Greedy Gossiping

Abstract

The renowned Gossiping Problem (1971) asks the following. There are $n$ people who each know an item of gossip. In a telephone call, two people share all the gossip they know. How many calls are needed for all of them to be informed of all the gossip? If $n\ge 4$, the answer is $2n-4$. We initiate and solve the related Greedy Gossiping Problem: given a fixed number $m<2n-4$ of calls, at most how much gossip can be known altogether? If every call increases the total knowledge of gossip as much as possible, the sum reaches $n^2$ only when $m=2n-3$. Our main result is that surprisingly, for each $m<2n-4$, this calling strategy is optimal.