Switching Networks: Recent Advances pp 257-258 | Cite as

# Remarks on Beneš Conjecture

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## Abstract

The shuffle-exchange network was initially proposed by Stone in 1971 [12]. Beneš conjectured in 1975 [1] that 2*n* - 1-stages are necessary and sufficient for shuffle-exchange networks to route all *N*! (*N* = 2^{ n }) perfect matchings from the *N* inputs to the *N* outputs, i.e., *m*(*n*) = 2*n* - 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*) ≤ 3*n*. Wu and Feng in 1981 [14] gave an explicit algorithm to route all matchings, proving *m*(*n*) ≤ 3*n* -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*) ≤ 3*n* - 4 [13]. They also showed 2*n* - 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*) ≤ 3*n* - 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.

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