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    The string of diamonds is nearly tight for rumour spreading

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    For a rumour spreading protocol, the spread time is defined as the first time that everyone learns the rumour. We compare the synchronous push&pull rumour spreading protocol with its asynchronous variant, and show that for any nn-vertex graph and any starting vertex, the ratio between their expected spread times is bounded by O(n1/3log2/3n)O \left({n}^{1/3}{\log^{2/3} n}\right). This improves the O(n)O(\sqrt n) upper bound of Giakkoupis, Nazari, and Woelfel (in Proceedings of ACM Symposium on Principles of Distributed Computing, 2016). Our bound is tight up to a factor of O(logn)O(\log n), as illustrated by the string of diamonds graph. We also show that if for a pair α,β\alpha,\beta of real numbers, there exists infinitely many graphs for which the two spread times are nαn^{\alpha} and nβn^{\beta} in expectation, then 0α10\leq\alpha \leq 1 and αβ13+23α\alpha \leq \beta \leq \frac13 + \frac23 \alpha; and we show each such pair α,β\alpha,\beta is achievable.Comment: Will be presented at RANDOM'2017 conference. 14 pages, Theorem 2.5 added in this versio
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