[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85800-en":3,"doc-seo-85800-105":30,"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":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},85800,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","Bichromatic Geometric Spanners","Bichromatic geometric spanners study edge-weighted graphs derived from red–blue point sets in the plane, where stretch controls how well a subgraph preserves all pairwise shortest-path distances. For the complete bipartite graph K(R,B), general lower bounds rule out stretch t\u003C3 with subquadratic edges. A new construction achieves a (3+ε)-spanner with O(sqrt(1/ε)·n) edges, removing an open O(log n) factor for decades and clarifying tradeoffs with ε.","arXiv :2607 . 10062v1 [ cs .CG] 11 Jul 2026  \nBichromatic Geometric Spanners ∗  \nTheodore Fung† Csaba D. T´oth‡  \nAbstract  \nFor an edge-weighted graph G = (V, E) and a stretch parameter t ≥ 1, a t-spanner is asubgraph H ⊆ G such that the shortest path distances in G and H satisfy δH (u, v) ≤ tδG (u, v) for all u, v ∈ V. In metric spanners, V is a finite metric space, and G is the complete graph with edge weights corresponding to the distances between the endpoints. When G is the complete graph on n points in the plane, O (n)-size t-spanners are possible for any t > 1: For every ε > 0, there is an (1 + ε)-spanner with O (n/ε) edges (i.e., the stretch can be arbitrarily close to 1) .  \nWhen G = K (R, B) is the complete bipartite graph on n bichromatic points in the plane, in general, no spanner construction can guarantee stretch t \u003C 3 with o (n2 ) edges. Bose et al. (SICOMP 2009) constructed a (3 + ε)-spanner with O(nlog n) edges for any constant ε > 0. Our main result is a new construction for a (3+ε)-spanner with O( p 1/ε·n) edges. Eliminating the O(log n) factor resolves a problem left open for more than 17 years, and raises a new research problem about optimizing the dependence on ε . We also study spanners for G = K (R, B) on n bichromatic points on the real line: In this case, we show that the MST of K (R, B) is a 7-spanner, and we construct a 3-spanner with at most 2n − 3 edges.  \n1 Introduction  \nA t-spanner for an edge-weighted graph G = (S, E) is a subgraph H = (S, E′) that contains, for every edge ab ∈ E, an ab-path of weight at most t · w(ab) . Let S be a set of n bichromatic points in Euclidean plane, with coloring c : S → {red, blue} . Let R and B, resp., denote the set of red and blue points in S, and let K (R, B) denote the complete bipartite graph on the partite sets Rand B. The weight of an edge ab is the Euclidean length |ab| of the line segment ab.  \nFor a set S of n (uncolored) points in the plane and any ε > 0, there exists a (1 + ε)-spanner with O(n/ε) edges [Cla87, Kei88] . Bose et al. [BCC+09a] noted that for a complete bipartite graph K (R, B), in general, one cannot find a t-spanner with o(n2 ) edges and stretch t \u003C 3. Specifically, for every ε > 0, one can arrange ⌈n/2⌉ red and ⌊n/2⌋ blue points in two clusters of diameter ~~ε~~3 at unit distance apart: If a subgraph H ⊆ K (R, B) misses any bichromatic edge ab, then every ab-path in H has length at least (3 − ε)|ab| . On the positive side, Bose et al. [BCC+09a] constructed, for every bichromatic set S = R ∪ B ⊂ R2 and constant ε > 0, an (5 + ε)-spanner with O (n) edgesand a (3 + ε)-spanner with O (nlog n) edges. They did not analyze the dependency on the stretch parameter ε . The main problem left open in their work is whether a bichromatic (3 + ε)-spanner with O (n) edges always exists. Our main result settles this problem in the affirmative.  \nTheorem 1 . For every ε >  0 and every set of n bichromatic points in the plane, there exists a (3 + ε)-spanner with O ( p 1/ε · n) edges, which can be constructed in O( p 1/ε · nlog n) time.  \n∗ Research supported, in part, by the NSF award DMS-2154347 .  \n†Department of Mathematics, California State University Northridge, Los Angeles, CA, USA.  \n‡Department of Mathematics, California State University Northridge, Los Angeles, CA; and Department of Computer Science, Tufts University, Medford, MA, USA.  \nWhile we settle the problem posed by Bose et al. [BCC+09a], our result raises a new problem to determine the optimal tradeoff between ε > 0 and the size of a bichromatic (3 + ε)-spanner in the plane. Theorem 1 generalizes to multichromatic point sets in Euclidean d-space for constant dimension d ≥ 2, and yields a (3 + ε)-spanner with O (n/ε(d−1)/2) edges. We describe the spanner construction and analyze it in detail in the plane, and then briefly sketch the generalization to higher dimensions.  \nWe also study spanners for K (R, B) in the real line. The lower bound construction by Bose et al. [BCC+09a] can be r","cbCaif9oolpWP3XM","https://ap.wps.com/l/cbCaif9oolpWP3XM","pdf",728172,4,1,19,"English","en",105,"# Abstract\n# Introduction\n## Definitions and model\n## Prior work and open problem\n## Main theorem and results\n## Comparison to previous work and technical highlights","[{\"question\":\"什么是 t-spanner，它如何度量“保真度”？\",\"answer\":\"t-spanner 是原图 G 的子图 H，使得任意两点 u,v 的最短路距离在 H 中不超过 t 倍的原图距离。也可等价为：对每条原图边都存在一条在 H 中的路径，其权重不超过 t 乘该边权重。\"},{\"question\":\"在平面上的二色点情形，已知的主要下界结论是什么？\",\"answer\":\"对完全二分图 K(R,B)，一般不能在保持 stretch t\\u003c3 的同时用 o(n^2) 条边完成稀疏化。文中给出通过在两团簇之间布置点来实现的下界直观。\"},{\"question\":\"本文的核心改进结果带来了什么提升？\",\"answer\":\"给出新的构造：对任意 ε\\u003e0，任意平面二色点集都存在 (3+ε)-spanner，其边数为 O(sqrt(1/ε)·n)，并可在 O(sqrt(1/ε)·n log n) 时间构造。该结果消除了先前工作中遗留的 O(log n) 因子，并提出新的 ε-依赖最优化研究问题。\"}]",1784206351,48,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":28},"bichromatic-geometric-spanners","",{"@graph":36,"@context":85},[37,53,68],{"@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":20},"https://docshare.wps.com/document/bichromatic-geometric-spanners/85800/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-25","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},"什么是 t-spanner，它如何度量“保真度”？","Question",{"text":75,"@type":76},"t-spanner 是原图 G 的子图 H，使得任意两点 u,v 的最短路距离在 H 中不超过 t 倍的原图距离。也可等价为：对每条原图边都存在一条在 H 中的路径，其权重不超过 t 乘该边权重。","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"在平面上的二色点情形，已知的主要下界结论是什么？",{"text":80,"@type":76},"对完全二分图 K(R,B)，一般不能在保持 stretch t\u003C3 的同时用 o(n^2) 条边完成稀疏化。文中给出通过在两团簇之间布置点来实现的下界直观。",{"name":82,"@type":73,"acceptedAnswer":83},"本文的核心改进结果带来了什么提升？",{"text":84,"@type":76},"给出新的构造：对任意 ε>0，任意平面二色点集都存在 (3+ε)-spanner，其边数为 O(sqrt(1/ε)·n)，并可在 O(sqrt(1/ε)·n log n) 时间构造。该结果消除了先前工作中遗留的 O(log n) 因子，并提出新的 ε-依赖最优化研究问题。","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":22,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},"General","general"]