Approximation Algorithm for Estimating Distances in Distributed Virtual Environments

This article deals with the issue of guaranteeing properties in Distributed Virtual Environments (DVEs) without a server. This issue is particularly relevant in the case of online games, that operate in a fully distributed framework and for which network resources such as bandwidth are the critical resources. Players typically need to know the distance between their character and other characters, at least approximately. They all share the same position estimation algorithm but, in general, do not know the current positions of others. We provide a synchronized distributed algorithm \(\mathcal {A}_{lc}\) to guarantee, at any time, that the estimated distance \(d_{est}\) between any pair of characters A and B is always a \(1+\varepsilon \) approximation of the current distance \(d_{act}\), regardless of movement pattern, and then prove that if characters move randomly on a \(d\)-dimensional grid, or follow a random continuous movement on up to three dimensions, the number of messages of \(\mathcal {A}_{lc}\) is optimal up to a constant factor. In a more practical setting, we also show that the number of messages of \(\mathcal {A}_{lc}\) for actual game traces is much less than the standard algorithm sending actual positions at a given frequency.