Gossip Latin square and the meet-all gossipers problem

Given a network of n=2ˆk gossipers, we want to schedule a cyclic calendar of meetings between all of them, such that: (1) each gossiper meets (gossips) only once a day, with one other gossiper, (2) in every (n-1) consecutive days, each gossiper meets all other gossipers, and (3) every gossip, initiated by any gossiper, will reach all gossipers within k=log@(n) days. In this paper we study the above stated meet-all gossipers problem, by defining and constructing the Gossip Latin Square (GLS), a combinatorial structure which solves the problem. We then present an efficient construction of GLS, based on maximal Fibonacci LFSR.