In light of recent advances in non-volatile main memory technology, Golab and Ramaraju reformulated the traditional mutex problem into the novel Recoverable Mutual Exclusion (RME) problem. In the best known solution for RME, due to Golab and Hendler from PODC 2017, a process incurs at most O( log n log log n ) remote memory references (RMRs) per passage, where a passage is an interval from when a process enters the Try section to when it subsequently returns to Remainder. Their algorithm, however, guarantees this bound only for cache-coherent (CC) multiprocessors, leaving open the question of whether a similar bound is possible for distributed shared memory (DSM) multiprocessors.We answer this question affirmatively by designing an algorithm that satisfies the same complexity bound as Golab and Hendler's for both CC and DSM multiprocessors. Our algorithm has some additional advantages over Golab and Hendler's: (i) its Exit section is wait-free, (ii) it uses only the Fetch-and-Store instruction, and (iii) on a CC machine our algorithm needs each process to have a cache of only O(1) words, while their algorithm needs O(n) words .

The disjoint set union problem is a basic problem in data structures with a wide variety of applications. We extend a known efficient sequential algorithm for this problem to obtain a simple and efficient concurrent wait-free algorithm running on an asynchronous parallel random access machine (APRAM). Crucial to our result is the use of randomization. Under a certain independence assumption, for a problem instance in which there are n elements, m operations, and p processes, our algorithm does Θ m α n, m np + log np m + 1 expected work, where the expectation is over the random choices made by the algorithm and α is a functional inverse of Ackermann's function. In addition, each operation takes O(log n) steps with high probability.Our algorithm is significantly simpler and more efficient than previous algorithms proposed by Anderson and Woll. Under our independence assumption, our algorithm achieves almost-linear speed-up for applications in which all or most of the processes can be kept busy. * This paper is a significantly revised version of a conference paper [JT16]

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We develop and analyze concurrent algorithms for the disjoint set union (“union-find” ) problem in the shared memory, asynchronous multiprocessor model of computation, with CAS (compare and swap) or DCAS (double compare and swap) as the synchronization primitive. We give a deterministic bounded wait-free algorithm that uses DCAS and has a total work bound of $$O\biggl ( m \cdot \left( \log {\left( \frac{np}{m} + 1 \right) } + \alpha {\left( n, \frac{m}{np} \right) } \right) \biggr )$$ O ( m · log np m + 1 + α n , m np ) for a problem with n elements and m operations solved by p processes, where $$\alpha $$ α is a functional inverse of Ackermann’s function. We give two randomized algorithms that use only CAS and have the same work bound in expectation. The analysis of the second randomized algorithm is valid even if the scheduler is adversarial. Our DCAS and randomized algorithms take $$O(\log n)$$ O ( log n ) steps per operation, worst-case for the DCAS algorithm, high-probability for the randomized algorithms. Our work and step bounds grow only logarithmically with p, making our algorithms truly scalable. We prove that for a class of symmetric algorithms that includes ours, no better step or work bound is possible. Our work is theoretical, but Alistarh et al (In search of the fastest concurrent union-find algorithm, 2019), Dhulipala et al (A framework for static and incremental parallel graph connectivity algorithms, 2020) and Hong et al (Exploring the design space of static and incremental graph connectivity algorithms on gpus, 2020) have implemented some of our algorithms on CPUs and GPUs and experimented with them. On many realistic data sets, our algorithms run as fast or faster than all others.