Yu Zhang

Yingqi Zhang

A uniformly random $m$-subset of $[n]=\{0,\ldots,n-1\}$ has entropy $\log_2\binom{n}{m}$. Standard without-replacement procedures often expose an additional ordering coordinate that is absent from the returned set. We show that Floyd's subset sampler admits an exact round-local factorization of this coordinate. In round $r$, let $S$ be an $(r-1)$-subset of $[j]$, let $T\sim\operatorname{Unif}([j+1])$, and let $S'$ be the result of Floyd's transition. If $D$ is the zero-based rank of the original draw $T$ in $S'$, then $(S,T)\leftrightarrow(S',D)$ is a bijection between $\binom{[j]}{r-1}\times[j+1]$ and $\binom{[j+1]}{r}\times[r]$. Consequently, $S'$ and $D$ are independent and uniform on their respective spaces. The digit $D$ can therefore be merged immediately into a residual uniform random state; an induction shows that the partial subset remains independent of that state after every round. For $k=\min(m,n-m)$, the sampling phase uses $O(k\log k)$ time and $O(k)$ auxiliary space with an order-statistic tree; explicitly materializing a complement incurs the unavoidable output cost. The combinatorial layer avoids binomial-coefficient arithmetic and recovers the complete $k!$ state-space factor exactly. We also give a finite counterexample showing that analogous immediate rank recycling in a partial Fisher-Yates array is invalid because the unselected suffix retains a correlated ordering. A 64-bit Rust implementation is checked by exhaustive state-space enumeration for all $n\leq 8$ and by an entropy-accounting trace for choosing $20{,}000$ of $30{,}000$ items. We make no claim of runtime superiority over existing subset samplers.

Wanying Yu, Boyang Ma, Zhibo Eric Sun et al.

Large language models are deployed in long-context, emotionally interactive environments like digital humans, AI companions, educational assistants, and counseling systems. Unlike jailbreak attacks with explicit adversarial prompts, these systems interact with emotionally charged narratives involving bullying, betrayal, loneliness, social hostility, and institutional unfairness. This raises an important question: can prolonged narrative exposure reshape the reasoning and alignment stability of LLMs? We present the first systematic study of narrative-induced alignment degradation in LLMs. We design BreakingBad, a three-stage framework that measures how negative narrative immersion affects moral reasoning, behaviors, and deployment risks. It combines ethical decision evaluation, behavioral probing, and digital-human interaction analysis. Our experiments reveal three findings. First, negative narrative exposure degrades moral accuracy across multiple LLMs, with average drops of 12%-31%, especially in ambiguous scenarios and those involving vulnerable individuals. Second, the degradation is structured: different narratives induce distinct shifts, and first-person narratives produce stronger effects than third-person. Third, these shifts propagate into real deployments. Across counseling, education, medical, and financial/legal scenarios, narrative-conditioned models increasingly normalize hopelessness, cynicism, emotional detachment, and ethically questionable reasoning while remaining superficially policy-compliant. More broadly, our findings suggest alignment robustness is not static but a dynamically conditioned state shaped by long-term semantic environments and interaction history. These results reveal a new class of alignment risk that existing safety defenses largely fail to capture.