Reconfiguring Arrays with Faults Part I: Worst-Case Faults

Abstract

In this paper we study the ability of array-based networks to tolerate worst-case faults. We show that an $N \times N$ two-dimensional array can sustain $N^{1-\epsilon}$ worst-case faults, for any fixed $\epsilon > 0$, and still emulate $T$ steps of a fully functioning $N \times N$ array in $O(T+N)$ steps, i.e., with only constant slowdown. Previously, it was known only that an array could tolerate a constant number of faults with constant slowdown. We also show that if faulty nodes are allowed to communicate, but not compute, then an $N$-node one-dimensional array can tolerate $\log^k N$ worst-case faults, for any constant $k > 0$, and still emulate a fault-free array with constant slowdown, and this bound is tight.