Optimal Multithreaded Batch-Parallel 2-3 Trees
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This paper presents a batch-parallel 2-3 tree T in the asynchronous PPM (parallel pointer machine) model that supports searches, insertions and deletions in sorted batches and has essentially optimal parallelism, even under the QRMW (queued read-modify-write) memory model where concurrent memory accesses to the same location are queued and serviced one by one. Specifically, if T has n items, then performing an item-sorted batch of b operations on T takes only 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 in the PPM model. The input batch can be any balanced binary tree, and hence T can also be used to implement sorted sets supporting optimal intersection, union and difference of sets with sizes m <= n in O( m * log(n/m+1) ) work and O( log m + log n ) span. To the author's knowledge, T is the first parallel sorted-set data structure with these performance bounds that can be used in an asynchronous multi-processor machine under a memory model with queued contention. This is unlike PRAM data structures such as the PVW 2-3 tree (by Paul, Vishkin and Wagener), which rely on lock-step synchronicity of the processors. In fact, T is designed to have bounded contention and satisfy the claimed work and span bounds regardless of the execution schedule. All data structures and algorithms in this paper fit into the dynamic multithreading paradigm. Also, as a consequence of working in the asynchronous PPM model, all their performance bounds are directly composable with those of other data structures and algorithms in the same model.
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