AI 中文总结
该研究针对CONGEST模型的围长估计，提出一种黑盒精确化策略的工作负载屏障，证明恒定精确概率需$\boldsymbol{\textit{Ω}}(n_t/\text{log }n_t)$的期望工作负载，为该类策略提供了约束。
AI 中文摘要
近期的多尺度最近源方法在CONGEST模型中可实现多项式次线性的围长近似。我们确定了使该框架达到精确性的直接黑盒途径：在新鲜的可交换源集上对同一估计器进行顺序调用，其中源基数和最近源容量根据先前的标量输出自适应选择，并采用自适应停止规则。对于有界度、对数直径的图族$H_t$，其包含$n_t$个顶点和唯一围长为$g_t=\theta(\text{log }n_t)$的环，精确性要求采样的环源在存在线性数量严格更近的竞争节点时，仍能在对跖边处存活。对于任何此类精确化方法$\boldsymbol{\textit{A}}$，排列秩论证可给出与实现无关的工作负载界$\text{Pr}[\boldsymbol{\textit{A}}(H_t)=g_t]\boldsymbol{\textit{≤}}(3g_t/n_t)\boldsymbol{\textit{E}}[\boldsymbol{\textit{Σ}}_{j=1}^{T}\text{min}\boldsymbol{\textit{\textit{Q}}}_j,\boldsymbol{\textit{k}}_j\rbrace]$，其中$T$为执行的调用次数，$Q_j$为源集基数，$k_j$为第$j$次调用的最近源容量。因此，恒定精确概率需要$\boldsymbol{\textit{Ω}}(n_t/g_t)=\boldsymbol{\textit{Ω}}(n_t/\text{log }n_t)$的期望保留源工作负载。我们正式证明，仅通过调整近期多尺度模板的尺度数量/顺序、伯努利或固定基数采样、容量及标量输出停止规则，即可将其归入此类。对于标准的顺序分组估计器实现，工作负载定理给出$\boldsymbol{\textit{Ω}}(n_t/\text{log }n_t)$的期望轮次推论。这是对定义明确的黑盒精确化策略的屏障，而非CONGEST模型中无限制精确围长的下界。
英文摘要
Recent multi-scale nearest-source methods give polynomially sublinear approximations for girth in the CONGEST model. We study exactification by adaptive black-box composition while preserving the same fresh exchangeable source-selection primitive. Our scalar-oracle model exposes the sampled source identities and the scalar estimate from every call, allows arbitrary persistent controller state, adaptive source cardinalities and capacities, adaptive stopping, and an arbitrary final decoder; the internal nearest-source tables remain encapsulated. We first construct, for infinitely many $n$, a same-size pair $G_t^0,G_t^1$ of maximum-degree-three, logarithmic-diameter graphs whose girths are distinct and both $Θ(\log n)$. A length-transfer construction makes every bulk source contribute identically on the two graphs. The scalar transcripts can separate the pair only when one of $O(\log n)$ interface sources survives a linear nearest-source rank competition. Coupling the adaptive executions with a conditional permutation-rank bound yields an $Ω(n/\log n)$ expected retained-source workload requirement for constant exactness probability, even with arbitrary final decoding. For the standard sequential packetized realization, the same scale is an expected-round barrier. A complementary bridgeless family $\widehat H_t$ shows the same $Ω(n/\log n)$ direct-retuning barrier on bounded-degree graphs with minimum degree at least two, no bridges, and $2$-core equal to the whole graph. Finally, under a known promise $g\ge h$, one full-source call from $Θ(n/h)$ uniformly sampled sources computes exact girth with constant probability in $O(n/h+D)$ rounds, matching the $n/g$ source scale.
