Institution: Stanford University, Department of Computer Science

Title: A class of solutions to the gossip problem

Author: West, Douglas B.

Date: November 1978

Abstract: We characterize and count optimal solutions to the gossip problem in which no one hears his own information. That is, we consider graphs with n vertices where the edges have a linear ordering such that an increasing path exists from each vertex to every other, but there is no increasing path from any vertex to itself. Such graphs exist only when n is even, in which case the fewest number of edges is 2n-4, as in the original gossip problem. We characterize optimal solutions of this sort (NOHO-graphs) using a correspondence with a set of permutations and binary sequences. This correspondence enables us to count these solutions and several subclasses of solutions. The numbers of solutions in each class are simple powers of 2 and 3, with exponents determined by n. We also show constructively that NOHO-graphs are planar and Hamiltonian, and we mention applications to related problems.

http://i.stanford.edu/pub/cstr/reports/cs/tr/78/693/CS-TR-78-693.pdf