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CS-106 · computer science · mathematics · Algorithms and data structures · Computational geometry

Dynamic Planar Nearest Neighbors

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Maintain a finite set $P\subset\mathbb R^2$ under insertions of new sites and deletions of existing sites. A query supplies $q\in\mathbb R^2$ and must return a site $p\in P$ minimizing $\|q-p\|_2$. Any minimizer is acceptable in a tie; when $P$ is empty, the structure reports this fact. Distances are exact, not approximate.

Is there a data structure supporting insertions, deletions, and nearest-neighbor queries all in $O(\log(n+2))$ time, where $n$ is the current number of sites? Use the usual real-arithmetic model for planar geometric data structures, with polynomial space and preprocessing. The strongest reading requests a worst-case bound for every operation. An amortized or randomized solution must state its guarantee explicitly; these are distinct performance formulations of the source's general target.

Updates may occur in arbitrary order, and their locations are not assumed to follow a random distribution. A structure allowing only insertions does not meet the fully dynamic requirement. Nor does logarithmic query time alone suffice when one update operation is slower. For an amortized formulation, the running time is assessed over whole update sequences, rather than charging every deletion its own worst-case cost. The source's related convex-hull reduction can guide constructions but does not itself supply the requested simultaneous bounds.

Short English statement

Can a fully dynamic planar point set support exact nearest-neighbor queries, insertions, and deletions in logarithmic time per operation?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-106-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-106

Research-attempt provenance

Definitions and mathematical statement

Maintain a finite set $P\subset\mathbb R^2$ under insertions of new sites and deletions of existing sites. A query supplies $q\in\mathbb R^2$ and must return a site $p\in P$ minimizing $\|q-p\|_2$. Any minimizer is acceptable in a tie; when $P$ is empty, the structure reports this fact. Distances are exact, not approximate.

Is there a data structure supporting insertions, deletions, and nearest-neighbor queries all in $O(\log(n+2))$ time, where $n$ is the current number of sites? Use the usual real-arithmetic model for planar geometric data structures, with polynomial space and preprocessing. The strongest reading requests a worst-case bound for every operation. An amortized or randomized solution must state its guarantee explicitly; these are distinct performance formulations of the source's general target.

Updates may occur in arbitrary order, and their locations are not assumed to follow a random distribution. A structure allowing only insertions does not meet the fully dynamic requirement. Nor does logarithmic query time alone suffice when one update operation is slower. For an amortized formulation, the running time is assessed over whole update sequences, rather than charging every deletion its own worst-case cost. The source's related convex-hull reduction can guide constructions but does not itself supply the requested simultaneous bounds.

Short English statement

Can a fully dynamic planar point set support exact nearest-neighbor queries, insertions, and deletions in logarithmic time per operation?

Sources

Literature and context check

TOPP Problem 63 was consulted for the fully dynamic target and its three-dimensional convex-hull reduction. The argument below tests explicit Voronoi maintenance, not every possible implicit data structure. No insertion-only result is promoted to the fully dynamic worst-case guarantee.

Research attempt: Why explicit Voronoi maintenance cannot charge every update logarithmically

The usual lifting identity is immediate: for $p=(a,b)$ and $q=(u,v)$, \[ \|q-p\|^2=\|q\|^2+(-2u,-2v,1)\cdot(a,b,a^2+b^2). \] Thus a minimum linear-functional query on the lifted points returns a nearest site. Insertions and deletions transfer one-for-one. This is an exact reduction, but supplies no data structure by itself.

The attempted planar implementation stored every Voronoi adjacency explicitly and updated only a supposedly bounded neighborhood after an insertion. Let $p_1,\ldots,p_m$ be distinct sites on the unit circle, and insert the origin. For each $i$, at $q_i=p_i/2$ the origin and $p_i$ are both at distance $1/2$. For $j\ne i$, \[ \|q_i-p_j\|^2=\tfrac54-p_i\cdot p_j>\tfrac14. \] There is a strict positive gap to every other site. By continuity, a small segment of the origin--$p_i$ perpendicular bisector about $q_i$ belongs to their shared Voronoi boundary. Hence the inserted site has $m$ distinct new Voronoi adjacencies. A representation allocating a separate record for every new adjacency must write at least $m$ records on this one insertion. Deleting the origin reverses the example.

The initial circle creates a high-order Voronoi vertex, but the adjacency conclusion does not rely on retaining that degeneracy. For fixed finite $m$, the finitely many displayed strict gaps have a positive minimum. A sufficiently small perturbation of the sites preserves positivity at suitably adjusted bisector points. In particular the effect cannot be dismissed as a tie-handling artifact. The original problem in any event permits the circular input.

This defeats the proposed local-change bound for an explicit diagram, not the existence of a more economical representation. A lazy representation might describe the new cell without eagerly materializing all its adjacencies; then one must analyze exact queries and future deletions using that representation. Similarly dividing sites into binary-sized static blocks gives at most $\sum_{k\le\log_2n}O(k+1)=O(\log^2(n+2))$ query work if each nonempty block is queried separately in logarithmic time. This upper-bound calculation does not become $O(\log n)$ merely because there are logarithmically many blocks.

Verification and adversarial audit

The circle inequality was checked for every pair $i\ne j$, including antipodal points. The $m$ new adjacencies are certified by separate strict local distance comparisons. Empty sets and ties remain governed by the source convention. The write-count argument is expressly representation-specific.
Audit corrections.
The linear explicit-diagram update was not stated as a lower bound for implicit structures, and the lifting reduction was not mistaken for an algorithm.
External computation.
None; symbolic arguments and boundary checks only.

Scope and remaining gaps

An implicit fully dynamic structure with all three logarithmic guarantees has not been constructed. The first gap is handling updates with many geometric changes without paying for all changes, while keeping exact worst-case queries and arbitrary later deletions efficient.
Integration review.
Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% The calculations below the preserved source statement investigate a concrete route, but establish no substantiated new progress toward the entire original target. Known special cases and failed sufficient conditions are not credited as a solution.

References

The original source references and catalogue attribution are preserved in the Sources section above. The following URLs were actually consulted for this attempt; retention of an additional original reference does not claim fresh consultation of it.

Model and contributor disclosure

OpenAI gpt-6-astra, requested reasoning configuration ultra, generated this research attempt on 2026-10-11. The exact serving revision is not exposed. Codex GPT-6 performed orchestration and repository-format review; its exact serving revision is also not exposed. The mathematical route, boundary checks and final adversarial self-audit followed contribution prompts A then B in docs/contribution_prompts.md, after reading CONTRIBUTING.md. Python 3.14.2 assembled the standalone documents; any mathematical computation is specified in Verification. No theorem prover was used. No human mathematical authorship, substantial human proof correction, or independent human proof checking is asserted. Model-assisted review, source retrieval, and structural checks do not establish correctness or novelty. Full private reasoning transcripts are unavailable as reproducibility artifacts; the complete arguments, inputs and outcomes actually used are stated here or in the companion batch audit.

Final status

Unresolved. This is a research attempt with an explicit failed route or restricted calculation, not a full solution, not a current-status certification, and not a claim that the elementary material is new. The catalogue's original source-status label is preserved independently of this attempt status.

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