Poly-logarithmic adaptive algorithms require revealing primitives

Abstract

This paper studies the step complexity of adaptive algorithms, depending on the revealing properties of the primitives used, namely, how many processes are revealed when concurrently applying the primitive (in specific situations).When only 0- or 1-revealing primitives are used, e.g., reads, writes, test&set, compare&swap and LL/SC, then for any collect algorithm there is an execution in which k processes collectively perform Ω(k2) steps, provided k∈O(loglogn). This implies that any adaptive collect algorithm has an Ω(k) amortized (and hence, worst-case) step complexity in an execution with total contention k∈O(loglogn). The lower bound applies for snapshot and renaming, both one-shot and long-lived. While there are snapshot algorithms whose step complexity is polylogarithmic in n using only reads and writes, there is no adaptive algorithm whose step complexity is polylogarithmic in the contention, even when compare&swap and LL/SC are used.Primitives like fetch&inc are more revealing, and using them admits snapshot algorithms with O(logk) step complexity, where k is the total or the point contention. These algorithms combine a renaming algorithm with a mechanism for propagating values so they can be quickly collected.The main implication of these results is that the step complexity of adaptive algorithms depends on the revealing power of the primitives used. Even conditional primitives that allow to solve consensus for any number of processes, like compare&swap and LL/SC, do not improve the step complexity of adaptive algorithms.