AbstarctFor a rumour spreading protocol, the spread time is defined as the first time everyone learns the rumour. We compare the synchronous push&pull rumour spreading protocol with its asynchronous variant, and show that for any n-vertex graph and any starting vertex, the ratio between their expected spread times is bounded by $O({n^{1/3}}{\log ^{2/3}}n)$. This improves the $O(\sqrt n)$ upper bound of Giakkoupis, Nazari and Woelfel (2016). Our bound is tight up to a factor of O(log n), as illustrated by the string of diamonds graph. We also show that if, for a pair α, β of real numbers, there exist infinitely many graphs for which the two spread times are nα and nβ in expectation, then $0 \le \alpha \le 1$ and $\alpha \le \beta \le {1 \over 3} + {2 \over 3} \alpha $; and we show each such pair α, β is achievable.

中文翻译：

---

谣言传播的钻石串几乎紧绷

摘要对于谣言传播协议，传播时间定义为每个人第一次得知谣言的时间。我们将同步推拉谣言传播协议与其异步变体进行比较，并表明对于任何n-顶点图和任何起始顶点，它们的预期传播时间之间的比率为$O({n^{1/3}}{\log ^{2/3}}n)$. 这提高了$O(\sqrt n)$Giakkoupis、Nazari 和 Woelfel (2016) 的上限。我们的界限是紧到一个因素○（日志n)，如钻石串图所示。我们还证明了如果，对于一对α,β对于实数，存在无限多个图，其两个传播时间为nα和nβ在期待中，那么$0 \le \alpha \le 1$和$\alpha \le \beta \le {1 \over 3} + {2 \over 3} \alpha $; 我们展示每一对这样的对α,β是可以实现的。