[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86502-en":3,"doc-seo-86502-105":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},86502,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","Deterministic Online Embedding of Metric Spaces into Low Dimensional Spaces","Deterministic online embedding studies how metric space points are mapped, one-by-one, into Euclidean space of constant dimension d>1 under an adaptive adversary while preserving distances with low distortion. For d=1 the behavior is well understood, but higher dimensions remain unclear, including the planar (d=2) growth rate in the worst case. The work proves polynomial distortion for fixed solid-graph metrics (e.g., K5) into R2, refuting an exponential lower-bound conjecture. It also shows that specific tree-like metrics (ultrametrics, HSTs) admit online embeddings with worst-case distortion comparable to offline: nΘ(1/d), even for d=Θ(log n), enabling near-optimal transfers from probabilistic HST results to low-dimensional Euclidean embeddings.","arXiv :2607 . 10624v 1 [ cs .DS] 12 Jul 2026  \nDeterministic Online Embedding of Metric Spaces into Low  \nDimensional Spaces  \nNoam Licht∗ Ilan Newman† Yuri Rabinovich‡  \nJuly 14, 2026  \nAbstract  \nWe study online embeddings of metric spaces into Euclidean spaces of a constant dimension d > 1, against an adaptive adversary. While the case of d = 1 is well understood, for higher dimensions little is known. In particular, even for d = 2 it remains unknown whether the worst-case distortion grows exponentially with the number of exposed points, as it does in the case for the line, or whether it is polynomial, as in the case for unbounded d.  \nOur first result is about fixed solid graphs, i.e. , K5 , whose edges are solid intervals, equipped with the shortest-path metric. We show that if the input points arrive from such a metric space, they can indeed be online-embedded into R2 with a polynomial distortion. This refutesthe previously believed conjecture that the topological non-embeddability of K5 into the plane could be exploited for establishing exponential lower bounds.  \nThe second results is about online embeddings of tree metrics of a certain type, including, e.g., ultrametrics and HST’s. Somewhat surprisingly, we show that for metrics from this class the worst-case online embedding into Rd is not much worse that the offline embedding, both being nΘ(1/d), and this holds even when d = Θ(log n) . This is in a stark contrast to the more common situation where the online-offline gap is typically huge, and even exponential. This result allows us to transfer results about probabilistic embeddings of metrics into HST’s to low-dimensional Euclidean spaces, in an almost optimal possible manner.  \n1 Introduction  \nThe modern theory of low-distortion embeddings of finite metric spaces began to take shape with the appearance of classical results of Johnson and Lindenstrauss [8]1 and Bourgain [6]2 , in the last decades of the 20’th century. It soon became clear that this theory provides powerful tools for numerous theoretical and practical algorithmic problems. Today, it has evolved into a well-established discipline employing advanced mathematical concepts, and playing a crucial role in solving complex algorithmic problems across a variety of domains.  \n∗ Dept. of CS, University of Haifa, Israel  \n†University of Haifa, Israel. Email: [ilan@cs.haifa.ac.il](ilan@cs.haifa.ac.il).  \n‡Dept. of CS, University of Haifa, Israel, Email: [yuri@cs.haifa.ac.il](yuri@cs.haifa.ac.il)  \n1Any n-point Euclidean metric can be efficiently embedded into ℓlo2g n/ϵ2 with (1 + ϵ)-distortion. 2Any n-point metric can be efficiently embedded into Euclidean space with distortion O(log n) .  \nThe study of online metric embedding is a relatively recent development. In this setting, data points are presented one by one, and decisions must be made as they arrive. The goal, as before, is to preserve the distances or metric properties as much as possible. While the online setting is better suited for needs of modern real-time data processing, achieving good guarantees on metric distortion in this setting becomes significantly harder.  \nThe first publication explicitly dedicated to online embeddings was [7] from 2010 . One of the key observations of this paper was that a large part of Bartal’s probabilistic offline embedding procedure into Hierarchically Well Separated Trees (henceforth, HST’s) of [2] can be implemented online. This line of research, continued in [3], and strengthened in [5], yields, e.g., that: Any metric space (X, d) can be probabilistically online embedded into a distribution of a non-contracting ultrametrics (and hence tree-metrics) with dilation O (log q · log min(q,∆)) . Here q is the number of exposed points, and it does not need to be known in advance. ∆ stands for the aspect ratio of (X, d), i.e., the ratio between the largest and the smallest distances therein. The result is essentially tight.  \nImportantly, in this probabilistic setting one c","cbCaitwCUkz2efBv","https://ap.wps.com/l/cbCaitwCUkz2efBv","pdf",1331003,5,1,34,"English","en",105,"# Introduction\n## Background on low-distortion embeddings\n## Online metric embedding setting\n## Probabilistic vs deterministic adversaries\n## Paper focus and results overview","[{\"question\":\"What adversary model is considered in deterministic online embeddings here?\",\"answer\":\"The adversary is adaptive: the input metric and the choice of the next exposed point can depend on the embedding produced so far.\"},{\"question\":\"What is proven for fixed solid graphs such as K5 with the shortest-path metric?\",\"answer\":\"Points arriving from such a metric space can be embedded online into R2 with polynomial distortion.\"},{\"question\":\"How does the paper’s result for tree metrics (ultrametrics and HSTs) compare online vs offline?\",\"answer\":\"For this class, the worst-case online embedding distortion is not much worse than the offline one, scaling as nΘ(1/d), including even when d=Θ(log n).\"}]",1784212237,86,{"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},"deterministic-online-embedding-of-metric-spaces-into-low-dimensional-spaces","",{"@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/deterministic-online-embedding-of-metric-spaces-into-low-dimensional-spaces/86502/",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","2026-07-27","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 adversary model is considered in deterministic online embeddings here?","Question",{"text":76,"@type":77},"The adversary is adaptive: the input metric and the choice of the next exposed point can depend on the embedding produced so far.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What is proven for fixed solid graphs such as K5 with the shortest-path metric?",{"text":81,"@type":77},"Points arriving from such a metric space can be embedded online into R2 with polynomial distortion.",{"name":83,"@type":74,"acceptedAnswer":84},"How does the paper’s result for tree metrics (ultrametrics and HSTs) compare online vs offline?",{"text":85,"@type":77},"For this class, the worst-case online embedding distortion is not much worse than the offline one, scaling as nΘ(1/d), including even when d=Θ(log 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