Remarks on Beneš Conjecture

Abstract

The shuffle-exchange network was initially proposed by Stone in 1971 [12]. Beneš conjectured in 1975 [1] that 2n - 1-stages are necessary and sufficient for shuffle-exchange networks to route all N! (N = 2ⁿ) perfect matchings from the N inputs to the N outputs, i.e., m(n) = 2n - 1, where m(n) is the minimum number of stages for a shuffle-exchange network to be able to rooute all permutations. Parker in 1980 [9] showed that n + 1 ≤ m(n) ≤ 3n. Wu and Feng in 1981 [14] gave an explicit algorithm to route all matchings, proving m(n) ≤ 3n -1. Huang and Tripathi in 186 [4] improved the bound to m(n) ≤ 3n - 3. Raghavendra and Varma in 1987 [11] verified the conjecture for n = 3 and then showed m(n) ≤ 3n - 4 [13]. They also showed 2n - 1 ≤ m(n). Linial and Tarsi in 1989 [7] proposed a different approach to prove the sanie results of Raghavendra and Vanna. Hwang [5] provided an alternative formulation of the conjecture. Feng and Seo in 1994 [3] gave a proof of the conjecture, which was incomplete as pointed out by Kim, Yooii, and Maeng in 1997 [6]. Ngo and Du in 2000 [8] verified the conjecture for n = 4, which implies m(n) ≤ 3n - 5. Two year ago, Çam [2] claimed another proof of the conjecture. So far, it is still an open problem whether Çam’s new proof is complete or riot.