[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81996-en":3,"doc-seo-81996-105":30,"detail-sidebar-cat-0-en-105":92},{"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":13,"seo_description":14,"update_tm":28,"read_time":29},81996,687197207639,"Asher","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Benchmarking and Engineering Data Structures for Spherical Range Queries","Spherical range queries are essential for spatial data processing, yet selecting an optimal index is difficult because worst-case theoretical guarantees are often overly pessimistic and existing average-case analyses are limited and difficult to compare. The work provides a comprehensive benchmark of state-of-the-art spatial indices using diverse datasets, including a new graph-embedding-based benchmark at scales up to 10M points and 2–960 dimensions. It also introduces the Sorted-Projection Radius KD-tree (SPRK-tree), engineered for aggressive pruning and fast leaf evaluations.","arXiv :2607 .07367v 1 [ cs .CG] 8 Jul 2026  \nBenchmarking and Engineering Data Structures for Spherical Range Queries  \nThomas Bläsius \\#   \nKarlsruhe Institute of Technology (KIT), Karlsruhe, Germany Jean-Pierre von der Heydt \\#   \nKarlsruhe Institute of Technology (KIT), Karlsruhe, Germany Tobias Kempf \\#   \nKarlsruhe Institute of Technology (KIT), Karlsruhe, Germany Dennis Kobert \\#   \nKarlsruhe Institute of Technology (KIT), Karlsruhe, Germany Nikolai Maas \\#   \nKarlsruhe Institute of Technology (KIT), Karlsruhe, Germany  \n~~ Abstract ~~  \nSpherical range queries are a fundamental primitive for working with spatial data. Many spatial data structures have been developed to answer these queries, but choosing the optimal one for a specific application is a difficult task. This is because theoretical worst-case bounds are often overly pessimistic, and existing average-case analyses are rather restricted and hard to compare.  \nWe address this problem with two main contributions. First, we present a comprehensive evaluation of state-of-the-art spatial indices across a diverse set of benchmarks. This includes a new benchmark based on graph embeddings alongside multiple real-world datasets from the literature. Our benchmark covers instances scaling up to 10 M points and ranging between 2 and 960 dimensions.  \nSecond, we introduce the Sorted-Projection Radius KD-tree (SPRK-tree), a high-performance KD-tree variant. The SPRK-tree combines aggressive subtree pruning via radius reduction, sorted projection-based leaf nodes, and careful implementation optimizations. It consistently achieves the fastest query times in almost all benchmarks, and ranks second in the few remaining cases.  \n2012 ACM Subject Classification Theory of computation → Computational geometry; Theory of computation → Data structures design and analysis  \nKeywords and phrases Spherical Range Queries, Fixed-Radius Nearest Neighbor Search, Spatial Indexing, KD-tree, Benchmarking, Graph Embedding, SPRK-Tree  \nSupplementary Material Source Code: [https://github.com/wembed-pdf/sprk](https://github.com/wembed-pdf/sprk)  \nBenchmarking Code: [https://github.com/wembed-pdf/rembed](https://github.com/wembed-pdf/rembed)  \nEmbedding Dataset: [https://doi.org/10.5281/zenodo.21243483](https://doi.org/10.5281/zenodo.21243483)  \nFunding Jean-Pierre von der Heydt: This work was supported by funding from the pilot program Core–Informatics of the Helmholtz Association (HGF) .  \n2 Benchmarking and Engineering Data Structures for Spherical Range Queries  \n 1  Introduction  \nSpherical range queries are a fundamental primitive for working with spatial data. Given a set of n points in d-dimensional Euclidean space, the objective is to retrieve all points that lie within the specified radius of a query point. In low-dimensional spaces, spatial indices significantly accelerate these queries by pruning large portions of the search space. However, in higher dimensions a brute-force approach that checks the distance to every point is often the only viable option due to the curse of dimensionality [27] .  \nOne application of spatial indices is the computation of graph embeddings [7, 8 , 31 , 46], which play an important role in machine learning tasks such as community detection, link prediction, and node classification [18, 23 , 34 , 38 , 45] . These methods map the vertices of a graph into a continuous space such that spatial distance reflects structural similarity [35] . This is often achieved by force-directed optimization, which applies attracting forces to adjacent vertices that are too distant from each other and repelling forces to non-adjacent vertices that are too close. The attracting forces can be computed in linear time O (m) by iterating over all m edges. However, computing repelling forces requires identifying all non-adjacent vertices within a specific distance of each other. A naive brute-force implementation would check all O (n2 ) pairs of vertices, which is particularly prohib","cbCainJrVCVvEQ10","https://ap.wps.com/l/cbCainJrVCVvEQ10","pdf",1871526,6,1,25,"English","en",105,"# Introduction\n## Spherical range queries as a core primitive\n## Motivation: limitations of theory and comparison\n## Benchmarking approach and scale\n## SPRK-tree: engineering and performance strategy\n## Applications and role in graph embeddings","[{\"question\":\"Why is choosing a spatial index for spherical range queries challenging?\",\"answer\":\"Worst-case bounds are often too pessimistic, and average-case analyses are limited and hard to compare, making it difficult to identify the best option for a specific application.\"},{\"question\":\"What does the benchmarking contribution include?\",\"answer\":\"A comprehensive evaluation of state-of-the-art spatial indices across diverse benchmarks, including a new graph-embedding-based benchmark and multiple real-world datasets, scaling up to 10M points and 2–960 dimensions.\"},{\"question\":\"What is the SPRK-tree and what makes it fast?\",\"answer\":\"The Sorted-Projection Radius KD-tree (SPRK-tree) is a KD-tree variant that combines aggressive subtree pruning via radius reduction, sorted projection-based leaf nodes, and implementation optimizations to achieve the fastest query times in most 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is choosing a spatial index for spherical range queries challenging?","Question",{"text":76,"@type":77},"Worst-case bounds are often too pessimistic, and average-case analyses are limited and hard to compare, making it difficult to identify the best option for a specific application.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What does the benchmarking contribution include?",{"text":81,"@type":77},"A comprehensive evaluation of state-of-the-art spatial indices across diverse benchmarks, including a new graph-embedding-based benchmark and multiple real-world datasets, scaling up to 10M points and 2–960 dimensions.",{"name":83,"@type":74,"acceptedAnswer":84},"What is the SPRK-tree and what makes it fast?",{"text":85,"@type":77},"The Sorted-Projection Radius KD-tree (SPRK-tree) is a KD-tree variant that combines aggressive subtree pruning via radius reduction, sorted projection-based leaf nodes, and implementation optimizations to achieve the fastest query times in most 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