A geometric Achlioptas process

The random geometric graph is obtained by sampling n points from the unit square (uniformly at random and independently), and connecting two points whenever their distance is at most r, for some given r = r(n). We consider the following variation on the random geometric graph: in each of n rounds in total, a player is offered two random points from the unit square, and has to select exactly one of these two points for inclusion in the evolving geometric graph. We study the problem of avoiding a linear-sized (or “giant”) component in this setting. Specifically, we show that for any r ≪ (n log log n) −1/3 there is a strategy that succeeds with high probability in keeping all component sizes sublinear. We also show that this is tight in the following sense: for any r ≫ (n log log n) −1/3, with high probabiliy the player will be forced to create a component of size (1 − o(1))n, no matter how he plays. We also prove that the corresponding offline problem exhibits a similar threshold behavior at r(n) = Θ(n −1/3). These findings should be compared to the existing results for the (ordinary) random geometric graph: there a giant component arises with high probability once r is of order n−1/2. Thus our results show, in particular, that in the geometric setting the power of choices can be exploited to a much larger extent than in the classical Erdős-Rényi random graph, where the appearance of a giant component can only be delayed by a constant factor.