How to reconcile fault-tolerant interval intersection with the Lipschitz condition

Summary.

We present a new fault-tolerant intersection function ${\boldmath{\cal F}}$, which satisfies the Lipschitz condition for the uniform metric and is optimal among all functions with this property. ${\boldmath{\cal F}}$ thus settles Lamport's question about such a function raised in [5]. Our comprehensive analysis reveals that ${\boldmath{\cal F}}$ has exactly the same worst-case performance as the optimal Marzullo function ${\boldmath{\cal M}}$, which does not satisfy a Lipschitz condition. The utilized modelling approach in conjunction with a powerful hybrid fault model ensures compatibility of our results with any known application framework, including replicated sensors and clock synchronization.