Routing Partial Permutations in General Interconnection Networks based on Radix Sorting

Abstract. In general, sorting networks can be used as interconnection networks in that the input messages are simply sorted according to their target addresses. If the target addresses form a permutation of all addresses, this is obviously correct since then the sorting algorithm routes each message to its target address. However, if not all inputs need a connection to one of the outputs, then some output addresses do not appear as target addresses, and thus, partial permutations have to be implemented. In this case, sorting networks work no longer correctly as interconnection networks since all messages with target addresses larger than the smallest missing target address will be routed to the wrong outputs. For merge-based sorting networks, there is a well-known general solution called the Batcher-Banyan network. However, for the larger class of radix-based sorting networks this does not work, and there is only one solution known for a particular permutation network [28]. In this paper, we present three general constructions to convert any binary sorter into a ternary split module which is the key to construct a radix-based interconnection network that can cope with partial permutations. We compare the sizes and depths of the circuits obtained by our constructions for six known binary sorters and show this way that the obtained circuits are of practical interest.