Periodic Sorting Using Minimum Delay, Recursively Constructed Merging Networks

Abstract

Let $\alpha$ and $\beta$ be a partition of $\{1,\ldots,n\}$ into two blocks. A merging network is a network of comparators which allows as input arbitrary real numbers and has the property that, whenever the input sequence $x_1,x_2,\ldots,x_n$ is such that the subsequence in the positions $\alpha$ and the subsequence in the positions $\beta$ are each sorted, the output sequence will be sorted. We study the class of "recursively constructed" merging networks and characterize those with delay $\lceil\log_2 n\rceil$ (the best possible delay for all merging networks). When $n$ is a power of 2, we show that at least $3^{n/2-1}$ of these nets are log-periodic sorters; that is, they sort any input sequence after $\log_2n$ passes through the net. (Two of these have appeared previously in the literature.)