Randomness vs. Time in Anonymous Networks

Abstract

In an anonymous network, symmetry breaking tasks can only be solved if randomization is available. But how many random bits are required to solve any such task? As it turns out, the answer to this question depends on the desired runtime of the algorithm.

Since any randomized anonymous network algorithm can be decomposed into a randomized 2-hop coloring stage and a deterministic stage, we tackle the question by focusing on the randomized stage. We establish that for any reasonable target function f, there is a randomized 2-hop coloring scheme running in $\mathcal {O} (f(n))$ time. Our coloring scheme allows to trade an increase in runtime by a factor of d for a decrease by the $d^\text {th}$ root in the random bit complexity.

To show that the achieved trade-off is asymptotically optimal for any choice of f, we establish a trade-off lower bound. Our bounds yield that it is sufficient to consider the cases when f is between $\Omega (\log ^*n)$ and $\mathcal {O} (\log \log n)$. We obtain that for the two extreme cases, i.e., where $f \in \Theta (\log ^*n)$ and $f \in \Theta (\log \log n)$, the random bit complexity is $\Theta (\root d \of {n})$ and $\Theta (\log n)$, respectively, for any constant d. The trade-off achieved by our scheme is asymptotically optimal for any f, i.e., reducing the runtime must lead to an increase in the random bit complexity.

Due to space constraints, in this extended abstract all proofs had to be omitted. The full version of this paper is available at http://disco.ethz.ch/publications/DISC2015-coloring.pdf.