  Relaxed concurrent data structures have become increasingly popular, due to
their scalability in graph processing and machine learning applications.
Despite considerable interest, there exist families of natural, high performing
randomized relaxed concurrent data structures, such as the popular MultiQueue
pattern for implementing relaxed priority queue data structures, for which no
guarantees are known in the concurrent setting. Our main contribution is in
showing for the first time that, under a set of analytic assumptions, a family
of relaxed concurrent data structures, including variants of MultiQueues, but
also a new approximate counting algorithm we call the MultiCounter, provides
strong probabilistic guarantees on the degree of relaxation with respect to the
sequential specification, in arbitrary concurrent executions. We formalize
these guarantees via a new correctness condition called distributional
linearizability, tailored to concurrent implementations with randomized
relaxations. Our result is based on a new analysis of an asynchronous variant
of the classic power-of-two-choices load balancing algorithm, in which
placement choices can be based on inconsistent, outdated information (this
result may be of independent interest). We validate our results empirically,
showing that the MultiCounter algorithm can implement scalable relaxed
timestamps, which in turn can improve the performance of the classic TL2
transactional algorithm by up to 3 times, for some settings of parameters.
