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The string of diamonds is nearly tight for rumour spreading
Published online by Cambridge University Press: 04 November 2019
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For 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.
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Footnotes
†
A preliminary version of this paper appeared in Proceedings of the 21st International Workshop on Randomization and Computation (RANDOM’2017). This full version includes a new result, namely Theorem 2.5.
§
Supported by NSERC.
¶
Supported by an NSERC Postdoctoral Fellowship and a Simons–Berkeley Research Fellowship. Part of this work was done while this author was at the Simons Institute for the Theory of Computing at UC Berkeley. Currently at McGill University.
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