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We study randomized test-and-set (TAS) implementations from registers in theasynchronous shared memory model with n processes. We introduce the problem ofgroup election, a natural variant of leader election, and propose a frameworkfor the implementation of TAS objects from group election objects. We thenpresent two group election algorithms, each yielding an efficient TASimplementation. The first implementation has expected max-step complexity$O(\log^\ast k)$ in the location-oblivious adversary model, and the second hasexpected max-step complexity $O(\log\log k)$ against any read/write-obliviousadversary, where $k\leq n$ is the contention. These algorithms improve theprevious upper bound by Alistarh and Aspnes [2] of $O(\log\log n)$ expectedmax-step complexity in the oblivious adversary model. We also propose amodification to a TAS algorithm by Alistarh, Attiya, Gilbert, Giurgiu, andGuerraoui [5] for the strong adaptive adversary, which improves its spacecomplexity from super-linear to linear, while maintaining its $O(\log n)$expected max-step complexity. We then describe how this algorithm can becombined with any randomized TAS algorithm that has expected max-stepcomplexity $T(n)$ in a weaker adversary model, so that the resulting algorithmhas $O(\log n)$ expected max-step complexity against any strong adaptiveadversary and $O(T(n))$ in the weaker adversary model. Finally, we prove thatfor any randomized 2-process TAS algorithm, there exists a schedule determinedby an oblivious adversary such that with probability at least $(1/4)^t$ one ofthe processes needs at least t steps to finish its TAS operation. Thiscomplements a lower bound by Attiya and Censor-Hillel [7] on a similar problemfor $n\geq 3$ processes.

Efficient Randomized Test-And-Set Implementations