[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85758-en":3,"doc-seo-85758-105":29,"detail-sidebar-cat-0-en-105":91},{"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85758,2336464648322,"Aria","https://ap-avatar.wpscdn.com/avatar/2200025388227c56fec?_k=1778556882303663488",8,"Research & Report","The Quick Dog Jumps the Log","We present linear-time, optimal (1+ε)-approximation algorithms for multiple variants of the Fréchet distance between c-packed curves, with c in O(1). The work removes an extra logarithmic factor present in prior methods by building a linear-size approximation of the elevation function using a rectangle decomposition of the domain and an implicit dynamic programming over it. The algorithms extend to strong, weak, discrete, and continuous Fréchet distances with running time about O(cn/ε).","arXiv :2607 .09917v1 [ cs .CG] 10 Jul 2026  \nThe Quick Dog Jumps the Log  \nLotte Blank∗ Anne Driemel† Sariel Har-Peled‡ Marena Richter§  \nJuly 14, 2026  \nAbstract  \nWe give linear-time, and thus optimal,(1+ε)-approximation algorithms for numerous variants of the Fr´echet distance between c-packed curves (where c ∈ O(1)), removing an additional log factor that was present in previous algorithms. The key to our new algorithms is a linear-size approximation of the elevation function, which uses a decomposition of the domain into rectangles, and a careful implicit dynamic programming on this decomposition. The algorithm extends to the strong, weak, discrete, and continuous Fr´echet distances with a running time of roughly O (cn/ε) . The c-packedness assumption is used only in the analysis, and the algorithm is simple and should work efficiently for other inputs.  \n1. Introduction  \nIn his seminal work, Fr´echet [] defined a distance measure between curves in Euclidean R3 (in the same paper, he introduced the notion of metric spaces) . Alt and Godau [] provided an algorithm for computing this distance between polygonal curves, and it is widely used as a standard metric between curves [] . Alt and Godau [] gave the following analogy for the Fr´echet distance: Imagine walking a dog with a fixed-length leash, where the owner is walking along one curve, and the dog is walking along the other curve. Both have to walk along their respective curve from the beginning to the end while coordinating to stay close together. The length of the shortest possible leash that permits such a walk is the Fr´echet distance between the two curves.  \nAlt and Godau [] showed how to compute the distance measure for polygonal curves with a total of n vertices in O(n2 log n) time by considering a path-finding problem in the sublevelset of the distance function defined on the joint parametric space of the two curves. For this, the critical threshold value when such a path exists is the Fr´echet distance, and the algorithm performs a parametric search for this value. Most follow-up work uses a similar framework of performing an implicit or explicit binary search over the set of critical values, thereby retaining a logarithmic factor in the running time of the decision algorithm. Alternatively, one can compute the distance function over the parametric space explicitly and propagate through this space the  \n∗ University of Bonn. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)– 459420781 (FOR AlgoForGe) .  \n†University of Bonn and Lamarr Institute for Machine Learning and Artificial Intelligence.  \n‡School of Computing and Data Science; University of Illinois; 201 N. Goodwin Avenue; Urbana, IL, 61801, USA; ;  .  \n§ University of Bonn. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)– 459420781 (FOR AlgoForGe) .  \nminimal elevation (the leash length) required to reach that point. Following this approach, one can obtain a (1 + ε)-approximation algorithm that runs in O (n2 log ~~1~~ε) time Buchin et al. [], thus removing the logarithmic factor in n.  \nTo avoid the quadratic running time, various approximation algorithms were suggested for special cases of curves, such as approximate shortest paths and locally bounded curves [ , ] . The authors of Driemel et al. Driemel et al. [, ] introduced the notion of packedness: a curve is c-packed if its length inside any ball is at most c times the radius of the ball. They showed that a (1 + ε)-approximation to the Fr´echet distance can be computed in O (cn/ε + cn log n) time for c-packed curves. This was later improved to O ( ~~ ~~ log ~~1~~ε + cn log n) by Bringmann and K¨unnemann [] which is (essentially) tight in the parameters c and ε, assuming SETH (i.e., strong exponential time hypothesis) .  \nOur work revisits the setting of c-packed curves and aims to remove the logarithmic factor in n from the running time. The resulting running time is linear in n and t","cbCaioPopbtH7S0w","https://ap.wps.com/l/cbCaioPopbtH7S0w","pdf",841357,1,31,"English","en",105,"# Abstract\n# Introduction\n## Related work","[{\"question\":\"What problem does the document address?\",\"answer\":\"It develops linear-time (1+ε)-approximation algorithms for computing variants of the Fréchet distance between c-packed curves.\"},{\"question\":\"How do the proposed algorithms achieve linear dependence on n?\",\"answer\":\"They approximate the elevation function with a linear-size representation derived from a rectangle decomposition and solve the path-finding via careful implicit dynamic programming on that decomposition.\"},{\"question\":\"Which variants of the Fréchet distance are covered?\",\"answer\":\"The methods extend to strong, weak, discrete, and continuous Fréchet distances, with running time roughly O(cn/ε) for c-packed curves.\"}]",1784206054,78,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"the-quick-dog-jumps-the-log","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/the-quick-dog-jumps-the-log/85758/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the document address?","Question",{"text":75,"@type":76},"It develops linear-time (1+ε)-approximation algorithms for computing variants of the Fréchet distance between c-packed curves.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do the proposed algorithms achieve linear dependence on n?",{"text":80,"@type":76},"They approximate the elevation function with a linear-size representation derived from a rectangle decomposition and solve the path-finding via careful implicit dynamic programming on that decomposition.",{"name":82,"@type":73,"acceptedAnswer":83},"Which variants of the Fréchet distance are covered?",{"text":84,"@type":76},"The methods extend to strong, weak, discrete, and continuous Fréchet distances, with running time roughly O(cn/ε) for c-packed curves.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]