The String of Diamonds Is Tight for Rumor Spreading

For a rumor spreading protocol, the spread time is defined as the first time that everyone learns the rumor. We compare the synchronous push&pull rumor 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 (in Proceedings of ACM Symposium on Principles of Distributed Computing, 2016). Our bound is tight up to a factor of O(log n), as illustrated by the string of diamonds graph.