[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83816-en":3,"doc-seo-83816-105":29,"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":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},83816,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Near-Optimal and Efficient Encoding for Two-Dimensional Range Minimum Queries","Near-optimal and efficient encoding is developed for the 2D range minimum query (2D RMQ) encoding problem. The method targets an m×n array over a total order and returns the position of a query rectangle’s maximum without accessing the original array. Previous work achieves good space but leaves query-time guarantees unclear. This work designs an encoding with near-optimal space using a parameter κ∈[1, log log n], giving O(κmn(log m+log log n)) bits and O(log1/κ n) query time.","arXiv :2607 .04509v 1 [ cs .DS] 5 Jul 2026  \nNear-Optimal and Efficient Encoding for Two-Dimensional  \nRange Minimum Queries  \nPaweł Gawrychowski∗1, Adam Górkiewicz∗1, and Srinivasa Rao Satti2  \n1 Institute of Computer Science, University of Wrocław, Poland  \n2 Department of Computer Science, Norwegian University of Science and Technology, Trondheim, Norway  \nAbstract  \nWe consider the 2D RMQ encoding problem: given an m × n array of mn elements over a total order, encode it such that, for any query rectangle, the position of its maximum element can be reported without accessing the original array. For m ≤ n, it is known how to encode the array in O (mn min{m, log n}) bits with O(1)-time queries [Brodal et al. , Algorithmica 2012], and also how to obtain an asymptotically optimal encoding consisting of O (mn log m) bits [Brodal et al., ESA 2013] . However, the latter approach does not prove any guarantee on the query time, and it appears to be inherently sequential: it requires scanning the whole encoding to answer a query. We design a different encoding that uses near-optimal space while allowing for efficient queries. More concretely, for every parameter κ ∈ [1 , log log n], our encoding uses O (κmn(log m + log log n)) bits and answers 2D RMQ queries in O(log1/κ n) time.  \n1 Introduction  \nRange minimum/maximum data structures are important substructures in the design of succinct and compressed data structures. The range maximum query (RMQ) problem asks us topreprocess an array so that, given a query range, we can return the position of a maximum element in that range (traditionally, RMQ refers to range minimum queries; we phrase the problem in terms of maxima, which is more convenient for our presentation, and the two variants are equivalent by negating all elements) . In the one-dimensional setting, the input is an array of length n and a query is an interval [i.. j] . In two dimensions, the input is an m × n array anda query is an axis-parallel rectangle [i1.. i2] × [j1.. j2] .  \nThe RMQ problem has been studied in the literature in two distinct models: indexing and encoding models [46] . In the indexing model, the input array remains available at query time, and the data structure stores only auxiliary information, referred to as the index, to support the queries efficiently. In the encoding model, which is more relevant to this paper, queries are answered without access to the original input, and hence only the information relevant to answer queries is encoded within the data structure. The motivation behind the encoding model is twofold. First, it allows us to better understand the mathematical properties of the particular queries, and bound their effective entropy. Second, in some applications we need to support queries on an implicitly defined input that is well-defined but not stored at all (and not easy to compute in the query time) . There has been a significant amount of research  \n∗ Partially supported by the Polish National Science Centre grant number 2023/51/B/ST6/01505 .  \nin the recent past on designing indexes and encodings for various range queries on arrays, such as 1D RMQ [2 , 8 , 12 , 17–19 , 23 , 35 , 39–41, 50], 2D RMQ [1 , 6–8 , 13 , 24 , 27 , 34 , 36 , 52], range min&max [25 , 31 , 33 , 51], range median and range mode [5 , 9 , 15 , 28 , 38 , 43 , 44], range selection and range top-k [11 , 15 , 25 , 29 , 32], and range majority and range minority [3 , 10 , 14 , 21 , 22 , 26 , 37 , 42] . See [34] for a survey on encodings for range queries on arrays.  \nRMQ is a basic primitive in data structures and algorithms. In one dimension, it is tightly connected to the lowest common ancestor (LCA) problem on trees, and has applications in suffix-array based text indexing, compressed suffix trees, document retrieval, geometric indexing and many others. In two dimensions, it is a special case of the orthogonal range-searching problem and is relevant as a subroutine in geometric indexing applications. The 1D prob","cbCaidUzVDzn0AZO","https://ap.wps.com/l/cbCaidUzVDzn0AZO","pdf",584980,1,21,"English","en",105,"# Abstract\n# Introduction\n## Range maximum/minimum queries\n## Indexing vs encoding models\n## Previous results\n# 1D case\n# 2D case","[{\"question\":\"What problem does the paper address in 2D RMQ encoding?\",\"answer\":\"It studies encoding an m×n array so that, for any query rectangle, the position of the maximum element can be reported without accessing the original array.\"},{\"question\":\"How does the paper’s encoding differ from prior near-optimal encodings?\",\"answer\":\"Earlier asymptotically optimal encodings for 2D RMQ do not provide a query-time guarantee and seem inherently sequential. This paper proposes an alternative that supports efficient queries while keeping near-optimal space.\"},{\"question\":\"What are the space and query-time bounds of the proposed method?\",\"answer\":\"For any κ∈[1, log log n], the encoding uses O(κmn(log m+log log n)) bits and answers 2D RMQ queries in O(log1/κ n) time.\"}]",1784190602,53,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":27},"near-optimal-and-efficient-encoding-for-two-dimensional-range-minimum-queries","",{"@graph":35,"@context":85},[36,53,68],{"@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/near-optimal-and-efficient-encoding-for-two-dimensional-range-minimum-queries/83816/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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},"What problem does the paper address in 2D RMQ encoding?","Question",{"text":75,"@type":76},"It studies encoding an m×n array so that, for any query rectangle, the position of the maximum element can be reported without accessing the original array.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper’s encoding differ from prior near-optimal encodings?",{"text":80,"@type":76},"Earlier asymptotically optimal encodings for 2D RMQ do not provide a query-time guarantee and seem inherently sequential. This paper proposes an alternative that supports efficient queries while keeping near-optimal space.",{"name":82,"@type":73,"acceptedAnswer":83},"What are the space and query-time bounds of the proposed method?",{"text":84,"@type":76},"For any κ∈[1, log log n], the encoding uses O(κmn(log m+log log n)) bits and answers 2D RMQ queries in O(log1/κ n) time.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]