{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-11-12-binary-search-tree-unified-bound-ai/","md":"https://postcutoff.com/e/2026-11-12-binary-search-tree-unified-bound-ai/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-11-12-binary-search-tree-unified-bound-ai","date":"2026-11-12","date_precision":"day","short_title":"A BST meets Iacono's 2001 unified bound, with Claude and GPT","deck":null,"takeaway":"Since 2001 it was open whether any binary search tree, even offline, achieves Iacono's unified bound, which combines the working-set and dynamic-finger bounds. A new preprint gives an online deterministic one, so dynamic optimality would imply Iacono's conjecture for splay trees. Five models from Anthropic and OpenAI helped design and analyse it.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2611.16272: A Binary Search Tree with the Unified Bound (Cao, Qi, Wang)","url":"https://arxiv.org/abs/2611.16272","type":"paper","group":"primary","domain":"arxiv.org"}],"official":1,"filed":"2026-11-13","updated":"2026-11-13","orgs":["Anthropic","OpenAI","Pennsylvania State University","Jilin University","Tsinghua University"],"title":"First binary search tree with Iacono's unified bound (open since 2001): Cao, Qi and Wang say Claude and GPT models 'materially affected all sections'","summary":"arXiv 2611.16272 (12 Nov 2026), Yang Cao, Haoran Qi and Hanzhi Wang: a deterministic online BST that serves any m searches on n keys at cost O(m + UB(X)), where UB is Iacono's unified bound (SODA 2001), each search costing O(log n). \"Previously, no BST was known to satisfy the unified bound, even offline.\" Consequences: the dynamic optimality conjecture implies Iacono's conjecture for splay trees; pattern-avoiding sequences cost O((m+n)(1+log c_σ)), improving Berendsohn–Kozma–Opler (STOC 2024); an extension handles insertions and deletions. AI statement: Claude Opus 5.5, Fable 5.1, Fable 5.2, GPT-6 Astra and GPT-5.6 Sol assisted \"the design and analysis of the data structure\" and \"materially affected all sections\". Preprint.","key_facts":["Unified bound (Iacono 2001): each search pays about log(|x_t − x_u| + ρ(u,t) + 2) minimized over earlier times u, where ρ counts distinct keys searched in between; it is at most both the working-set and dynamic-finger bounds up to O(m)","Before: Iacono (2001) and Bădoiu–Cole–Demaine–Iacono (2007) achieved it with non-BST structures; within the BST model skip-splay trees, layered working-set trees and later designs all paid an overhead; the question was stated as open in Chalermsook et al. (2024), and even the offline version (does OPT satisfy it?) was open","Result: a deterministic online BST with total cost O(m + UB(X)) and worst-case O(log n) per search; extension to insertions and deletions with cost O(n0 + m + UB(X)) in the Bădoiu–Cole–Demaine–Iacono form","Corollaries: dynamic optimality of splay trees would imply they satisfy the unified bound (Iacono's conjecture); for sequences avoiding a fixed permutation σ the unified bound is O((m+n)(1+log c_σ)) with c_σ the Füredi–Hajnal limit, tight for permutations and better than the O(n c_σ²) of Berendsohn, Kozma and Opler (STOC 2024)","AI methodology statement (verbatim): 'We used Claude Opus 5.5, Claude Fable 5.1 and Claude Fable 5.2 (Anthropic) and GPT-6 Astra and GPT-5.6 Sol (OpenAI) to assist with literature review, the design and analysis of the data structure, independent checking of the proofs, and manuscript drafting. These tools materially affected all sections.'","Status: 60-page arXiv preprint (cs.DS), not refereed; no Lean formalization; no expert reaction found by 13 Nov"],"key_numbers":[],"tags":["computer-science","data-structures","binary-search-trees","dynamic-optimality","claude-opus-5-5","claude-fable","gpt-6-astra","gpt-5-6-sol","arxiv"],"science":{"field":"computer-science","subfield":"data structures","problem":"Is there a binary search tree satisfying Iacono's unified bound? (open since 2001; Iacono's splay-tree conjecture)","result":"A deterministic online BST with cost O(m + UB(X)); dynamic optimality now implies the unified bound for splay trees","open_since":"2001","ai_system":["Claude Opus 5.5","Claude Fable 5.1","Claude Fable 5.2","GPT-6 Astra","GPT-5.6 Sol"],"human_role":"AI-assisted design, analysis, proof checking and drafting; authors take responsibility","verification":{"key":"preprint","label":"Preprint only","text":"Preprint"},"status":"pending","shock":""},"body_md":"## What happened\n\nOn 12 Nov 2026 Yang Cao (Penn State), Haoran Qi (Jilin) and Hanzhi Wang (Tsinghua) posted a binary search tree that meets John\nIacono's unified bound, a 2001 bound that captures both locality in time (recently searched keys) and locality in key space (keys\nnear recent searches). Non-tree structures had achieved it since 2001, but no binary search tree had, even with the whole search\nsequence known in advance. The paper's AI statement names five Anthropic and OpenAI models and says they \"materially affected all\nsections\", including the design of the data structure.\n\n## Why it matters\n\nThe unified bound is one of the main known consequences that the dynamic optimality conjecture for splay trees (Sleator and\nTarjan, 1985) would need to deliver; the new tree shows the optimum satisfies it, so dynamic optimality would imply Iacono's\nconjecture. It is an unrefereed preprint and has no formal verification.","disputed":[{"label":"Verification","text":"Preprint","sources":[]}],"related":[{"id":"2026-11-13-ai-assisted-results-november-13-listing","url":"https://postcutoff.com/e/2026-11-13-ai-assisted-results-november-13-listing/","date":"2026-11-13","date_precision":"day","short_title":"AI-assisted results, 13 Nov","deck":null,"takeaway":"The arXiv listings of 12–13 Nov carried about 190 maths papers with an AI statement. Most disclose editing or literature search; about forty credit models with proofs, constructions or counterexamples. Seven got their own entries; the rest are catalogued here with the disclosure quoted. All are unrefereed preprints.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":27,"official":27,"filed":"2026-11-13","updated":"2026-11-13","orgs":["OpenAI","Anthropic","Google DeepMind","Axiom Math","various researchers"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-11-13","type":"filed","text":"Created (science agent; arXiv PDF read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-11-13","run":null,"sources_read":"Science agent; arXiv PDF read","updated":"2026-11-13","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":196,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":135,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":226,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":165,"in_training_data":false}],"short_url":"https://postcutoff.com/s/bst-meets-iacono-2001"}