Sorting by Parallel Insertion on a One-Dimensional Subbus Array

We consider the problem of sorting on a one-dimensional sub-bus array of processors. The sub-bus broadcast operation makes possible a new class of parallel sorting algorithms whose complexity we analyze with the parallel insertion model. A sorting method, or sorting strategy, in the parallel insertion model, uses a sequence of left and right insertion steps, of which we give two types: greedy insertion steps and simple insertion steps. For two restricted classes of parallel insertion sorting, the one-way and the alternating sorting strategies, we give lower bounds and optimal sorting strategies that exactly match the lower bounds. Optimal alternating sorting strategies are demonstrated to use a factor of two fewer insertion steps on average than oddeven transposition sort and any optimal one-way sorting strategy. For general sorting strategies, we give a weak lower bound and consider a sorting strategy that uses the fewest greedy insertion steps. Finally, we discuss the issues involved in implementing parallel insertion sorting strategies on sub-bus machines. We evaluate the performance of our sorting strategies by applying them to shearsort, a common two-dimensional mesh sorting algorithm, and by contrasting the results with our theoretical results from the parallel insertion model. Sorting by Parallel Insertion on a One-Dimensional Sub-Bus Array