{"paper":{"title":"A Dynamic, Self-balancing k-d Tree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A k-d tree can be made dynamic and self-balancing with specialized insertion, deletion, and rebalancing algorithms that preserve its alternating-dimension invariant.","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Russell A. Brown","submitted_at":"2025-09-09T21:10:15Z","abstract_excerpt":"The original description of the k-d tree recognized that rebalancing techniques, used for building an AVL or red-black tree, are not applicable to a k-d tree, because these techniques involve cyclic exchange of tree nodes that violates the invariant of the k-d tree. For this reason, a static, balanced k-d tree is often built from all of the k-dimensional data en masse. However, it is possible to build a dynamic k-d tree that self-balances when necessary after insertion or deletion of each k-dimensional datum. This article describes insertion, deletion, and rebalancing algorithms for a dynamic,"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"It is possible to build a dynamic k-d tree that self-balances when necessary after insertion or deletion of each k-dimensional datum.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Rebalancing steps can be designed that restore balance after each update while strictly preserving the k-d tree's alternating-dimension partitioning invariant at every node.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The paper introduces insertion, deletion, and rebalancing algorithms for a dynamic self-balancing k-d tree and measures their performance.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A k-d tree can be made dynamic and self-balancing with specialized insertion, deletion, and rebalancing algorithms that preserve its alternating-dimension invariant.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b5f8a474c13ed3ac1eca12aa76da064dce8d116df71db439137214b0f5b9d0fc"},"source":{"id":"2509.08148","kind":"arxiv","version":18},"verdict":{"id":"b552f260-991f-4077-a2af-28fde04d753b","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-18T17:44:52.223312Z","strongest_claim":"It is possible to build a dynamic k-d tree that self-balances when necessary after insertion or deletion of each k-dimensional datum.","one_line_summary":"The paper introduces insertion, deletion, and rebalancing algorithms for a dynamic self-balancing k-d tree and measures their performance.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Rebalancing steps can be designed that restore balance after each update while strictly preserving the k-d tree's alternating-dimension partitioning invariant at every node.","pith_extraction_headline":"A k-d tree can be made dynamic and self-balancing with specialized insertion, deletion, and rebalancing algorithms that preserve its alternating-dimension invariant."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2509.08148/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"7786e3fb70db14f7b9a9303f73404163cf68ecb8ab809d8a5e8634b0cb8ca884"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}