少量共享随机比特足以实现常数轮的几乎稳定匹配
A Few Shared Random Bits Suffice for Constant-Round Almost Stable Matching
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中文总结 AI 辅助
本研究提出度数保护冻结规则，仅用少量共享随机比特就在一般二分图上实现了常数轮的几乎稳定匹配求解，突破了以往算法受图度数和规模限制的瓶颈，并拓展到无共享随机的CONGEST模型与MPC模型。
中文摘要 AI 辅助
我们证明，在一般二分图$G=(V,E)$上，仅需使用少量共享随机比特，就能在常数分布式轮次内求解几乎稳定匹配问题。具体而言，在$\textsf{CONGEST}$（拥塞）模型中，我们计算得到一个匹配，其阻塞对的期望数量至多为$\tau |E|$，所需轮次为$O\left(\frac{\log(1/\tau)}{\tau^4}\right)$，仅使用$O\left(\log(1/\tau)\right)$个共享随机比特。因此，对于任意常数$\tau>0$，轮次复杂度为$O(1)$，与顶点数量和最大度数无关。\n 以往的算法仅在有界度数或几乎正则图上能达到常数轮次复杂度；在一般图上，它们的轮次复杂度与$n$呈多重对数关系。我们的核心技术思路是一种度数保护冻结规则，该规则允许通过单一全局计费论证处理差异极大的度数，避免了以往工作中使用的$\Theta(\log n)$个连续度数阈值。共享随机比特仅用于选择公共的随机输出迭代。\n 作为推论，通过低直径分解，我们得到了一个无需预共享随机性的$O\left(\frac{\log(1/\tau)}{\tau^4} + \frac{\log n}{\tau} \right)$轮$\textsf{CONGEST}$算法；以及一个在总内存为线性的完全可扩展大规模并行计算（$\textsf{MPC}$）模型下，轮次复杂度为$O\left(\frac{\log(1/\tau)}{\tau^4}\right)$的算法。
英文摘要
We show that almost stable matching can be solved in constant distributed rounds on general bipartite graphs $G=(V,E)$ using only a few shared random bits. Specifically, in the $\congest$ model, we compute a matching whose expected number of blocking pairs is at most $\varepsilon |E|$ in $O\left(\frac{\log(1/\varepsilon)}{\varepsilon^4}\right)$ rounds using $O\left(\log(1/\varepsilon)\right)$ shared random bits. Thus, for every constant $\varepsilon>0$, the round complexity is $O(1)$, independent of the number of vertices and the maximum degree. Previous algorithms achieve constant round complexity only for bounded-degree or almost-regular graphs; on general graphs, their round complexity depends polylogarithmically on $n$. Our main technical idea is a degree-guarded freezing rule that allows widely varying degrees to be handled by a single global charging argument, avoiding the $Θ(\log n)$ successive degree thresholds used in previous work. The shared random bits are used only to select a common random output iteration. As consequences, we obtain an $O\left( \frac{\log(1/\varepsilon)}{\varepsilon^4} + \frac{\log n}{\varepsilon} \right)$-round $\congest$ algorithm without pre-shared randomness, via a low-diameter decomposition, and an $O\left(\frac{\log(1/\varepsilon)}{\varepsilon^4}\right)$-round algorithm in the fully-scalable Massively Parallel Computation ($\mpc$) model with linear total memory.
