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Wide-sense Nonblocking for Multi-log(d^n,m,k) Networks under the Minimum Index Strategy

Wide-sense Nonblocking for Multi-log(d^n,m,k) Networks under the Minimum Index Strategy. Speaker: Fei-Huang Chang Coauthers: Ding-An Hsien, Chih-Hung Yen. Definition: Multi-stage Inter-connectional Networks. Crossbars. Output stage. Input stage. C(2,4,3). 1. 1. 1. n. 2. 2. 2. r. r.

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Wide-sense Nonblocking for Multi-log(d^n,m,k) Networks under the Minimum Index Strategy

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  1. Wide-sense Nonblocking for Multi-log(d^n,m,k) Networks under the Minimum Index Strategy Speaker: Fei-Huang Chang Coauthers: Ding-An Hsien, Chih-Hung Yen

  2. Definition:Multi-stage Inter-connectional Networks Crossbars Output stage Input stage

  3. C(2,4,3) 1 1 1 n 2 2 2 r r m Definition:3-stage Clos network---C(n,m,r) Input stage Output stage Middle stage

  4. Definition:Request An order pair of (input-crossbar, output-crossbar) is a request. 1 1 1 n (1,2) request 2 2 2 r r m

  5. Definition:The Corresponding Matrix 1 1 1 2 1 2 2 2 3 2,3 3 3 m

  6. Definition:Strictly Nonblocking (SNB)Wide-sense Nonblocking (WSNB) A network is strictly nonblocking if a request can always be routed regardless of how the previous pairs are routed. A network is said to be wide-sense nonblocking with respect to a routing strategy M if every request is routable under M.

  7. Definition: Two routing strategies for Clos networksPacking(P)Minimum Index (MI) P: Route through anyone of the busiest middle crossbars. MI: Route through the smallest index of middle crossbars if possible.

  8. Theorem:Clos (1953)C(n,m,r) is SNB if and only if m>2n-2. Proof: 1 1 1 n-1 co-inlet n-1 co-outlet 2 2 2 r r m

  9. Theorem: Benes (1965)C(n,m,2) is WSNB under P if and only if m≧[3n/2]. Theorem: Smith(1977)C(n,m,r) is not WSNB under P or MI if m≦[2n-n/r]. Theorem: Du et al.(2001)C(n,m,r) is not WSNB under P or MI if m≦[2n-n/2^(r-1)]. Theorem: Chang et al.(2004)C(n,m,r) is WSNB under P(r≠2), MI if and only if m>2n-2.

  10. Theorem: Du et al.(2001)C(n,m,r) is not WSNB under MI if m≦[2n-n/2^(r-1)]. For C(8,m,3) , 2n-n/2^(r-1)=16-2=14 [1,4] [1,8] [1,2] [9,12] [13,14] [9,12] [1,8] [5,8] [6,8] 15 [3,8] [1,8] [1,5]

  11. [14,16] [25,28] [17,24] [1,16] [29,30] [29,30] [1,13] 31 Chang (2002.10)C(n,m,r) is not WSNB under MI if m≦[2n-n/2^(2r-2)].

  12. Chang (2003.2)C(n,m,r) is WSNB under MI if and only if m>2n-2 For C(16,m,2) by induction on n. n=15 is true. When n=16 [3,7] [1,7] [3,4] [17,20] [25,28] [15,21] [9,12] [5,8] [8,14] 29 [29,30] [22,28] [22,24] [21,24] [13,16] [1,2]

  13. Definition:Banyan-type networks (Log d^n networks)

  14. Definition:Base Line Networks (Banyan-type) BL2(4) BL2(3) BL2(2)

  15. Banyan Banyan Banyan Definition:Multi-log N networks with p copies Middle crossbar of middle stage Input stage

  16. 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7

  17. Theorem: Shyy and Lea (1991), Hwang (1998)

  18. I1 O1 I2 O2 Theorem: Chang et al. (2006)Multi-log N networks is WSNB under MI if and only if p≧p(n).

  19. Theorem: Chang et al. (2006)Multi-log N networks is WSNB under MI if and only if p≧p(n). I’1 O’1 O1 I1 O2 I2 I’2 O’2

  20. Definition:Extra Stage of Banyan-type networks

  21. BL(n,k) BL(n,k) BL(n,k) Definition:Multi-log (N=d^n,p,k) Networks (Log_d(N,p,k))

  22. Theorem: Hwang (1998) Chang et al. (2006)Log_d(N,p,k) is SNB if and only if p>p(n,k).

  23. Theorem: Chang et al. (2006)Log_d(N,p,k) is WSNB under CD, CS, STU, P if and only if p≧p(n,k).

  24. Proposition:BL(n, k) contains d copies of BL(n-1, k-1).

  25. Theorem: Hwang (1998) Chang et al. (2006)Log_2(N,p,k) is SNB if and only if p≧p(n,k).

  26. Theorem: Log_2(N,p,1) is WSNB under MI if and only if p≧p(n,1).

  27. I1 O1 BL(4,1) I2 O2 BL(4,1) n’=3 n’=3 n”=4 n”=4 Theorem: Log_2(N,p,1) is WSNB under MI if and only if p≧p(n,1). BL(3,0) BL(3,0)

  28. Theorem: Log_2(N,p,k>1) is WSNB under MI if and only if p≧p(n,k).

  29. References:[1] C. Clos, A study of nonblocking switching networks, Bell System Technol. J. 32 (1953) 406-424.[2] F. K. Hwang, The Mathematical Theory of Nonblocking Switching Networks, World Scientific, Singapore, first ed. 1998; second ed. 2004.[3] D. Z. Du et al., Wide-sense nonblocking for 3-stage Clos networks, in: D. Z. Du, H. Q. Ngo(Eds.), Switching Networks: Recent Advances, Kluwer, Boston, (2001) 89-100.[4] F. K. Hwang, Choosing the best log_k(N,m,p) strictly nonblocking networks, IEEE Trans. Comm. 46 (4) (1998) 454-455.[5] D.-J. Shyy., C.-T. Lea, log_2(N,m,p) strictly nonblocking networks, IEEE Trans. Comm. 39 (10) (1991) 1502-1510.[6] D.G. Smith, Lower bound in the size of a 3-stage wide-sense nonblocking network, Elec. Lett. 13 (1977) 215-216.[7] F. H. Chang et al., Wide-sense nonblocking for symmetric or asymmetric 3-stage Clos networks under various routing strategies, Theoret. Comput. Sci. 314 (2004) 375-386.[8] F. H. Chang et al., Wide-sense nonblocking for multi-log_d N networks under various routing strategies, Theoret. Comput. Sci. 352 (2006) 232- 239.

  30. The End. Thank you for your attention!!

  31. 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7

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