ANN for Time Series Under the Fréchet Distance

Static and Streaming Data Structures for Fréchet Distance Queries

“…Lower bound. Driemel and Psarros [DP20] proved a cell probe lower bound for decision distance oracle, providing evidence that our Theorem 1 might be tight. In the cell probe model, one construct a data structure which is divided into cells of size w. Given a query, one can probe some cells of the data structure and preform unbounded local computation.…”

“…Consider a cell probe distance oracle O for curves in R d where d = Θ(log m), that has word size w < m λ , and provide answers for queries of length k < m γ , with approximation factor < 3 /2, while using only constant number of probes. Driemel and Psarros [DP20] showed that O must use space 2 Ω(kd) .…”

Tight Bounds for Approximate Near Neighbor Searching for Time Series under the Fréchet Distance

“…Recently, Driemel and Psarros [DP21] obtained bounds for the continuous Fréchet distance that are similar to the bounds of Filtser et al, albeit at the expense of a higher approximation factor and only for curves in R. They present a (5 + ε)-ANN data structure which uses space in n • O 1 ε k + O(nm) and has query time in O (k), and a (2 + ε)-ANN data structure, which uses space in n • O m kε k + O(nm) and has query time in O k • 2 k . Even more efficient data structures can be obtained at the expense of an even larger approximation factor, see the work of Driemel, Silvestri, and Psarros [DS17] and [DP21] which uses locality-sensitive hashing. In these results neither the space nor the query time is exponential in the complexity of the curves (neither input nor query), but the approximation factor is linear in the query complexity k.…”

“…We review some efforts in answering this question and discuss the limitations of the current techniques. Driemel and Psarros [DP21,DP20] approach this question using a technique by Miltersen [Mil94] for proving cell-probe lower bounds. Their results indicate that any data structure answering a query for a near neighbor under the continuous Fréchet distance by using only a constant number of probes to memory cells cannot have a space usage that is independent of the arclength of the input curves (assuming a query radius of 1).…”

“…During query time, we apply a simple transformation to the query curve (such as rounding vertices to a scaled integer grid) and look up the answer in the dictionary. Filtser et al [FFK20] used this technique for the discrete Fréchet distance and Driemel and Psarros [DP21] showed that it can also be applied for the continuous Fréchet distance of one-dimensional curves. A particular challenge that appears in the continuous case is that the doubling dimension can be unbounded, even if the complexity of the curves is small.…”