Graph Theoretical Characterizations of Bit Permutation Networks

The bit permutation networks (BPNs), proposed by Chang, Hwang and Tong (Networks, 33 (1999) 261-267), are a class of digraphs which include the underlying topological structure of almost all the commonly used switching networks or sorting networks. Many problems about BPNs have been intensively studied. Our work here is to present several graph theoretical characterizations of BPNs, which can be naturally divided into two parts. One part follows the approach of several researchers in France and German, in which we characterize BPNs in terms of their distinguished component structure. The other part combines the techniques used by several researchers in Israel and Taiwan, where layered cross product (Networks, 29 (1997) 219-223) and channel graph play a critical role. Our work confirms the observation that a high degree of regularity is the reason why most of the networks in use have BPNs as underlying topologies. Our results have implications in many kinds of problems about BPNs, such as devising algorithms for checking topological equivalence, verifying useful network representations suiting specific need, revealing the rich intrinsic combinatorial properties of BPNs, and so on.