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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

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

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

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

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

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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. Definition: Strictly Nonblocking (SNB) Wide-sense Nonblocking (WSNB)

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

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

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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.

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

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

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

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

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BL 2 (4) Definition: Base Line Networks (Banyan-type) BL 2 (3) BL 2 (2)

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

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

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

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Theorem: Chang et al. (2006) Multi-log N networks is WSNB under MI if and only if p ≧ p(n). I1I1 I2I2 O1O1 O2O2

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I1I1 I2I2 O1O1 O2O2 I’ 1 O’ 1 O’ 2 I’ 2

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Definition: Extra Stage of Banyan-type networks

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

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

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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).

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

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

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

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BL(4,1) BL(3,0) I1I1 I2I2 O1O1 O2O2 n’=3 n”=4

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

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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.

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The End. Thank you for your attention!!

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

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