The Information Flow Problem on Clock Networks

TL;DR

This paper investigates the information flow problem on clock networks, demonstrating solutions for specific network configurations and establishing bounds for when the problem can be solved.

Contribution

It provides new results on the solvability of the information flow problem on clock networks with particular edge sets, including explicit bounds for general cases.

Findings

Solved the problem for R = {1,2,...,r} for all n ≥ r

Established solvability for networks with gcd(R)=1 when n ≥ 3r^3

Provided bounds for the information flow problem on general clock networks

Abstract

The information flow problem on a network asks whether $r$ senders, $v_1,v_2, \ldots ,v_r$ can each send messages to $r$ corresponding receivers $v_{n+1}, \ldots ,v_{n+r}$ via intermediate nodes $v_{r+1}, \ldots ,v_n$. For a given finite $R \subset \mathbb{Z}^+$, the clock network $N_n(R)$ has edge $v_iv_k$ if and only if $k>r$ and $k-i \in R$. We show that the information flow problem on $N_n(\{1,2, \ldots ,r\})$ can be solved for all $n \geq r$. We also show that for any finite $R$ such that $\gcd(R)=1$ and $r = \max(R)$, we show that the information flow problem can be solved on $N_n(R)$ for all $n \geq 3r^3$.