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<p><it>Abstract</it>—An <it>N</it>×<it>N k</it>-Omega network is obtained by adding <it>k</it> more stages in front of an Omega network. An <it>N</it>-permutation defines a bijection between the set of <it>N</it> sources and the set of <it>N</it> destinations. Such a permutation is said to be admissible to a <it>k</it>-Omega if <it>N</it> conflict-free paths, one for each source-destination pair defined by the permutation, can be established simultaneously. When an <it>N</it>-permutation is not admissible, it is desirable to divide the <it>N</it> pairs into a minimum number of groups (passes) such that the conflict-free paths can be established for the pairs in each group. Raghavendra and Varma solved this problem for BPC (<it>Bit Permutation Complement</it>) permutations on an Omega without extra stage. This paper generalizes their result to a <it>k</it>-Omega where <it>k</it> can be any integer between 0 and <it>n</it>  1. An <it>O</it>(<it>NlgN</it>) algorithm is given which realizes any BPC permutation in a minimum number of passes on a <it>k</it>-Omega (0 ≤<it>k</it>≤<it>n</it>  1).</p>
BPC permutation, conflict graph, graph coloring, k-Omega network, permutation realization.

X. Shen, "Optimal Realization of Any BPC Permutation on K-Extra-Stage Omega Networks," in IEEE Transactions on Computers, vol. 44, no. , pp. 714-719, 1995.
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