[NOI 2025] Sequence Transformation

The question

Can the source-listed reference complexity O(t n^2) be substantially improved?

Algorithmic frontier

Reference frontier: $\mathcal{O}\!\left(t n^{2}\right)$
Latest reported AI result: $\mathcal{O}\!\left(t (n + \log P)\right)$

Normalized growth proxy · not benchmark data

This compares source-reported bounds; it is not a correctness or novelty finding.

Results

2026-09-12

Claimed result: $\mathcal{O}\!\left(t (n + \log P)\right)$

AI / model: ChatGPT

Reported method

The page unifies both queries as one function DP over a (max, +) / (sum, product) semiring, proves only O(n) breakpoints are ever created, and uses lazy reflection plus three whole-range tags on a two-stack deque to make each step amortised O(1), giving O(n + log P) time and O(n) space.

Verification

Source: Source reviewed
Theory: Theory not checked
Implementation: Theory only · code pending
Prior art: Prior art not reviewed

The result page was opened and compared with the collection claim. Legacy audit state: audited_public.

The page cites a submission record at https://qoj.ac/submission/2932920, but that URL 302-redirects to the QOJ login page; P02 did not authenticate, so neither verdict nor code was observed and the record cannot be judge_verified. The page itself contains no code block at all.