[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81930-en":3,"doc-seo-81930-105":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},81930,8796095461564,"Liam","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Color Voronoi Diagrams and Circular Sequences of Color Permutations","Color Voronoi diagrams partition the plane using distances to colored abstract sites, unifying many Voronoi variants on generalized sites and metrics. The work defines higher-order abstract color Voronoi diagrams for n colored abstract sites and analyzes all concrete instances under the same abstraction. It proves an upper bound of 4k(n−k)−2n on the vertex count for the order-k diagram and gives an iterative construction algorithm. The bound extends to families of disjoint simple polygons and improves for simple polygons via circular color-permutation sequences.","arXiv :2607 .05383v 1 [ cs .CG] 6 Jul 2026  \nAbstract Color Voronoi Diagrams and Circular Sequences of Color Permutations  \nSang Won Bae \\#   \nDivision of AI Computer Science and Engineering, Kyonggi University, Suwon, Republic of Korea Nicolau Oliver \\#   \nFaculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland Evanthia Papadopoulou \\#   \nFaculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland  \n~~ Abstract ~~  \nAbstract Voronoi diagrams are defined in terms of a given system of planar bisecting curves satisfying some simple combinatorial properties. They offer a unifying framework for a wide range of concrete Voronoi instances on generalized sites and metrics. In this paper, we formulate higher-order abstract color Voronoi diagrams of a set S of n colored abstract sites, simultaneously considering all concrete instances under their umbrella. We prove that the number of vertices in the order-k abstract color Voronoi diagram is at most 4k(n − k) − 2n, and present an iterative construction algorithm. The bound directly applies to a family of m disjoint simple polygons of total complexity n. For simple polygons the bound can further improve to O(min{k(n − k),(m − k)2 n}) . A critical ingredient of our proof is a combinatorial analysis on circular sequences of color permutations derived from the unbounded edges of these diagrams, which is interesting in its own right.  \n2012 ACM Subject Classification Theory of computation → Computational Geometry  \nKeywords and phrases higher-order Voronoi diagrams, abstract Voronoi diagrams, color Voronoi diagrams, circular sequences, allowable sequences, generalized sites, simple polygons  \n2 Abstract Color Voronoi Diagrams  \n 1  Introduction  \nVoronoi diagrams are versatile and influential space partitioning structures. Given a set Sof n sites in R2 and an underlying distance function, the ordinary Voronoi diagram VD (S) partitions the plane into maximal regions by the nearest site relation. The order-k Voronoi diagram VDk (S) partitions R2 into regions by the k nearest sites for 1 ⩽ k ⩽ n − 1, where VD 1 (S) = VD(S) . The farthest-site Voronoi diagram FVD (S) is equal to VDn−1(S) . Sites may often be assumed to be points in the Euclidean plane, however, generalized sites, such as disks, line segments and polygons, under generalized metrics may also constitute the input sites. See [7, 36] for extensive information.  \nLee [31] proved the tight bound O (k(n − k)) on the combinatorial complexity of VDk (S) of point sites in the Euclidean plane, and presented an iterative algorithm that computes the diagrams order 1 up to k in O(k2 nlog n) time. The O (k(n − k)) bound on the complexity of VDk (S) has been extended to line segments under any Lp metric [41] and to abstract Voronoi diagrams [13] . For point sites in the L1 /L∞ metric a better bound O(min{k(n−k),(n−k)2 }) is known [32] . The problem of constructing the Euclidean order-k Voronoi diagram VDk (S) for point sites S had been one of the most interesting open problems in computational geometry, and the first optimal O (nlog n + k(n − k))-time algorithm was presented recently by Chan et al. [18], after a series of algorithmic advances for over four decades [2, 3, 8, 17, 19, 35, 42] . For generalized sites, however, there is a notable scarcity of corresponding results.  \nIn color Voronoi diagrams, colors are assigned to the sites in S, modeling a common property that sites of the same color share; let K be the set of these m ⩽ n colors. The color assignment aggregates simple sites, such as points, segments, or disks, into compound ones of non-constant complexity, such as simple polygons, arc polygons, and site clusters. The color, a non-spatial property, facilitates the modeling of diverse applications, including facility location [1], shape matching [27], spatial databases [20], wireless sensor networks [30], nearestneighbor classification [16], fault detection and analysis in VLSI networks [39","cbCaia2Vk5TukvJ7","https://ap.wps.com/l/cbCaia2Vk5TukvJ7","pdf",1114083,1,34,"English","en",105,"# Abstract Color Voronoi Diagrams\n## Introduction","[{\"question\":\"What problem does the paper address in color Voronoi diagrams?\",\"answer\":\"It formulates higher-order abstract color Voronoi diagrams for colored abstract sites and provides vertex-count bounds and an iterative construction algorithm. It also analyzes concrete instances encompassed by the abstraction.\"},{\"question\":\"What is the main theoretical result about the number of vertices?\",\"answer\":\"For the order-k abstract color Voronoi diagram, the paper proves the number of vertices is at most 4k(n−k)−2n. The result applies to relevant families of disjoint simple polygons under the stated modeling.\"},{\"question\":\"What key combinatorial idea supports the proof?\",\"answer\":\"A combinatorial analysis on circular sequences of color permutations is derived from the unbounded edges of the diagrams. This circular-sequence structure is presented as interesting in its own right.\"}]",1784177106,86,{"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},"color-voronoi-diagrams-and-circular-sequences-of-color-permutations","",{"@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/color-voronoi-diagrams-and-circular-sequences-of-color-permutations/81930/",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 in color Voronoi diagrams?","Question",{"text":74,"@type":75},"It formulates higher-order abstract color Voronoi diagrams for colored abstract sites and provides vertex-count bounds and an iterative construction algorithm. It also analyzes concrete instances encompassed by the abstraction.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"What is the main theoretical result about the number of vertices?",{"text":79,"@type":75},"For the order-k abstract color Voronoi diagram, the paper proves the number of vertices is at most 4k(n−k)−2n. The result applies to relevant families of disjoint simple polygons under the stated modeling.",{"name":81,"@type":72,"acceptedAnswer":82},"What key combinatorial idea supports the proof?",{"text":83,"@type":75},"A combinatorial analysis on circular sequences of color permutations is derived from the unbounded edges of the diagrams. 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