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Discrete Fréchet Distance Oracles

Boris Aronov, Tsuri Farhana, Matthew J. Katz, Indu Ramesh

TL;DR

This work studies exact and approximate distance oracles for the discrete Fréchet distance when a fixed polygonal curve $P$ is preprocessed. It develops near-linear space data structures that support sublinear-time queries for constant-size query curves $Q$, and extends to subcurves of $P$ and to vertex-to-vertex subpaths in trees and certain geometric graphs. By combining heavy-path decompositions, per-path curve oracles, and a black-box framework for polygonal curves, the authors obtain tight bounds for various query sizes: $O(\log^2 n)$ for $k=2$, $O(\log^3 n)$ for $k=3$, and $O^*(\sqrt{n})$ for $k=4$ on curves, with similar sublinear performance for trees and 1-local graphs. The results advance map-matching and curve-similarity tasks by enabling fast, exact or approximate Fréchet-distance queries against a fixed geometric structure with provably near-linear preprocessing.

Abstract

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 $t>1$.

Discrete Fréchet Distance Oracles

TL;DR

This work studies exact and approximate distance oracles for the discrete Fréchet distance when a fixed polygonal curve is preprocessed. It develops near-linear space data structures that support sublinear-time queries for constant-size query curves , and extends to subcurves of and to vertex-to-vertex subpaths in trees and certain geometric graphs. By combining heavy-path decompositions, per-path curve oracles, and a black-box framework for polygonal curves, the authors obtain tight bounds for various query sizes: for , for , and for on curves, with similar sublinear performance for trees and 1-local graphs. The results advance map-matching and curve-similarity tasks by enabling fast, exact or approximate Fréchet-distance queries against a fixed geometric structure with provably near-linear preprocessing.

Abstract

It is unlikely that the discrete Fréchet distance between two curves of length can be computed in strictly subquadratic time. We thus consider the setting where one of the curves, , 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 in sublinear time. Since there is evidence that this is impossible for query curves of length , for any , 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, -local graphs (which strictly contains the family of geometric spanners with constant stretch), for which a similar oracle exists: we can preprocess a graph in the family, so that, given a query segment and a pair of vertices in , one can quickly compute the smallest discrete Fréchet distance between the segment and any -path in . The answer is exact, if , and approximate if .
Paper Structure (8 sections, 8 theorems, 3 equations, 2 figures)

This paper contains 8 sections, 8 theorems, 3 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Distance oracles for trees
  3. Data structures
  4. The Algorithm
  5. Distance oracles for local graphs
  6. Black box revealed: Distance oracles for curves
  7. k=2
  8. k=3

Key Result

Theorem 1

For a geometric tree on $n$ vertices, one can construct discrete Fréchet distance oracles with query times $O(\log^3 n)$, $O(\log^3 n)$, $O(\log^4 n)$, and $O^*(\sqrt{n})$ for query sizes one, two, three, and four, respectively. The structures require $O(n\log n)$ space for query sizes up to three a

Figures (2)

  • Figure 1: Distance oracles for curves. Decision problem answers questions of the form: "given $Q$ and $r$, is $d_{\textnormal{dF}}(P,Q)\leq r$?" Optimization problem computes the discrete Fréchet distance. We also offer a variant where at query time one can restrict the query to an arbitrary vertex-to-vertex subcurve of $P$.
  • Figure 2: The heavy-path decomposition. The heavy paths are drawn in bold; the path $\Pi_{uv}$ is the concatenation of the subpaths $P_1,\ldots,P_4$.

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 3
  • Claim 5
  • Theorem 6
  • Theorem 7
  • Corollary 8
  • Lemma 10
  • Theorem 11
  • Claim 12: Feasibility Test
  • Lemma 14