Linearizable implementations do not suffice for randomized distributed computation

Abstract

Linearizability is the gold standard among algorithm designers for deducing the correctness of a distributed algorithm using implemented shared objects from the correctness of the corresponding algorithm using atomic versions of the same objects. We show that linearizability does not suffice for this purpose when processes can exploit randomization, and we discuss the existence of alternative correctness conditions. This paper makes the following contributions: 1. Various examples demonstrate that using well-known linearizable implementations of objects (e.g., snapshots) in place of atomic objects can change the probability distribution of the outcomes that the adversary is able to generate. In some cases, an oblivious adversary can create a probability distribution of outcomes for an algorithm with implemented, linearizable objects, that not even a strong adversary can generate for the same algorithm with atomic objects. 2. A new correctness condition for shared object implementations, called strong linearizability, is defined. We prove that a strong adversary (i.e., one that sees the outcome of each coin flip immediately) gains no additional power when atomic objects are replaced by strongly linearizable implementations. In general, no strictly weaker correctness condition suffices to ensure this. We also show that strong linearizability is a local and composable property. 3. In contrast to the situation for the strong adversary, for a natural weaker adversary (one that cannot see a process' coin flip until its next operation on a shared object) we prove that there is no correspondingly general correctness condition. Specifically, any linearizable implementation of counters from atomic registers and load-linked/store-conditional objects, that satisfies a natural locality property, necessarily gives the weak adversary more power than it has with atomic counters.

Randomized mutual exclusion with sub-logarithmic RMR-complexity

“…As observed by Golab, Higham, and Woelfel [12], replacing atomic objects with linearizable implementations can increase the power of the strong adversary and thus might increase the expected complexity of a randomized algorithm. For example, if a non-atomic but linearizable operation o overlaps with a coin flip made by another process, then the adversary might schedule other events in such a way that the linearization point of o occurs either before the coin flip or after it, depending on the outcome of the coin flip.…”

“…It follows from [13] and [11], that these variables can be implemented from only reads and writes while maintaining asymptotic worst-case RMR complexities. Although expected RMR complexity is not necessarily preserved if one replaces atomic operations with linearizable ones (see [12] for a discussion of these issues), any linearizable CAS implementation with O(1) RMR complexity can be used in our algorithm.…”

Tight Bounds for Asynchronous Renaming

“…Recent results by Golab et al [Golab et al 2011] show that linearizability is not a sufficient correctness condition when randomization is employed. More precisely, they show that the adversary can gain extra power whenever a randomized algorithm uses other (deterministic or randomized) linearizable implementations as sub-algorithms.…”

A tight RMR lower bound for randomized mutual exclusion

“…It follows from [14], that these implementations can be used in a randomized adaptive adversary model in place of atomic compare&swap objects without increasing the expected RMR complexity. Therefore, our lower bound holds even when the system provides atomic compare&swap objects.…”

“…We assume that schedules are generated by an adaptive adversary (see, e.g., [5,14]), and thus can depend on the random values generated by the processes. In the adaptive adversary model, after each step the adversary decides which process takes the next step, and in order to make this decision it can take all preceding events into account, including the results of past coin flips, but not the results of any of the future coin flips.…”