eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-10-04
6:1
6:18
10.4230/LIPIcs.DISC.2021.6
article
Ruling Sets in Random Order and Adversarial Streams
Assadi, Sepehr
1
Dudeja, Aditi
1
Department of Computer Science, Rutgers University, Piscataway, NJ, USA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol209-disc2021/LIPIcs.DISC.2021.6/LIPIcs.DISC.2021.6.pdf
Symmetry breaking
Ruling sets
Lower bounds
Communication Complexity