[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81975-en":3,"doc-seo-81975-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},81975,1099514068035,"Ezra","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","(5 + ε)-Approximation of Fréchet Distance in Strongly Subquadratic Time","Randomized (5 + ϵ)-approximation algorithms are presented for both continuous and discrete Fréchet distance between two arbitrary polygonal curves τ and σ in Rd (fixed dimension). The method certifies long boundary-to-boundary reachability using auxiliary surrogate curves, avoiding an extra conversion back to input subcurves and thereby removing an additional triangle-inequality loss. The running time improves via a two-scale macro-surrogate search with dyadic auxiliary-transfer structures; the discrete case gains faster bounds from exact planar reachability in the discrete free-space graph.","arXiv :2607 .06864v 1 [ cs .CG] 7 Jul 2026  \n(5 + ϵ)-Approximation of Fréchet Distance in Strongly Subquadratic  \nTime ∗  \nLenny Liu Jihan Wang  \nJuly 9, 2026  \nAbstract  \nWe give randomized (5 + ϵ)-approximation algorithms for both the continuous and discrete Fréchet distances on arbitrary two polygonal curves τ and σ in Rd for fixed d, with n and rs[vtCeraroHlticgongZ2esritly5],rhsespm fubqwhiecoruchtaivddrgiiesavlycrtie.ec(t7uFn-lgchanapoetprt-fritraohuctximnsormfiaornaptioodos,ntϵx(io4tiu/5osn) FtarilécmgoheretTithruhmnessseoifnboCdn,ϵsg(,ni8pu/ra9one,timthaneed,  \nThe approximation improvement comes from certifying long boundary-to-boundary reachability directly through auxiliary surrogate curves, avoiding an extra conversion back to input subcurves and hence removing one triangle-inequality loss. The running-time improvement comes from a two-scale macro-surrogate search combined with dyadic auxiliary-transfer structures, with the discrete case gaining a faster bound from exact planar reachability in the discrete free-space graph.  \n∗ University of Illinois Urbana-Champaign, Urbana, IL 61801. Email: {hengyu2, [jihanw2}@illinois.edu](jihanw2}@illinois.edu).  \n1 Introduction  \nFréchet distance is a classical measure of similarity between curves. In the standard “man and dog”interpretation, two agents traverse their respective curves from start to finish without backtracking, and the distance is the minimum leash length that permits such a traversal. Because the definition respects the order of points along each curve, the Fréchet distance is well suited to comparing trajectories, paths, and time-series data.  \nLet τ, σ : [0 , 1] → Rd be polygonal curves. A continuous matching between τ and σ is a pair of continuous nondecreasing maps ρ, ϱ : [0, 1] → [0 , 1] satisfying ρ(0) = ϱ(0) = 0 and ρ(1) = ϱ(1) = 1 . The continuous Fréchet distance is  \ndF (τ, σ) = min max ∥τ(ρ(t)) − σ(ϱ(t))∥ .  \nρ,ϱ t∈[0 , 1]  \nFor the discrete variant, write the vertex sequences as τ = (v1 , . . . , v n) and σ = (w1 , . . . , wm) . A discrete matching is a sequence  \nC = ((i1 , j 1 ) , . . . ,(iL , jL ))  \nsuch that (i1 , j 1 ) = (1 , 1) , (iL , jL ) = (n, m), and for every ℓ \u003C L,  \n(i ℓ+1 − i ℓ, j ℓ+1 − j ℓ) ∈ {(1, 0) ,(0 , 1) ,(1 , 1)} .  \nThe discrete Fréchet distance is  \nddF (τ, σ) = mCin (mi,j ∥vi − wj∥ .  \nExact algorithms. Alt and Godau [AG95] initiated the algorithmic study of the continuous Fréchet distance and gave the classical algorithm, which computes the distance between curves of complexities m and n in O (mn log(mn)) time. Eiter and Mannila [EM94] introduced the discrete Fréchet distance and gave an O (mn)-time dynamic programming algorithm. These quadratic-type bounds remain the natural baseline for arbitrary input curves.  \nSeveral works have obtained subpolynomial improvements for exact computation. For the discrete Fréchet distance in the plane, Agarwal, Ben Avraham, Kaplan, and Sharir [AAKS14] gave a word-RAM algorithm running in O (mn log log n/ log n) time, assuming m ≤ n. For the continuous Fréchet distance, Cheng and Huang [CH25] recently gave an exact randomized algorithm in arbitrary fixed dimension with expected running time O(mn(log log n)2+µ log n/ log1+µ m) for some constant µ ∈ (0 , 1) .  \nHardness and lower bounds. Conditional lower bounds suggest that truly subquadratic exact algorithms are unlikely. Bringmann [Bri14], assuming the Strong Exponential Time Hypothesis (SETH) of Impagliazzo and Paturi [IP01], ruled out strongly subquadratic time exact algorithms for both the continuous and the discrete Fréchet distance, already for curves in the plane. His lower bound holds even for imbalanced complexities, so, it is likely no algorithm runs in O((nm)1−γ ) time for any constant γ > 0. The same work also rules out strongly subquadratic time 1.001-approximation. Buchin, Ophelders, and Speckmann [BOS19] raised the inapproximability threshold, under SETH, no strongly subquadratic algorithm approximates the continuous or th","cbCaisf9vbPV6MI8","https://ap.wps.com/l/cbCaisf9vbPV6MI8","pdf",584613,7,1,32,"English","en",105,"# Abstract\n# Introduction\n## Exact algorithms and baselines\n## Hardness and lower bounds\n## Approximation algorithms and prior work\n## Main contributions and runtime improvements","[{\"question\":\"What problem does the paper address?\",\"answer\":\"It addresses computing approximate Fréchet distances between two arbitrary polygonal curves for both the continuous and discrete variants.\"},{\"question\":\"How does the approximation quality improve to (5+ϵ)?\",\"answer\":\"It improves the approximation by certifying long boundary-to-boundary reachability through auxiliary surrogate curves, which avoids an extra conversion step and removes one triangle-inequality loss.\"},{\"question\":\"What technique yields the strongly subquadratic running time?\",\"answer\":\"A two-scale macro-surrogate search is combined with dyadic auxiliary-transfer structures; additionally, the discrete case uses exact planar reachability in the discrete free-space graph for a faster bound.\"}]",1784177364,81,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"5-approximation-of-frechet-distance-in-strongly-subquadratic-time","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/5-approximation-of-frechet-distance-in-strongly-subquadratic-time/81975/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-29","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What problem does the paper address?","Question",{"text":76,"@type":77},"It addresses computing approximate Fréchet distances between two arbitrary polygonal curves for both the continuous and discrete variants.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the approximation quality improve to (5+ϵ)?",{"text":81,"@type":77},"It improves the approximation by certifying long boundary-to-boundary reachability through auxiliary surrogate curves, which avoids an extra conversion step and removes one triangle-inequality loss.",{"name":83,"@type":74,"acceptedAnswer":84},"What technique yields the strongly subquadratic running time?",{"text":85,"@type":77},"A two-scale macro-surrogate search is combined with dyadic auxiliary-transfer structures; 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