eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
10:1
10:14
10.4230/LIPIcs.SoCG.2024.10
article
Discrete Fréchet Distance Oracles
Aronov, Boris
1
https://orcid.org/0000-0003-3110-4702
Farhana, Tsuri
2
Katz, Matthew J.
2
https://orcid.org/0000-0002-0672-729X
Ramesh, Indu
1
https://orcid.org/0009-0008-9967-0819
Department of Computer Science and Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, USA
Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel
It is unlikely that the discrete Fréchet distance between two curves of length n can be computed in strictly subquadratic time. We thus consider the setting where one of the curves, P, is known in advance. In particular, we wish to construct data structures (distance oracles) of near-linear size that support efficient distance queries with respect to P in sublinear time. Since there is evidence that this is impossible for query curves of length Θ(n^α), for any α > 0, we focus on query curves of (small) constant length, for which we are able to devise distance oracles with the desired bounds.
We extend our tools to handle subcurves of the given curve, and even arbitrary vertex-to-vertex subcurves of a given geometric tree. That is, we construct an oracle that can quickly compute the distance between a short polygonal path (the query) and a path in the preprocessed tree between two query-specified vertices. Moreover, we define a new family of geometric graphs, t-local graphs (which strictly contains the family of geometric spanners with constant stretch), for which a similar oracle exists: we can preprocess a graph G in the family, so that, given a query segment and a pair u,v of vertices in G, one can quickly compute the smallest discrete Fréchet distance between the segment and any (u,v)-path in G. The answer is exact, if t = 1, and approximate if t > 1.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.10/LIPIcs.SoCG.2024.10.pdf
discrete Fréchet distance
distance oracle
heavy-path decomposition
t-local graphs