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pith:CM6AIW6I

pith:2026:CM6AIW6IWQ6FAN2AFIFA7NDTWZ

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Near-tight Bounds for Computing the Fr\'echet Distance in d-Dimensional Grid Graphs and the Implications for {\lambda}-low Dense Curves

Eva Rotenberg, Frederikke Uldahl, Ivor van der Hoog, Jacobus Conradi

Fréchet distance between n-vertex paths in d-dimensional grids can be (1+ε)-approximated in Õ((n/ε)^{2-2/d} + n) time.

arxiv:2604.24135 v1 · 2026-04-27 · cs.CG · cs.DS

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Claims

C1strongest claim

We give an algorithm, that for two simple paths on n vertices, (1+ε)-approximates the Fréchet distance in time Õ((n/ε)^{2-2/d} +n). We complement this by a near-matching fine-grained lower bound: for constant dimensions d ≥ 3, there is no O((ε^{2/d}(n/ε)^{2-2/d})^{1-δ}) algorithm for any δ>0 unless the Orthogonal Vector Hypothesis fails.

C2weakest assumption

The Orthogonal Vector Hypothesis must hold for the conditional lower bound to apply; the curves must be simple paths on the grid graph.

C3one line summary

Near-tight (1+ε)-approximation algorithms and OVH-based lower bounds are given for Fréchet distance on d-dimensional grid graphs, with tightness results for λ-low dense curves.

References

7 extracted · 7 resolved · 0 Pith anchors

[1] Subtrajectory clustering: Models and algorithms.ACM SIGMOD-SIGACT- SIGAI Symposium on Principles of Database Systems (PODS), pages 75–87, 2018.doi: 10.1145/3196959.3196972 2018 · doi:10.1145/3196959.3196972

[2] 5 Alessandro Bombelli, Lluis Soler, Eric Trumbauer, and Kenneth D Mease 2017 · doi:10.2514/1.g002308

[3] 11 Kevin Buchin, Maike Buchin, Joachim Gudmundsson, Maarten Löffler, and Jun Luo 2011 · doi:10.1145/3423334.3431451

[4] pages 58-74. J. Conradi, I. van der Hoog, F. Uldahl, E. Rotenberg 15 19 Mark de Berg, Marcel Roeloffzen, and Bettina Speckmann. Kinetic compressed quadtrees in the black-box model with applications to 2012 · doi:10.1007/978-3-642-33090-2_34

[5] 21 Anne Driemel, Amer Krivošija, and Christian Sohler 2016 · doi:10.1007/s00454-012-9402-z

Receipt and verification

First computed	2026-05-20T00:00:39.429366Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

133c045bc8b43c5037402a0a0fb473b66387288873c47ec957ee82f0532d0c3f

Aliases

arxiv: 2604.24135 · arxiv_version: 2604.24135v1 · doi: 10.48550/arxiv.2604.24135 · pith_short_12: CM6AIW6IWQ6F · pith_short_16: CM6AIW6IWQ6FAN2A · pith_short_8: CM6AIW6I

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/CM6AIW6IWQ6FAN2AFIFA7NDTWZ \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 133c045bc8b43c5037402a0a0fb473b66387288873c47ec957ee82f0532d0c3f

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "2fc967aea039ca9a9fdf3abbce2d18ca4283a8d7a47e50ec892a01a03f0de749",
    "cross_cats_sorted": [
      "cs.DS"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.CG",
    "submitted_at": "2026-04-27T07:42:51Z",
    "title_canon_sha256": "7476946eccdb7c3c2a2c2797a0bb6a87c3e8f1bcc40612244ba9a7f6076882ff"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.24135",
    "kind": "arxiv",
    "version": 1
  }
}