The goal of this paper is to understand the complexity of a key symmetry breaking problem, namely the (α,β)-ruling set problem in the graph streaming model. Given a graph G = (V,E), an (α, β)-ruling set is a subset I ⊆ V such that the distance between any two vertices in I is at least α and the distance between a vertex in V and the closest vertex in I is at most β. This is a fundamental problem in distributed computing where it finds applications as a useful subroutine for other problems such as maximal matching, distributed colouring, or shortest paths. Additionally, it is a generalization of MIS, which is a (2,1)-ruling set.

Our main results are two algorithms for (2,2)-ruling sets:

1) In adversarial streams, where the order in which edges arrive is arbitrary, we give an algorithm with Õ(n^{4/3}) space, improving upon the best known algorithm due to Konrad et al. [DISC 2019], with space Õ(n^{3/2}).

2) In random-order streams, where the edges arrive in a random order, we give a semi-streaming algorithm, that is an algorithm that takes Õ(n) space. Finally, we present new algorithms and lower bounds for (α,β)-ruling sets for other values of α and β. Our algorithms improve and generalize the previous work of Konrad et al. [DISC 2019] for (2,β)-ruling sets, while our lower bound establishes the impossibility of obtaining any non-trivial streaming algorithm for (α,α-1)-ruling sets for all even α > 2.