Isomorphism of conflict graphs in multistage interconnection networks and its application to optimal routing

A study on the isomorphism of conflict graphs in multistage interconnection networks (MINs) and its applications is outlined. A concept called group-transformation is introduced for the baseline network, which induces an equivalence partition on the set of all permutations. All members belonging to the same equivalence class have isomorphic conflict graphs. Thus, determination of conflict resolution of one permutation results in determination of conflict resolution of all other equivalent members. The BPCL (bit-permute-closure) class of permutations is defined, for which the conflict resolution problem can be settled in linear time by an earlier algorithm developed only for the BPC (bit-permute-complement) permutations. It is proved that for an N×N MIN, |BPCL|⩾n 2N-1, in contrast to |BPC| n 2n, (nlog² N). Conflict graphs for BPCL permutations are also characterized. An O(N) time algorithm to check membership of a given permutation to the BPCL class is described. All of these results are generalized to extend their applicability to other unique-path full-access MINs