Perfectly load-balanced, optimal, stable, parallel merge

We present a simple, work-optimal and synchronization-free solution to the problem of stably merging in parallel two given, ordered arrays of m and n elements into an ordered array of m+n elements. The main contribution is a new, simple, fast and direct algorithm that determines, for any prefix of the stably merged output sequence, the exact prefixes of each of the two input sequences needed to produce this output prefix. More precisely, for any given index (rank) in the resulting, but not yet constructed output array representing an output prefix, the algorithm computes the indices (co-ranks) in each of the two input arrays representing the required input prefixes without having to merge the input arrays. The co-ranking algorithm takes O(log min(m,n)) time steps. The algorithm is used to devise a perfectly load-balanced, stable, parallel merge algorithm where each of p processing elements has exactly the same number of input elements to merge. Compared to other approaches to the parallel merge problem, our algorithm is considerably simpler and can be faster up to a factor of two. Compared to previous algorithms for solving the co-ranking problem, the algorithm given here is direct and maintains stability in the presence of repeated elements at no extra space or time cost. When the number of processing elements p does not exceed (m+n)/log min(m,n), the parallel merge algorithm has optimal speedup. It is easy to implement on both shared and distributed memory parallel systems.