Fault Tolerant Network Constructors

In this work we examine what graphs (networks) can be stably and distributedly formed if adversarial crash failures may happen. Our dynamic graphs are constructed by fixed memory protocols, which are like population protocols but also allow nodes to form/delete links when pairwise interactions occur (Network Constructors). First, we consider standard Network Constructors (i.e. without fault notifications) and we partially characterize the class of such protocols that are fault-tolerant. We show that the class is non-empty but small. Then, we assume a minimal form of fault notifications (N-NET protocols) and we give fault-tolerant protocols for constructing graphs such as spanning star and spanning line. We show a fault tolerant construction of a Turing Machine $M$ that allows a fault tolerant construction of any graph accepted by $M$ in linear space with a population waste of $min\{n/2 + f(n), \; n\}$ (due to the construction of $M$), where $f(n)$ is an upper bound on the number of faults. We then extend the class of graphs to any graph accepted in $O(n^2)$ space, by allowing $min\{2n/3 + f(n), \; n\}$ waste. Finally, we use non-constant memory to achieve a general fault-tolerant restart of any N-NET protocol with no waste.