Remarks on Beneš Conjecture

Part of the Network Theory and Applications book series (NETA, volume 5)


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 = 2 n ) 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.


  1. [1]
    V.E. Beneš, Proving the rearrangeability of connecting networks by group calculations, Bell System Tech. J., 45 (1975) 421–434.Google Scholar
  2. [2]
    Hasan çam, Rearrangeability of (2n - 1)-Stage Shuffle-Exchange Networks, submitted to SIAM Journal on Computing.Google Scholar
  3. [3]
    T.-Y. Feng and S.-W. Seo, A new routing algorithm for a class of rearrangeable networks, IEEE Trans. Comput.43 (1994) 1270–1280.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    S.T. Huang and S.K. Tripathi, Finite state model and compatibility theory: new analysis tools for permutation networks, IEEE Trans. on Comput., C-35 (1986) 591–601.Google Scholar
  5. [5]
    F.K. Hwang, A mathematical abstraction of the rearrangeability conjecture for shuffle-exchange networks, Operations Research Letters, 8 (1989) 85–87.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    M.K. Kim, H. Yoon and S.R. Maeng, On the correctness of inside-out routing algorithm, IEEE Trans. on Comput., 46 (1997) 820–823.CrossRefGoogle Scholar
  7. [7]
    N. Linial and M. Tarsi, Interpolation between bases and the Shuffle-Exchange network, Europ. J. Combinatorics, 10 (1989) 29–39.MathSciNetzbMATHGoogle Scholar
  8. [8]
    H.Q. Ngo and D.-Z. Du, On the rearrangeability of shuffle-exchange networks, TR-045, Department of Computer Science and Engineering, University of Minnesota, 2000.Google Scholar
  9. [9]
    D.S. Parker, Notes on shuffle/exchange-type switching networks, IEEE Trans. Comput., C-29 (1980) 213–222.CrossRefGoogle Scholar
  10. [10]
    C.S. Raghavendra, “On the rearrangeability conjecture of (2log2N -1)-stage shuffle/exchange network (Position Paper),” Computer Architecture Technical Committee Newsletter, pp. 10–12, Winter 1994–1995.Google Scholar
  11. [11]
    C.S. Raghavendra and A. Varma, Rearrangeability of the five stage Shuffle-Exchange network for N=8, IEEE Trans. on Commun., com-35 (1987) 808–812.MathSciNetGoogle Scholar
  12. [12]
    H. Stone, Parallel processing with perfect shuffle, IEEE Trans. Comput.20 (1971) 153–161.zbMATHCrossRefGoogle Scholar
  13. [13]
    A. Varma and C. Raghavendra, Rearrangeability of multistage shuffle/exchange networks, IEEE Trans. Comm. COM-36 (1988) 1138–1147.Google Scholar
  14. [14]
    C. Wu and T.-Y. Feng, The universality of the shuffle-exchange network, IEEE Trans. on Comput., C-30 (1981) 324–331.MathSciNetCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringUniversity of MinnesotaMinneapolisUSA

Personalised recommendations