Worst-Case Inputs for Smoothed Caching Complexity
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Definitions and mathematical statement
Shared notation.
Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.A cache of capacity $k$ stores at most $k$ pages. A request absent from the cache causes a miss. Competitive ratio compares an online algorithm's cost with an offline optimum, while smoothed analysis first perturbs the request sequence using a specified distribution.
Statement recorded in the supplied source.
Characterize the input sequences that have the worst performance under the smoothed complexity of the caching problem, whose competitive-ratio complexity is of order $\log k$ for cache size $k$. [S2]Scope qualification.
The caching algorithm, perturbation distribution, and performance functional require the report.Source status.
The supplied catalogue labels this record open. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.Short English statement
Characterize caching sequences that remain worst under the source' s smoothed complexity model.Sources
- S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30002324 (OWR-12481-013). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
- S2 Combinatorics and Probability (2014), Oberwolfach Reports, DOI 10.4171/owr/2013/18, pp. 1087--1152 https://doi.org/10.4171/owr/2013/18. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.
Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30002324 and source code OWR-12481-013 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.Statement licence: CC-BY-SA-4.0 · Source file
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gpt 6 astra ultra attempt on CS-316
Worst-Case Inputs for Smoothed Caching Complexity
Definitions and mathematical statement
Shared notation.
Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.A cache of capacity $k$ stores at most $k$ pages. A request absent from the cache causes a miss. Competitive ratio compares an online algorithm's cost with an offline optimum, while smoothed analysis first perturbs the request sequence using a specified distribution.
Statement recorded in the supplied source.
Characterize the input sequences that have the worst performance under the smoothed complexity of the caching problem, whose competitive-ratio complexity is of order $\log k$ for cache size $k$. [S2]Scope qualification.
The caching algorithm, perturbation distribution, and performance functional require the report.Source status.
The supplied catalogue labels this record open. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.Short English statement
Characterize caching sequences that remain worst under the source' s smoothed complexity model.Sources
- S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30002324 (OWR-12481-013). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
- S2 Combinatorics and Probability $2014$, Oberwolfach Reports, DOI 10.4171/owr/2013/18, pp. 1087--1152 https://doi.org/10.4171/owr/2013/18. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.
Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30002324 and source code OWR-12481-013 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.Research attempt
Literature, source context and scope
The full original catalogue body and source qualifications are preserved above. The relevant primary workshop passages were read in the cached publisher report. Historical claims and subsequent attributed results are distinguished from the mathematical work below. A restricted example, standard lemma or known theorem is not counted as new progress on the entire original target. Missing assumptions and repeated or rhetorical extracts are not silently turned into different solved problems.A specified perturbation can reverse a ranking
The source leaves both the small-perturbation distribution and the online caching algorithm unspecified. We choose an explicit limited experiment rather than assert a characterization under every smoothing law. The cache has capacity two and starts empty. The online algorithm is least-recently-used (LRU); the comparison is the offline minimum number of misses, also starting empty. Given a six-request sequence, with probability $1-p$ leave it unchanged and with probability $p$ swap one uniformly selected adjacent pair, with $0\le p\le1$. An adjacent pair of equal requests may be chosen, in which case the sequence is unchanged. The performance criterion is the expected ratio of misses, not the ratio of expected misses.Proposition: For $A=(0,1,2,0,2,1)$ and $B=(0,2,0,1,1,2)$, both having two occurrences of each page, the respective expected ratios are \[
R_A(p)=\frac54+\frac{p}{15},\qquad R_B(p)=\frac43-\frac{p}{12}.
\] Thus $A$ is better at $p=0$ but worse whenever $p>5/9$.
Proof: Here are the complete miss counts. Column 0 is the original sequence; column $j$ swaps positions $j$ and $j+1$, for $1\le j\le5$. \[
\begin{array}{c|rrrrrr}
&0&1&2&3&4&5\\\hline
\mathrm{LRU}(A)&5&4&6&4&5&6\\
\mathrm{OPT}(A)&4&4&4&3&4&4\\
\mathrm{LRU}(B)&4&4&3&5&4&4\\
\mathrm{OPT}(B)&3&3&3&4&3&3
\end{array}
\] The LRU row follows by moving a requested resident page to the most-recent position, or evicting the least-recent page on a miss in a full cache. To verify every optimum count, let $F(s,C)$ be the minimum remaining misses on suffix $s$ with cache $C$. For its first request $a$, the exact recursion is \[
F((),C)=0,\quad
F((a,s'),C)=\begin{cases}
F(s',C),&a\in C,\\
1+F(s',C\cup\{a\}),&a\notin C,\ |C|<2,\\
1+\min_{v\in C}F(s',(C\setminus\{v\})\cup\{a\}),&a\notin C,\ |C|=2.
\end{cases}
\] This is valid by considering the first miss and every permitted victim, followed by an optimal suffix decision; induction on suffix length proves it. Applying it to the displayed twelve length-six inputs gives the table. Averaging the five ratios in columns 1 through 5 gives $79/60$ for $A$ and $5/4$ for $B$. Mixing with column 0 gives the stated formulas. Their difference is $-1/12+3p/20$, which changes sign at $5/9$. ∎
Verification and remaining gaps
The exact recursion was executed with rational averaging, and all 90 distinct sequences with multiset $\{0,0,1,1,2,2\}$ were searched for such a reversal. The displayed pair is one verified outcome. This refutes a proposed shortcut of ranking these inputs by their unperturbed ratio alone, even after the smoothing law is fixed. Equal page histograms also fail to determine the smoothed ratio. It is not a characterization for arbitrary request lengths, cache sizes, online algorithms or perturbations, and it does not establish the source's asymptotic smoothed competitive guarantees. The chosen law is an explicit auxiliary model, not a reconstruction of an unspecified intended law.References and source attribution
Primary workshop report: https://doi.org/10.4171/owr/2013/18. Relevant passages read: 2013/18 p. 1118, smoothed caching performance and unspecified small perturbations.All retrievals were checked on 11 October 2026. Original question attribution remains in the preserved Sources section. External theorems are attributed and not claimed as new discoveries or fully verified merely by citation.
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% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.Final status and attribution
Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAIgpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch, reviewed the written mathematical arguments, and contributed formatting/repository review. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch019/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.