
algebraic geometric methods. Let and be the
spaces of all polygonal curves of and vertices in ,
respectively. We assume that . Let be the set
of ranges in for all possible metric balls of polygonal curves
in under the Fr\'{e}chet distance. We prove a nearly optimal
bound of on the VC dimension of the range space
, improving on the previous
upper bound and approaching the current
lower bound. Our upper bound also holds for the weak Fr\'{e}chet distance. We
also obtain exact solutions that are hitherto unknown for curve simplification,
range searching, nearest neighbor search, and distance oracle.Comment: To appear at SODA24, correct some reference
Map matching queries on realistic input graphs under the Fr\'echet distance
Map matching is a common preprocessing step for analysing vehicle
trajectories. In the theory community, the most popular approach for map
matching is to compute a path on the road network that is the most spatially
similar to the trajectory, where spatial similarity is measured using the
Fr\'echet distance. A shortcoming of existing map matching algorithms under the
Fr\'echet distance is that every time a trajectory is matched, the entire road
network needs to be reprocessed from scratch. An open problem is whether one
can preprocess the road network into a data structure, so that map matching
queries can be answered in sublinear time.
In this paper, we investigate map matching queries under the Fr\'echet
distance. We provide a negative result for geometric planar graphs. We show
that, unless SETH fails, there is no data structure that can be constructed in
polynomial time that answers map matching queries in query
time for any , where and are the complexities of the
geometric planar graph and the query trajectory, respectively. We provide a
positive result for realistic input graphs, which we regard as the main result
of this paper. We show that for -packed graphs, one can construct a data
structure of size that can answer -approximate
map matching queries in time, where hides lower-order factors and dependence of .Comment: To appear in SODA 202