{"paper":{"title":"Near-tight Bounds for Computing the Fr\\'echet Distance in d-Dimensional Grid Graphs and the Implications for {\\lambda}-low Dense Curves","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Fréchet distance between n-vertex paths in d-dimensional grids can be (1+ε)-approximated in Õ((n/ε)^{2-2/d} + n) time.","cross_cats":["cs.DS"],"primary_cat":"cs.CG","authors_text":"Eva Rotenberg, Frederikke Uldahl, Ivor van der Hoog, Jacobus Conradi","submitted_at":"2026-04-27T07:42:51Z","abstract_excerpt":"The Fr\\'echet distance is a popular distance measure between trajectories or curves in space, or between walks in graphs. We study computing the Fr\\'echet distance between walks in the $d$-dimensional grid graphs, i.e. $\\mathbb{Z}^d$ where points share an edge if they differ by one in one coordinate.\n  We give an algorithm, that for two simple paths on $n$ vertices, $(1+\\varepsilon)$-approximates the Fr\\'echet distance in time $\\widetilde{O}((\\frac{n}{\\varepsilon})^{2-2/d} +n)$. We complement this by a near-matching fine-grained lower bound: for constant dimensions $d \\geq 3$, there is no $O(("},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The Orthogonal Vector Hypothesis must hold for the conditional lower bound to apply; the curves must be simple paths on the grid graph.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Fréchet distance between n-vertex paths in d-dimensional grids can be (1+ε)-approximated in Õ((n/ε)^{2-2/d} + n) time.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c698842565fe63189aac2fa88bc2bf063bd63b9a347e9e784597377b3a85d0dc"},"source":{"id":"2604.24135","kind":"arxiv","version":1},"verdict":{"id":"0a38e0fb-5494-4656-bda0-9704170eab89","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T17:25:31.400822Z","strongest_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.","one_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.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The Orthogonal Vector Hypothesis must hold for the conditional lower bound to apply; the curves must be simple paths on the grid graph.","pith_extraction_headline":"Fréchet distance between n-vertex paths in d-dimensional grids can be (1+ε)-approximated in Õ((n/ε)^{2-2/d} + n) time."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.24135/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T22:23:56.680699Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"811d1f37b28c115c90863647a10cb614f6c313e569a28b59c87818519618c573"},"references":{"count":7,"sample":[{"doi":"10.1145/3196959.3196972","year":2018,"title":"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","work_id":"f5788104-6928-4331-89b7-5911b5f9757e","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.2514/1.g002308","year":2017,"title":"5 Alessandro Bombelli, Lluis Soler, Eric Trumbauer, and Kenneth D Mease","work_id":"07bad610-b381-4624-b78d-43986dcc0b46","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1145/3423334.3431451","year":2011,"title":"11 Kevin Buchin, Maike Buchin, Joachim Gudmundsson, Maarten Löffler, and Jun Luo","work_id":"9833ac29-5054-4d73-b4ad-774d8787ff90","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/978-3-642-33090-2_34","year":2012,"title":"pages 58-74. 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