Communication Round and Computation Efficient Exclusive Prefix-Sums Algorithms (for MPI_Exscan)

Parallel scan primitives compute element-wise inclusive or exclusive prefix sums of input vectors contributed by $p$ consecutively ranked processors under an associative, binary operator $\oplus$. In message-passing systems with bounded, one-ported communication capabilities, at least $\lceil\log_2 p\rceil$ or $\lceil\log_2 (p-1)\rceil$ communication rounds are required to perform the scans. While there are well-known, simple algorithms for the inclusive scan that solve the problem in $\lceil\log_2 p\rceil$ communication rounds with $\lceil\log_2 p\rceil$ applications of $\oplus$ (which could be expensive), the exclusive scan appears more difficult. Conventionally, the problem is solved with either $\lceil\log_2 (p-1)\rceil+1$ communication rounds (e.g., by shifting the input vectors), or in $\lceil\log_2 p\rceil$ communication rounds with $2\lceil\log_2 p\rceil-1$ applications of $\oplus$ (by a modified inclusive scan algorithm). We give a new, simple algorithm that computes the exclusive prefix sums in $q=\lceil\log_2 (p-1)+\log_2\frac{4}{3}\rceil$ simultaneous send-receive communication rounds with $q-1$ applications of $\oplus$. We compare the three algorithms implemented in MPI against the MPI library native MPI\_Exscan primitive on a small, $36$-node cluster with a state-of-the-art MPI library, indicating possible and worthwhile improvements to standard implementations. The algorithms assume input vectors to be small so that performance is dominated by the number of communication rounds. For large input vectors, other (pipelined, fixed-degree tree) algorithms must be used.