771 research outputs found
Recovery from Non-Decomposable Distance Oracles
A line of work has looked at the problem of recovering an input from distance
queries. In this setting, there is an unknown sequence , and one chooses a set of queries and
receives for a distance function . The goal is to make as few
queries as possible to recover . Although this problem is well-studied for
decomposable distances, i.e., distances of the form for some function , which includes the important cases of
Hamming distance, -norms, and -estimators, to the best of our
knowledge this problem has not been studied for non-decomposable distances, for
which there are important special cases such as edit distance, dynamic time
warping (DTW), Frechet distance, earth mover's distance, and so on. We initiate
the study and develop a general framework for such distances. Interestingly,
for some distances such as DTW or Frechet, exact recovery of the sequence
is provably impossible, and so we show by allowing the characters in to be
drawn from a slightly larger alphabet this then becomes possible. In a number
of cases we obtain optimal or near-optimal query complexity. We also study the
role of adaptivity for a number of different distance functions. One motivation
for understanding non-adaptivity is that the query sequence can be fixed and
the distances of the input to the queries provide a non-linear embedding of the
input, which can be used in downstream applications involving, e.g., neural
networks for natural language processing.Comment: This work has been presented at conference The 14th Innovations in
Theoretical Computer Science (ITCS 2023) and accepted for publishing in the
journal IEEE Transactions on Information Theor
Fréchet Distance for Uncertain Curves
In this article, we study a wide range of variants for computing the (discrete and continuous) Fréchet distance between uncertain curves. An uncertain curve is a sequence of uncertainty regions, where each region is a disk, a line segment, or a set of points. A realisation of a curve is a polyline connecting one point from each region. Given an uncertain curve and a second (certain or uncertain) curve, we seek to compute the lower and upper bound Fréchet distance, which are the minimum and maximum Fréchet distance for any realisations of the curves. We prove that both problems are NP-hard for the Fréchet distance in several uncertainty models, and that the upper bound problem remains hard for the discrete Fréchet distance. In contrast, the lower bound (discrete [5] and continuous) Fréchet distance can be computed in polynomial time in some models. Furthermore, we show that computing the expected (discrete and continuous) Fréchet distance is #P-hard in some models.On the positive side, we present an FPTAS in constant dimension for the lower bound problem when Δ/δis polynomially bounded, where δis the Fréchet distance and Δbounds the diameter of the regions. We also show a near-linear-time 3-approximation for the decision problem on roughly δ-separated convex regions. Finally, we study the setting with Sakoe-Chiba time bands, where we restrict the alignment between the curves, and give polynomial-time algorithms for the upper bound and expected discrete and continuous Fréchet distance for uncertainty modelled as point sets.</p
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