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

Input stage Output stage

Crossbars

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 stage Output stage

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

Definition: The Corresponding Matrix

1

3

1

m

2

2

1

3

2

1I

2I

3I

1O 2O 3O

21

3

2,3

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)

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 networksPacking(P)Minimum Index (MI)

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-inlet n-1 co-outlet

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.

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)].

1I

2I

3I

1O 2O 3O

[1,8]

[1,8]

[1,8]

[5,8]

[9,12][1,2]

[3,8]

[13,14][1,4]

[9,12] [6,8]

[1,5]

15

Chang (2002.10)C(n,m,r) is not WSNB under MI if m≦ [2n-n/2^(2r-2)].

1I

2I

3I

1O 2O 3O

[1,13]

[29,30][1,16]

[29,30]31

[14,16]

[17,24]

[25,28]

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]

.output each path to unique a hasinput Each

)3( Baseline 2BL

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

BL2(4)

Definition:Base Line Networks (Banyan-type)

BL2(3)

BL2(2)

Definition:Multi-log N networks with p copies

Banyan

Banyan

Banyan

networks 16 log

of copies 3with

network 16 log-Multi

2

2

Input stage

Middle crossbar of middle stage

7

6

0

1

3

2

5

4

7

6

0

1

3

2

5

4

network )2,1,8(2Log

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

where

),( if SNB isnetwork log-Multi nppNd

odd. for 12

even, for 1)1()(

21-n

2 1

nd

nddnp

n

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

3.for 3122

4,for 512)3()(

21-3

24 1

n

nnp

21-n

d

odd. for 12

even, for 1)1(21-n

2 1

nd

nddn

I1

I2

O1

O2

21-n

' dn

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

12n

1 )d(d-I1

I2

O1

O2

12n d

1)1(

)1(121

11

2n

2n

2n

dd

ddd

I’1 O’1

O’2I’2

)1,3(),( 1F

BLknBL

stages.k last theof imagemirror the

toidentical are stagesk extra The:F

stages. extra hasnetwork asuch say we

then stages than more hasnetwork a If

1-

n

Definition:Extra Stage of Banyan-type networks

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

BL(n,k)

BL(n,k)

BL(n,k)

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

odd. for 22/)1(2

even, for 2)1(/)1(2),(

21-k-n

2 1

kndddk

knddddkknp

kn

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).≧

odd. for 12/)1(2

even, for 1)1(/)1(2),(

21-k-n

2 1

kndddk

knddddkknp

kn

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

)1,3( and )2,4( BLBL

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

odd. for 122

even, for 123),(

21-k-n

2 1

knk

knkknp

kn

.integer positive allfor

,1)1,1(),(

,2When

nn, k

knpknp

d

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

odd. for 1221

even, for 1231)1,(

21-k-n

2 1

kn

knnp

kn

3122)0,3(

41221)1,4(

p

p

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

BL(4,1)

BL(4,1)

BL(3,0)BL(3,0)

I1

I2

O1

O2

n’=3 n’=3

n”=4 n”=4

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

odd. for 122

even, for 123),(

21-k-n

2 1

knk

knkknp

kn

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.

The End.

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