On the Minimal Period of Fast Periodic Sorting Networks

We consider comparator networks M which are used repeatedly: while the output produced by M is not sorted, it is fed again into M. Sorting algorithms working in this way are called periodic. The number of parallel steps performed during a single run of M is called its period, the sorting time of M is the total number of parallel steps that are necessary to sort in the worst case. Periodic sorting networks have the advantage that they need little hardware (control logic, wiring, area) and that they are adaptive. In 9] we introduce a general method called the peri-odiication scheme that converts automatically an arbitrary sorting network that sorts n items in time T(n) into a sorting network that has period 5 and sorts (n T(n)) items in time O(T(n) log n). Applied to Batcher's algorithms, we get practical period 5 com-parator networks that sort in time O(log 3 n). For theoretical interest, one may use the AKS network resulting in a period 5 comparator network with runtime O(log 2 n). In this paper we address the problem of determining the minimal period necessary to get fast periodic sorting networks. We note that period 2 results in linear time sorting. Our main result is that period 3 suuces to get the same results as mentioned above for period 5. Futhermore our networks have a very simple design. The rst two steps are those of odd-even transposition sort, only the third step is designed depending on the embedded (non-periodic) network. Using the techniques necessary for reducing the period to 3, we construct a practial merging network of period 3 that merges n items in time O(log 2 n).