Efficient randomized test-and-set implementations

Abstract

We study randomized test-and-set (TAS) implementations from registers in the asynchronous shared memory model with n processes. We introduce the problem of group election, a natural variant of leader election, and propose a framework for the implementation of TAS objects from group election objects. We then present two group election algorithms, each yielding an efficient TAS implementation. The first implementation has expected max-step complexity O(log * k) in the location-oblivious adversary model, and the second has expected max-step complexity O(log log k) against any read/write-oblivious adversary, where k ≤ n is the contention. These algorithms improve the previous upper bound by Alistarh and Aspnes [2] of O(log log n) expected max-step complexity in the oblivious adversary model.We also propose a modification to a TAS algorithm by Alistarh, Attiya, Gilbert, Giurgiu, and Guerraoui [5] for the strong adaptive adversary, which improves its space complexity from superlinear to linear, while maintaining its O(log n) expected max-step complexity. We then describe how this algorithm can be combined with any randomized TAS algorithm that has expected max-step complexity T (n) in a weaker adversary model, so that the resulting algorithm has O(log n) expected max-step complexity against any strong adaptive adversary and O(T (n)) in the weaker adversary model.Finally, we prove that for any randomized 2-process TAS algorithm, there exists a schedule determined by an oblivious adversary such that with probability at least 1/4 t one of the processes needs at least t steps to finish its TAS operation. This complements a lower bound by Attiya and Censor-Hillel [7] on a similar problem for n ≥ 3 processes.can lose. I.e., if all participating processes finish the protocol, then exactly one of them returns win and all others return lose. Obviously, any TAS object immediately yields a leader election protocol: Each process executes a single TAS() operation and returns win if the TAS() call returns 0, or lose if TAS() returns 1. Similarly, a leader election algorithm, together with one additional register, can be used to implement a linearizable TAS object with just a constant increase in the number of steps [15]. Similar transformations from leader election to linearizable TAS objects are implicit in several TAS algorithms, e.g., [1,2].Early randomized TAS implementations assumed a strong adaptive adversary model, where the adversary bases its scheduling decisions on the entire past history of events, including the coin flips by processes. Tromp, and Vitányi [22,23] presented a randomized implementation for two processes which has constant expected max-step complexity and constant space complexity against any strong adaptive adversary. (The max-step complexity of an execution is the maximum number of steps any process needs to finish its algorithm in the execution. See Section 2.2 for formal definitions and a discussion.) Afek, Gafni, Tromp, and Vitányi [1] gave a deterministic implementation of a TAS object for n processes,...

Lower Bounds for Shared-Memory Leader Election under Bounded Write Contention

“…Follow-up work on time upper bounds has extended these results to the adaptive setting, showing logarithmic expected step complexity in the number of participating processors k [4,16]. Further, Giakkoupis and Woelfel [16] showed that, if the adversary is oblivious to the randomness used by the algorithm, O(log k) step complexity is achievable, improving upon a previous sublogarithmic upper bound by Alistarh and Aspnes [2]. Another related line of work has focused on the space complexity of this problem, which is now resolved.…”

“…Another related line of work has focused on the space complexity of this problem, which is now resolved. Specifically, it is known that Ω(log n) distinct registers are necessary [28,16], and a breakthrough result by Giakkoupis, Helmi, Higham, and Woelfel [15] provided the first asymptotically matching upper bound of O(log n), improving upon an O( √ n) algorithm by the same authors [14]. The clear gap in the complexity landscape for this problem concerns time complexity lower bounds.…”

“…1 Due to the simplicity of the problem, none of the classic lower bound approaches, e.g. [23,24,22], apply, and resolving the time complexity of shared-memory leader election is known to be a challenging open problem [2,16]. Moreover, given that the step complexities of shared-memory consensus [8] and renaming [3] have been resolved, leader election remains one of the last basic objects for which no tight complexity bounds are known.…”