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arxiv: 1907.04420 · v2 · pith:5FSUMD37new · submitted 2019-07-09 · 💻 cs.CG

Sublinear data structures for short Fr\'echet queries

Anne Driemel , Ioannis Psarros , Melanie Schmidt This is my paper

Pith reviewed 2026-05-24 23:49 UTC · model grok-4.3

classification 💻 cs.CG
keywords Fréchet distancedata structurescurvesapproximationdoubling spacesnearest neighborsstreaming
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The pith

Approximate distance oracles for discrete Fréchet distance on curves can have space independent of input curve length m.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies data structures for curves under discrete Fréchet distance in an asymmetric setting: stored curves have complexity m while query curves have complexity k much smaller than m. It shows that (1+ε)-approximate distance oracles exist whose space depends only on k, ε and constant dimension d rather than total input size N. The oracles answer queries deterministically in O(k²) time and can be maintained dynamically under a streaming input model. The techniques also produce an exponential improvement in the dependence on m for approximate nearest-neighbor structures over sets of curves.

Core claim

There exist deterministic (1+ε)-approximate distance oracles for the discrete Fréchet distance between a stored curve of complexity m and a query curve of complexity k that use space O((k log(ε^{-1}) ε^{-d})^k) in Euclidean R^d, answer queries in O(k²) time, and can be maintained in the stream with polylogarithmic extra memory while supporting queries in O(k⁴ log² m) time.

What carries the argument

The asymmetric distance oracle that stores a compressed version of each input curve whose size depends only on the query complexity k.

If this is right

  • Distance queries can be answered in time O(k²) independent of m.
  • The oracle can be maintained dynamically when the input arrives as a stream, using only polylogarithmic extra memory.
  • Approximate nearest-neighbor data structures for sets of curves achieve an exponential improvement in their dependence on input-curve complexity m.
  • All bounds hold for curves embedded in Euclidean R^d with constant d.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The streaming maintenance result implies that similarity queries can be answered against the prefix of a trajectory seen so far without storing the entire prefix.
  • The same compression technique may be combinable with other sketching methods to handle slowly growing k relative to m.
  • The independence from m suggests that trajectory databases could be indexed once for a fixed query length k and reused across many different stored curves.

Load-bearing premise

Query curves must have complexity k much smaller than the stored curves' complexity m, and the ambient space must be doubling such as constant-dimensional Euclidean space.

What would settle it

An explicit family of input curves of length m for which every (1+ε)-approximate oracle for k-length queries requires space that grows with m would falsify the independence from input size.

Figures

Figures reproduced from arXiv: 1907.04420 by Anne Driemel, Ioannis Psarros, Melanie Schmidt.

Figure 1
Figure 1. Figure 1: Exponential grid construction around point [PITH_FULL_IMAGE:figures/full_fig_p037_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example of a curve X (black), its simplification Xk (orange) and a query curve q (blue) with distance r = ddF (X, q) indicated by the radius of the blue disks. 38 [PITH_FULL_IMAGE:figures/full_fig_p039_2.png] view at source ↗
read the original abstract

We study metric data structures for curves in doubling spaces, such as trajectories of moving objects in Euclidean $\mathbb{R}^d$, where the distance between two curves is measured using the discrete Fr\'echet distance. We design data structures in an \emph{asymmetric} setting where the input is a curve (or a set of $n$ curves) each of complexity $m$ and the queries are with curves of complexity $k\ll m$. We show that there exist approximate data structures that are independent of the input size $N = d \cdot n \cdot m$ and we study how to maintain them dynamically if the input is given in the stream. Concretely, we study two types of data structures: (i) distance oracles, where the task is to store a compressed version of the input curve, which can be used to answer queries for the distance of a query curve to the input curve, and (ii) nearest-neighbor data structures, where the task is to preprocess a set of input curves to answer queries for the input curve closest to the query curve. In both cases we are interested in approximation. For curves embedded in Euclidean $\mathbb{R}^d$ with constant $d$, our distance oracle uses space in $\mathcal{O}((k \log(\epsilon^{-1}) \epsilon^{-d})^k)$ ($\epsilon$ is the precision parameter). The oracle performs $(1+\epsilon)$-approximate queries in time in $\mathcal{O}(k^2)$ and is deterministic. We show how to maintain this distance oracle in the stream using polylogarithmic additional memory. In the stream, we can dynamically answer distance queries to the portion of the stream seen so far in $\mathcal{O}(k^4 \log^2 m)$ time. We apply our techniques to the second problem, approximate near neighbor (ANN) data structures, and achieve an exponential improvement in the dependency on the complexity of the input curves compared to the state of the art.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript claims to construct (1+ε)-approximate distance oracles and nearest-neighbor data structures for the discrete Fréchet distance on curves in Euclidean R^d (d constant). In the asymmetric setting, input curves have complexity m while queries have complexity k ≪ m; the distance oracle uses space O((k log(ε^{-1}) ε^{-d})^k), answers queries in O(k^2) time, is deterministic, and extends to a streaming/dynamic setting with polylog additional memory and O(k^4 log^2 m) query time. The techniques also yield an exponential improvement in the dependence on m for approximate near-neighbor structures relative to prior work.

Significance. If the claimed constructions and bounds hold, the work is significant for metric data structures on trajectories: it achieves space independent of the total input size N = d·n·m, which is essential for large-scale curve datasets. The deterministic guarantee, the polylog-space streaming maintenance, and the exponential improvement for ANN are concrete strengths that advance the state of the art in asymmetric Fréchet query problems.

minor comments (2)
  1. [§1] §1 (Introduction): the definition of the discrete Fréchet distance is referenced but not restated; adding the standard recursive definition would improve readability for readers outside computational geometry.
  2. [Abstract / Theorem 1] The space bound O((k log(ε^{-1}) ε^{-d})^k) is stated for constant d; the dependence on d should be made explicit in the theorem statement even if d is treated as constant.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper presents algorithmic constructions for asymmetric distance oracles and ANN data structures under discrete Fréchet distance in doubling spaces. Space bounds O((k log(ε^{-1}) ε^{-d})^k) and query times are obtained by direct analysis of the proposed compressed representations and dynamic maintenance procedures in the streaming model. These are explicit algorithmic results depending on input parameters k, ε, d and m, with no fitted parameters, self-definitional reductions, or load-bearing self-citations that collapse the claimed bounds back to the inputs by construction. The derivation chain consists of standard algorithmic analysis and is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The abstract relies on standard properties of doubling metric spaces and the definition of discrete Fréchet distance; no explicit free parameters beyond the user-chosen approximation ε are named.

free parameters (1)
  • epsilon
    User-chosen approximation parameter appearing in the space bound.
axioms (1)
  • domain assumption Ambient space is a doubling metric space (explicitly Euclidean R^d with constant d)
    Invoked to obtain the concrete space bound O((k log(ε^{-1}) ε^{-d})^k).

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Reference graph

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