Exploration of Periodically Varying Graphs

TL;DR

This paper investigates the exploration problem in periodically varying graphs, establishing necessary conditions, lower bounds, and providing optimal algorithms for different system types, advancing understanding of dynamic network exploration.

Contribution

It introduces tight bounds and necessary conditions for exploration in PV graphs, along with simple, optimal algorithms for anonymous and id-equipped systems.

Findings

Necessary and sufficient conditions for exploration

Lower bounds on exploration time in PV graphs

Constructive algorithms for optimal exploration

Abstract

We study the computability and complexity of the exploration problem in a class of highly dynamic graphs: periodically varying (PV) graphs, where the edges exist only at some (unknown) times defined by the periodic movements of carriers. These graphs naturally model highly dynamic infrastructure-less networks such as public transports with fixed timetables, low earth orbiting (LEO) satellite systems, security guards' tours, etc. We establish necessary conditions for the problem to be solved. We also derive lower bounds on the amount of time required in general, as well as for the PV graphs defined by restricted classes of carriers movements: simple routes, and circular routes. We then prove that the limitations on computability and complexity we have established are indeed tight. In fact we prove that all necessary conditions are also sufficient and all lower bounds on costs are tight. We do so constructively presenting two worst case optimal solution algorithms, one for anonymous systems, and one for those with distinct nodes ids. An added benefit is that the algorithms are rather simple.