Feedback from Nature: Simple Randomised Distributed Algorithms for Maximal Independent Set Selection and Greedy Colouring

TL;DR

This paper introduces simple, efficient distributed algorithms for maximal independent set selection and greedy colouring that operate under extremely limited communication conditions, achieving optimal or near-optimal performance.

Contribution

It presents novel algorithms incorporating local feedback for maximal independent set selection and a new approach for greedy colouring under harsh conditions, improving efficiency and robustness.

Findings

Maximal independent set algorithm runs in expected O(log n) rounds with O(1) messages per node.

Greedy colouring algorithm runs in expected O(Δ + log n) time with O(1) messages per node.

Previous methods without feedback cannot achieve better than Ω(log^2 n) expected time.

Abstract

We propose distributed algorithms for two well-established problems that operate efficiently under extremely harsh conditions. Our algorithms achieve state-of-the-art performance in a simple and novel way. Our algorithm for maximal independent set selection operates on a network of identical anonymous processors. The processor at each node has no prior information about the network. At each time step, each node can only broadcast a single bit to all its neighbours, or remain silent. Each node can detect whether one or more neighbours have broadcast, but cannot tell how many of its neighbours have broadcast, or which ones. We build on recent work of Afek et al. which was inspired by studying the development of a network of cells in the fruit fly~\cite{Afek2011a}. However we incorporate for the first time another important feature of the biological system: varying the probability value used at each node based on local feedback from neighbouring nodes. Given any $n$-node network, our algorithm achieves the optimal expected time complexity of $O(\log n)$ rounds and the optimal expected message complexity of $O(1)$ single-bit messages broadcast by each node.We also show that the previous approach, without feedback, cannot achieve better than $\Omega(\log^2 n)$ expected time complexity, whatever global scheme is used to choose the probabilities. Our algorithm for distributed greedy colouring works under similar harsh conditions: each identical node has no prior information about the network, can only broadcast a single message to all neighbours at each time step representing a desired colour, and can only detect whether at least one neighbour has broadcast each colour value. We show that our algorithm has an expected time complexity of $O(\Delta+\log n)$, where $\Delta$ is the maximum degree of the network, and expected message complexity of $O(1)$ messages broadcast by each node.