[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81641-en":3,"doc-seo-81641-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"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},81641,8796095360427,"Lucas Martin","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","A dynamic (1 + ε)-spanner for disk intersection graphs","Maintaining an approximately distance-preserving sparse subgraph is addressed for disk intersection graphs under dynamic disk updates. All disks are restricted to diameters within a known interval [4, Ψ]. The work constructs a (1 + ε)-spanner whose size is O(nε−2 log Ψ log(ε−1)) and maintains it via persistent data structures. It uses O(ε−2n log4 n log Ψ) space with expected amortised update time O((Ψ/ε)^2 log4 n log2 Ψ log2(ε−1)). For ε\u003C1, the spanner also supports connectivity, improving dynamic connectivity bounds and extending results to d-dimensional hypercubes.","arXiv :2604 .25397v2 [ cs .CG] 10 Jul 2026  \nA dynamic (1 + ε)-spanner for disk intersection graphs  \nSarita de Berg \\#   \nIT University of Copenhagen, Denmark Ivor van der Hoog \\#   \nIT University of Copenhagen, Denmark Eva Rotenberg \\#   \nIT University of Copenhagen, Denmark Johanne Müller Vistisen \\#  IT University of Copenhagen, Denmark Sampson Wong \\#   \nUniversity of Copenhagen, Denmark  \n~~ Abstract ~~  \nWe maintain a (1 + ε)-spanner over the disk intersection graph of a dynamic set of disks. We restrict all disks to have their diameter in [4 , Ψ] for some fixed and known Ψ . The resulting (1 + ε)-spanner has size O(nε−2 log Ψ log(ε−1)), where n is the present number of disks.  \nWe develop a novel use of persistent data structures to dynamically maintain our (1 + ε)-spanner. Our approach requires O(ε−2n log4 n log Ψ) space and has an O( 􀀀 Ψε􀀁 2 log4 n log2 Ψ log2 (ε−1)) expected amortised update time. For constant ε and Ψ, this spanner has near-linear size, uses near-linear space and has polylogarithmic update time. Furthermore, we observe that for any ε \u003C 1 , our spanner also serves as a connectivity data structure. With a slight adaptation of our techniques, this leads to better bounds for dynamically supporting connectivity queries in a disk intersection graph. In particular, we improve the space usage when compared to the dynamic data structure of (Baumann et al., DCG’24), replacing the linear dependency on Ψ by a polylogarithmic dependency. Finally, we generalise our results to d-dimensional hypercubes.  \n2012 ACM Subject Classification Theory of computation → Computational geometry Keywords and phrases intersection graphs, dynamic data structures, spanners  \nFunding This research was supported by Danmarks Frie Forskningsfond Case no. 10.46540/3160- 00020B, the VILLUM Foundation grant (VIL37507) “Efficient Recomputations for Changeful Problems”, and the European Union’s Marie Skłodowska-Curie Actions Postdoctoral Fellowship Project number No. 101146276.  \n2 A dynamic (1 + ε)-spanner for disk intersection graphs  \n 1  Introduction  \nGiven a set of n geometric shapes, e.g. disks, the corresponding intersection graph has the shapes as its vertex set, and an edge between two shapes whenever they intersect (see Figure 1) . Disk intersection graphs are often used as a model for wireless ad-hoc networks [14,29,33], in particular, the radius of a disk equals the transmission range of the device. One of the central algorithmic challenges of disk intersection graphs is that, with n vertices, the graph may have Ω(n2 ) edges. Nonetheless, there are many problems that can be solved in subquadratic time in disk graphs [5,9,25,27,42,43,47] . Both weighted [5,23,32,34] and unweighted [9,11,13,47] disk intersection graphs have been studied; we will consider weighted disk intersection graphs where the weight of an edge is the Euclidean distance between the centres of the disks. We study two dynamic problems: maintaining a spanner for the disk intersection graph or a data structure for connectivity queries (for the latter, the edge weights are irrelevant) .  \nGeometric spanners. Given a graph, a spanner is a subgraph that approximately preserves the distances of the original graph. Formally, a (1+ε)-spanner for a graph G is a subgraph G′ such that for every pair of vertices u, v ∈ G, we have that dG′ (u, v) ≤ (1 + ε) · dG (u, v), where dX (·, ·) denotes the shortest path distance in X . Throughout the paper, we assume that ε isan arbitrary fixed constant with 0 \u003C ε \u003C 1. In computational geometry, there are many works that consider spanners of point sets, where the graph G is the complete graph over the point set and edge weights correspond to Euclidean distances. Results for such Euclidean spanners of point sets include a variety of (1 + ε)-spanners of size O(n/εd−1) [4,6,15,26,37,48,55] . Fora thorough treatment of spanners for Euclidean point sets, we reference the textbook [50] . Spanners are also well studied in other settings, s","cbCaiv3oNvAzQDbp","https://ap.wps.com/l/cbCaiv3oNvAzQDbp","pdf",1457396,1,39,"English","en",105,"# Introduction\n## Geometric spanners\n## Spanners in disk graphs\n## Dynamic connectivity in disk graphs","[{\"question\":\"What problem does the paper address for disk intersection graphs?\",\"answer\":\"It maintains a (1+ε)-spanner over the disk intersection graph of a dynamic set of disks, while also deriving implications for connectivity queries.\"},{\"question\":\"How is the disk set restricted in the spanner construction?\",\"answer\":\"All disks are restricted to have their diameters within a fixed and known range [4, Ψ].\"},{\"question\":\"What are the main size and update-time guarantees of the dynamic spanner?\",\"answer\":\"The (1+ε)-spanner has size O(nε−2 log Ψ log(ε−1)). Using persistent data structures, it requires O(ε−2n log4 n log Ψ) space and achieves an expected amortised update time of O((Ψ/ε)^2 log4 n log2 Ψ log2(ε−1)).\"}]",1784175065,98,{"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":85,"head_meta":87,"extra_data":89,"updated_unix":27},"a-dynamic-1-spanner-for-disk-intersection-graphs","",{"@graph":35,"@context":84},[36,53,67],{"@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/a-dynamic-1-spanner-for-disk-intersection-graphs/81641/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What problem does the paper address for disk intersection graphs?","Question",{"text":74,"@type":75},"It maintains a (1+ε)-spanner over the disk intersection graph of a dynamic set of disks, while also deriving implications for connectivity queries.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How is the disk set restricted in the spanner construction?",{"text":79,"@type":75},"All disks are restricted to have their diameters within a fixed and known range [4, Ψ].",{"name":81,"@type":72,"acceptedAnswer":82},"What are the main size and update-time guarantees of the dynamic spanner?",{"text":83,"@type":75},"The (1+ε)-spanner has size O(nε−2 log Ψ log(ε−1)). Using persistent data structures, it requires O(ε−2n log4 n log Ψ) space and achieves an expected amortised update time of O((Ψ/ε)^2 log4 n log2 Ψ log2(ε−1)).","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":105,"slug":137},19,"General","general"]