Stabilizing Maximal Independent Set in Unidirectional Networks is Hard

TL;DR

This paper investigates the complexity of self-stabilizing algorithms for constructing maximal independent sets in unidirectional networks, proving impossibility results for deterministic solutions and proposing probabilistic protocols with various trade-offs.

Contribution

It demonstrates the impossibility of deterministic self-stabilizing maximal independent set algorithms in uniform unidirectional networks and introduces probabilistic protocols with different complexity and scheduling guarantees.

Findings

Deterministic self-stabilization is impossible in uniform networks.

Silence property cannot be guaranteed for deterministic or probabilistic protocols.

Probabilistic protocols with polynomial expected time are proposed for different scheduling models.

Abstract

A distributed algorithm is self-stabilizing if after faults and attacks hit the system and place it in some arbitrary global state, the system recovers from this catastrophic situation without external intervention in finite time. In this paper, we consider the problem of constructing self-stabilizingly a \emph{maximal independent set} in uniform unidirectional networks of arbitrary shape. On the negative side, we present evidence that in uniform networks, \emph{deterministic} self-stabilization of this problem is \emph{impossible}. Also, the \emph{silence} property (\emph{i.e.} having communication fixed from some point in every execution) is impossible to guarantee, either for deterministic or for probabilistic variants of protocols. On the positive side, we present a deterministic protocol for networks with arbitrary unidirectional networks with unique identifiers that exhibits polynomial space and time complexity in asynchronous scheduling. We complement the study with probabilistic protocols for the uniform case: the first probabilistic protocol requires infinite memory but copes with asynchronous scheduling, while the second probabilistic protocol has polynomial space complexity but can only handle synchronous scheduling. Both probabilistic solutions have expected polynomial time complexity.