Optimal Multithreaded Batch-Parallel 2-3 Trees

This paper presents a batch-parallel 2-3 tree T in an asynchronous dynamic multithreading model that supports searches, insertions and deletions in sorted batches and has essentially optimal parallelism, even under the restrictive QRMW (queued read-modify-write) memory contention model where concurrent accesses to the same memory location are queued and serviced one by one. Specifically, if T has n items, then performing an item-sorted batch (given as a leaf-based balanced binary tree) of b operations on T takes O( b * log(n/b+1) + b ) work and O( log b + log n ) span (in the worst case as b,n -> inf). This is information-theoretically work-optimal for b <= n, and also span-optimal for pointer-based structures. Moreover, it is easy to support optimal intersection, union and difference of instances of T with sizes m <= n, namely within O( m * log(n/m+1) ) work and O( log m + log n ) span. Furthermore, T supports other batch operations that make it a very useful building block for parallel data structures. To the author's knowledge, T is the first parallel sorted-set data structure that can be used in an asynchronous multi-processor machine under a memory model with queued contention and yet have asymptotically optimal work and span. In fact, T is designed to have bounded contention and satisfy the claimed work and span bounds regardless of the execution schedule. Since all data structures and algorithms in this paper fit into the dynamic multithreading paradigm, all their performance bounds are directly composable with those of other data structures and algorithms in the same model. Finally, the pipelining techniques in this paper are also likely to be very useful in asynchronous parallelization of other recursive data structures.