[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81883-en":3,"doc-seo-81883-105":31,"detail-sidebar-cat-0-en-105":93},{"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":28,"seo_description":14,"update_tm":29,"read_time":30},81883,2336464648322,"Aria","https://ap-avatar.wpscdn.com/avatar/2200025388227c56fec?_k=1778556882303663488",8,"Research & Report","Fully Scalable MPC Algorithms for WSPD in Doubling and Euclidean Spaces","This paper studies constructing a (1/ε)-well-separated pair decomposition (WSPD) for n points in the Massively Parallel Computation (MPC) model. It presents an O(1)-round MPC algorithm that builds an O(1/ε)-WSPD in doubling metric spaces with constant doubling dimension, using high probability guarantees, total space (1/ε)O(ddim)·O(n), and O(nδ) space per machine. For d-dimensional Euclidean space, it improves WSPD size and total space to (1/ε)O(d)n, enabling O(1)-round MPC solutions for spanners, diameter approximation, closest pair, and k-nearest neighbors.","arXiv :2607 .038 1 1v 1 [ cs .CG] 4 Jul 2026  \nFully Scalable MPC Algorithms for WSPD in Doubling and Euclidean Spaces  \nEunjin Oh \\#   \nDepartment of Computer Science and Engineering, POSTECH, Pohang, Korea Hyeonjun Shin \\#   \nDepartment of Computer Science and Engineering, POSTECH, Pohang, Korea  \n~~ Abstract ~~  \nIn this paper, we study the problem of constructing a (1/ε)-well-separated pair decomposition (WSPD) for a point set of size n in the Massively Parallel Computation (MPC) model, where multiple machines work in parallel and communicate in synchronous rounds˜. We present an O(1)-round MPC algorithm that constructs a O(1/ε)-WSPD of size (1/ε)O (ddim) · O(n) for point sets ˜in a  \nmetric space of a constant doubling dimension ddim, with high probability, using (1/ε)O (ddim) · O(n) total space and O(nδ ) space per machine for a constant δ ∈ (0 , 1) . In the d-dimensional Euclidean space, we can improve the size of the WSPD and the total space to (1/ε)O (d)n. This improves the best-known algorithm [FOCS’93] for computing a WSPD which requires O (log n) rounds and works only in Euclidean spaces. As a consequence, the following problems can be solved in O(1) rounds in the MPC model: computing a (1 + ε)-spanner, a (1 − ε)-approximation of the diameter, the closest pair, and the k-nearest neighbors (k-NN) . While our k-NN algorithm is specific to Euclidean space, the other three problems can be solved in both Euclidean and doubling metric spaces.  \n2012 ACM Subject Classification Theory of computation → Computational geometry  \nKeywords and phrases MPC model, parallel algorithms, Computational Geometry, well-separated pair decomposition, doubling metric spaces  \nFunding Supported by Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No. RS-2024-00440239, Sublinear Scalable Algorithms for Large-Scale Data Analysis) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. RS-2024-00358505) .  \n 1  Introduction  \nIn this paper, we study geometric proximity problems in the Massively Parallel Computation (MPC) model, introduced by [39], now a standard framework for large-scale data processing in distributed systems such as MapReduce [23], Dryad [37], Hadoop [44], and Spark [46] . Here, the input data is distributed across multiple machines, and computation proceeds in synchronous rounds of local computation and communication. A central algorithmic goal is to minimize the number of communication rounds and to use small local memory while keeping the total work nearly linear. Over the past decade, significant progress has been made on fundamental MPC problems such as sorting, connectivity, and matching [7, 8 , 10 , 17 , 21 , 25 , 28 , 32 , 40 , 42] . While some geometric problems such as clustering, Euclidean MST, and spanners have been studied [4, 6 , 9 , 18 , 22 , 26 , 38], they represent only a limited portion of the rich landscape of computational geometry. In particular, only a few prior works consider fundamental proximity structures such as well-separated pair decompositions, which are central to many classic geometric algorithms.  \nWhile only a few papers explicitly address geometric proximity problems in the MPC model, such problems have been extensively studied in earlier parallel computation models, particularly the PRAM model [3, 5 , 13 , 15 , 20 , 43] . For additional references, see the survey [31] . These algorithms typically assume global memory access and fine-grained  \n2 Fully Scalable MPC Algorithms for WSPD in Doubling and Euclidean Spaces  \nsynchronization, but these assumptions are less realistic in modern distributed systems. In contrast, the MPC model more accurately captures the constraints of today’s large-scale computing environments by modeling limited local memory and emphasizing communicationefficient computation. Prior work [32, 39] has shown that PRAM algorithms can be simulatedin t","cbCaiixThQ6XsM5Y","https://ap.wps.com/l/cbCaiixThQ6XsM5Y","pdf",1076466,6,1,32,"English","en",105,"# Introduction\n## Problem setting and MPC framework\n## Well-separated pair decomposition (WSPD) and applications\n## Doubling metrics and Euclidean improvements","[{\"question\":\"What problem does the paper address in the MPC model?\",\"answer\":\"It addresses how to construct a (1/ε)-well-separated pair decomposition (WSPD) for a set of n points while minimizing communication rounds and local memory usage in the MPC framework.\"},{\"question\":\"What main complexity improvement does the paper achieve?\",\"answer\":\"It gives an O(1)-round MPC algorithm that constructs a (1/ε)-WSPD with near-linear total work and specified space bounds in metric spaces with constant doubling dimension.\"},{\"question\":\"Which geometric proximity problems can be solved in O(1) rounds after computing the WSPD?\",\"answer\":\"In the MPC model, the paper states that computing a (1+ε)-spanner, a (1−ε)-approximation of the diameter, the closest pair, and k-nearest neighbors can be done in O(1) rounds, with k-NN specifically noted for Euclidean space.\"}]","Fully Scalable MPC Algorithms for WSPD in Doubling and Euclidean Spaces | 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problem does the paper address in the MPC model?","Question",{"text":77,"@type":78},"It addresses how to construct a (1/ε)-well-separated pair decomposition (WSPD) for a set of n points while minimizing communication rounds and local memory usage in the MPC framework.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"What main complexity improvement does the paper achieve?",{"text":82,"@type":78},"It gives an O(1)-round MPC algorithm that constructs a (1/ε)-WSPD with near-linear total work and specified space bounds in metric spaces with constant doubling dimension.",{"name":84,"@type":75,"acceptedAnswer":85},"Which geometric proximity problems can be solved in O(1) rounds after computing the WSPD?",{"text":86,"@type":78},"In the MPC model, the paper states that computing a (1+ε)-spanner, a (1−ε)-approximation of the diameter, the closest pair, and k-nearest neighbors can be done in O(1) rounds, with k-NN specifically noted for Euclidean 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