A King in every two consecutive tournaments

TL;DR

This paper extends Landau's theorem by proving that in any sequence of two tournaments, there exists a processor whose initial input can reach all others within two communication rounds, regardless of adversarial choices.

Contribution

It generalizes the concept of a King in a single tournament to sequences of tournaments, showing the existence of a universally reachable processor after two rounds under adversarial conditions.

Findings

Existence of a processor reaching all others in two rounds despite adversarial tournament sequences

Generalization of Landau's theorem to multiple tournaments

Applicability to communication networks with dynamic topologies

Abstract

We think of a tournament $T=([n], E)$ as a communication network where in each round of communication processor $P_i$ sends its information to $P_j$, for every directed edge $ij \in E(T)$. By Landau's theorem (1953) there is a King in $T$, i.e., a processor whose initial input reaches every other processor in two rounds or less. Namely, a processor $P_{\nu}$ such that after two rounds of communication along $T$'s edges, the initial information of $P_{\nu}$ reaches all other processors. Here we consider a more general scenario where an adversary selects an arbitrary series of tournaments $T_1, T_2,\ldots$, so that in each round $s=1, 2, \ldots$, communication is governed by the corresponding tournament $T_s$. We prove that for every series of tournaments that the adversary selects, it is still true that after two rounds of communication, the initial input of at least one processor reaches everyone. Concretely, we show that for every two tournaments $T_1, T_2$ there is a vertex in $[n]$ that can reach all vertices via (i) A step in $T_1$, or (ii) A step in $T_2$ or (iii) A step in $T_1$ followed by a step in $T_2$. }