A systolic design for generating permutations in lexicographic order

A Cost-Optimal Pipeline Algorithm for Permutation Generation in Lexicographic Order

“…on a bidirectional linear array of size N and is therefore cost-optimal. Moreover, contrary to the solution proposed by Lee and Tsay [3], no arithmetic operation is performed on the elements to be permuted: these elements are only shifted in both directions during the execution of the algorithm. The authors end their presentation by saying that the problem of designing a costoptimal parallel algorithm for generating permutations of M elements out of N in lexicographic order is an open interesting question.…”

“…The algorithm proposed in 1906 by D. N. Lehmer and reported in 1964 by D. H. Lehmer in [4] for the generation of permutations in lexicographic order has been implemented recently by Lee and Tsay [3] on a two-dimensional systolic array. This algorithm is based on a very nice idea.…”

“…Section 2 is devoted to some basic notations and definitions. In Section 3, we present Lehmer's algorithm [4] for generating sequentially permutations of N objects in lexicographic order, together with the systolic implementation proposed by Lee and Tsay [3]. The fourth section shows how to obtain a cost-optimal solution by removing redundant computations in Lehmer's algorithm, and by introducing an anticipation strategy.…”

“…The first one which is due to Lee and Tsay [3] runs in time O(N !) on a two-dimensional triangular unidirectional array of size O(N 2 ).…”

“…Starting from the solution proposed by Lee and Tsay [3], we first eliminate redundant computations. However, this is not sufficient.…”