TY - GEN

T1 - Discrete Fréchet Distance Oracles

AU - Aronov, Boris

AU - Farhana, Tsuri

AU - Katz, Matthew J.

AU - Ramesh, Indu

N1 - Publisher Copyright:
© Boris Aronov, Tsuri Farhana, Matthew J. Katz, and Indu Ramesh.

PY - 2024/6/1

Y1 - 2024/6/1

N2 - 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(nα), 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.

AB - 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(nα), 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.

KW - discrete Fréchet distance

KW - distance oracle

KW - heavy-path decomposition

KW - t-local graphs

UR - http://www.scopus.com/inward/record.url?scp=85191971249&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SoCG.2024.10

DO - 10.4230/LIPIcs.SoCG.2024.10

M3 - Conference contribution

AN - SCOPUS:85191971249

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 40th International Symposium on Computational Geometry, SoCG 2024

A2 - Mulzer, Wolfgang

A2 - Phillips, Jeff M.

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 40th International Symposium on Computational Geometry, SoCG 2024

Y2 - 11 June 2024 through 14 June 2024

ER -