We consider comparator networks M that 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. We are interested in comparator networks of a constant period, due to their potential applications in hardware design. Previously, very little was known on such networks. The fastest solutions required time O(n ε ) where the depth was roughly 1/ε. We introduce a general method called periodification scheme that converts automatically an arbitrary sorting network that sorts n items in time T(n ) and that has layout area A(n ) into a sorting network that has period 5, sorts ***( n • T ( n ) items in time O(T( )• log n ), and has layout area O(A(n) ) • T(n )). In particular, applying this scheme to Batcher's algorithms, we get practical period 5 comparator networks that sort in time O (log 3 n ). For theoretical interest, one may use the AKS netork resulting in a period 5 comparator network with runtime O (log 2 n ).

A periodic comparator network has depth (or period) k, if for every t>k, the compare-exchange operations performed at step t are executed between exactly the same registers as at step t-k. We introduce a general method that converts an arbitrary comparator network that sorts n items in time T(n) and that has layout area A into a periodic sorting network of depth 5 that sorts /spl Theta/(n/spl middot/T(n)) items in time O(T(n)/spl middot/log n) and has layout area O(A/spl middot/T(n)). This scheme applied to the AKS network yields a depth 5 periodic comparator network that sorts in time O(log/sup 2/ n). More practical networks with runtime O(log/sup 3/ n) can be obtained from Batcher's networks. Developing the techniques for the main result, we improve some previous results: Let us fix a d/spl isin/N. Then we can construct a network of depth 3 based on a d-dimensional mesh sorting n items in time O(n/sup 1/d//spl middot/log/sup O(d/) n).>