The Power of Orientation in Symmetry-Breaking

Symmetry breaking is a fundamental operation in distributed computing. It has applications to important problems such as graph vertex and edge coloring, maximal independent sets, and the like. Deterministic algorithms for symmetry breaking that run in a polylogarithmic number of rounds are not known. However, randomized algorithms that run in poly-logarithmic number of rounds are known starting from Luby´s algorithm. Recently, orientation on edges was considered and it was shown that an O(Δ) coloring of the vertices of a given oriented graph can be arrived at using essentially O(log Δ + √log n) bits of communication. In this paper we further demonstrate the power of orientation on edges in symmetry-breaking. We present efficient algorithms to construct fractional independent sets in constant degree graphs using very low order communication between the vertices. For instance, we show that in bounded degree graphs and planar graphs, it is possible to construct a fractional independent set by exchanging O(1) bits. Further, we present algorithms to construct maximal independent sets in bounded degree graphs and oriented trees. Our algorithm for constructing an MIS of an oriented tree uses only O(log n) bits of communication.