[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-665474-105":59,"doc-detail-665474-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","on-the-average-complexity-of-3d-voronoi-diagrams-of-random-points-on-convex-polytopes","On the average complexity of 3D-Voronoi diagrams of random points on convex polytopes","","It is well known that the complexity of the 3-dimensional Voronoi diagram of n points, measured by the number of vertices, edges, and faces, can be as large as O(n^2). Under independently identically distributed points chosen uniformly from a full-dimensional region such as a cube or sphere, the expected complexity drops to O(n). This paper studies the intermediate case where points are drawn from a 2-dimensional region embedded in 3D space, and proves that when points come from a Poisson distribution on a convex polytope surface, the expected complexity is O(n).",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/on-the-average-complexity-of-3d-voronoi-diagrams-of-random-points-on-convex-polytopes/665474/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/on-the-average-complexity-of-3d-voronoi-diagrams-of-random-points-on-convex-polytopes/665474.png","ImageObject",300,407,{"name":92,"@type":93},"Melati","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-29",true,{"@type":102,"interactionType":103,"userInteractionCount":81},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is the worst-case complexity of a 3D Voronoi diagram of n points?","Question",{"text":112,"@type":113},"The complexity, defined by the number of vertices, edges, and faces, can be as bad as O(n^2).","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does expected complexity behave when points are uniformly sampled from a full-dimensional 3D region?",{"text":117,"@type":113},"If the points are independently identically distributed uniformly from a reasonably smooth full-dimensional region (e.g., a cube or sphere), the expected complexity falls to O(n).",{"name":119,"@type":110,"acceptedAnswer":120},"What expected complexity does the paper prove for points on the surface of a convex polytope?",{"text":121,"@type":113},"When points are drawn from a Poisson distribution with rate n on the surface of a convex polytope, the expected complexity of the resulting Voronoi diagram is O(n).","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},665474,1793304384,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":81,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},962085570644,"https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","Computational Geometry 25 (2003) 197–231  \n[www.elsevier.com/locate/comgeo](www.elsevier.com/locate/comgeo)  \nOn the average complexity of 3D-Voronoi diagrams of random points on convex polytopes ✩  \nMordecai J. Golin a, Hyeon-Suk Na b  \na Department of Computer Science, Hong Kong Univ. of Sci. & Technology, Kowloon, HongKong b Department of Mathematics, Pohang University of Science and Technology, Pohang, South Korea  \nReceived 1 August 2001; accepted 23 January 2002  \nCommunicated by J.D. Boissonnat  \nAbstract  \nIt is well known that the complexity, i.e. the number of vertices, edges and faces, of the 3-dimensional Voronoi diagram of n points can be as bad as 􀀂(n2 ) . It is also known that if the points are chosen Independently Identically Distributed uniformly from a 3-dimensional region such as a cube or sphere, then the expected complexity falls to O(n) . In this paper we introduce the problem of analyzing what occurs if the points are chosen from a 2-dimensional region in 3-dimensional space. As an example, we examine the situation when the points are drawn from a Poisson distribution with rate n on the surface of a convex polytope. We prove that, in this case, the expected complexity of the resulting Voronoi diagram is O(n) .  \n􀀁 2003 Elsevier Science B.V. All rights reserved.  \nKeywords: Voronoi diagrams; Convex polytopes; Random points; Average case analysis  \n1. Introduction  \nGiven a set Sn = {p1 , p2 , . . . , pn } ⊆ Rk of n points in k-dimensional Euclidean space, the Voronoi Diagram, VD(Sn), of Sn is a very well understood subdivision of Rk . For each point pi ∈ Sn there is an associated (convex) cell  \nCi = 􀀁x ∈ Rk: ∀j  i, d(x, pi) 􀀁 d(x, pj)􀀂 , where d( · , ·) is the Euclidean distance function. By definition these cells partition Rk . The complexity of VD(Sn) is the number of lower dimensional pieces that compose VD(Sn). For example, in the planar case,  \n✩ The work of both authors was partially supported by HK RGC CERG grants 6082/97E, 6137/98E and 6206/02E. E-mail addresses: [golin@cs.ust.hk](golin@cs.ust.hk) (M.J. Golin), [hsnaa@postech.ac.kr](hsnaa@postech.ac.kr) (H.-S. Na).  \n0925-7721/03/$ – see front matter 􀀁 2003 Elsevier Science B.V. All rights reserved.  \ndoi:10.1016/S0925-7721(02)00123-2  \n198 M.J. Golin, H.-S. Na / Computational Geometry 25 (2003) 197–231  \nk = 2, VD(Sn) contains edges E and vertices V. The complexity of VD(Sn) will be | VD(Sn)| = |E| + |V| . Since it is also known that the 2-dimensional Voronoi Diagram is a planar graph, Euler’s theorem immediately implies that | VD(Sn)| = 􀀂(n) [11] .  \nIf k = 3 then VD(Sn) is composed not only of edges E and vertices V but also of the faces F of the convex cells. The complexity of VD(Sn) will then be | VD(Sn)| = |E| + |V| + |F| . In three dimensions it can be proven that | VD(Sn)| = O(n2) . For some cases, such as when all of the points in Sn are on the moment curve {(t, t2 , t 3 ) : t ∈ R}, it can be easily proven that |V| = 􀀂(n 2) so | VD(Sn)| = 􀀂(n 2)  \n[7] . Another well known example of this worst-case behavior is built around two line segments L 1 ={(x, 0 , 0): x ∈ [0 , 12]} and L2 = { (1, y, 1): y ∈ [ 12 , 1]}; given any n 1 points on L 1 and any n2 points on L2 , |V| = 􀀂(n 1n2) . In particular if n 1 = n2 = ~~n~~2 then |V| = 􀀂(n 2) .  \nMoving away from worst-case behavior to average-case behavior it has been shown that if n points of Sn are independently identically distributed (IID) chosen from the uniform distribution over a“reasonably” smooth full dimensional bounded region P such as a cube or sphere then E(| VD(Sn)|) =􀀂(n) [4–6] .1  \nDropping the condition that P has full dimensionality dramatically changes the situation. For example, if we set P = L 1 ∪ L2 to be the union of the two 1-dimensional segments previously defined and choose n points Sn uniformly at ‘random’ from P then n 1 = |Sn ∩ L1 |, the number of points on L 1 , is a binomial random variable with parameters n, 12 , so E(n 1) = ~~n~~2 and E(n21) ∼ ~~n~~42 . Since n2 = |Sn","cbCaiag1i1MIxE6r","https://ap.wps.com/l/cbCaiag1i1MIxE6r","pdf",671285,35,"English","# Abstract\n# Introduction\n## Voronoi diagram and complexity definition\n## Worst-case behavior in 3D\n## Average-case behavior in full-dimensional regions\n## Points sampled from lower-dimensional sets\n## Motivation: 2D surfaces in 3D\n## Prior work and research gap","[{\"question\":\"What is the worst-case complexity of a 3D Voronoi diagram of n points?\",\"answer\":\"The complexity, defined by the number of vertices, edges, and faces, can be as bad as O(n^2).\"},{\"question\":\"How does expected complexity behave when points are uniformly sampled from a full-dimensional 3D region?\",\"answer\":\"If the points are independently identically distributed uniformly from a reasonably smooth full-dimensional region (e.g., a cube or sphere), the expected complexity falls to O(n).\"},{\"question\":\"What expected complexity does the paper prove for points on the surface of a convex polytope?\",\"answer\":\"When points are drawn from a Poisson distribution with rate n on the surface of a convex polytope, the expected complexity of the resulting Voronoi diagram is O(n).\"}]","On the average complexity of 3D-Voronoi diagrams of random points on convex polytopes | PDF",88]