Greedy Gossiping

TL;DR

This paper investigates the greedy gossiping problem, determining the maximum gossip knowledge achievable with a limited number of calls and proving the optimality of the greedy strategy for most cases.

Contribution

It introduces and solves the greedy gossiping problem, establishing the optimality of the greedy calling strategy for nearly all call counts less than the known total.

Findings

Maximum gossip knowledge reaches n^2 only at m=2n-3 calls.

For each m<2n-4, the greedy strategy is proven optimal.

The problem extends the classic Gossiping Problem with a focus on limited calls.

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.