# Lossy Gossip and Composition of Metrics

Discrete & Computational Geometry, Jun 2015

We study the monoid generated by $n \times n$ distance matrices under tropical (or min-plus) multiplication. Using the tropical geometry of the orthogonal group, we prove that this monoid is a finite polyhedral fan of dimension $\left( {\begin{array}{c}n\\ 2\end{array}}\right)$, and we compute the structure of this fan for $n$ up to $5$. The monoid captures gossip among $n$ gossipers over lossy phone lines, and contains the gossip monoid over ordinary phone lines as a submonoid. We prove several new results about this submonoid as well. In particular, we establish a sharp bound on chains of calls in each of which someone learns something new.

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Lossy Gossip and Composition of Metrics, Discrete & Computational Geometry, 2015, 890-913, DOI: 10.1007/s00454-015-9666-1