NC 論２ (No.2) 1

Indirect (dynamic) Networks

• Communication between any two nodes has to be carried through some switches.

• Classified into:– Crossbar network

– Multistage interconnection networks (MIN)• Nonblocking networks

• Rearrangeable nonblocking networks

• Blocking networks

– Single-stage interconnection networks

NC 論２ (No.2) 2

Crossbar (nonblocking, 1-hop)

An N x M crossbar

The cost increases as N x M.

Switch point

(crosspoint)

M outputs

N

inputs

N:1 multiplexer

NC 論２ (No.2) 3

A large crossbar0

1

2

3

NC 論２ (No.2) 4

MINs

Three-stage MIN

first stage

second stage

third stage

N

inputs

M outputs

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Nonblocking MIN

• Any input port can be connected to any free output port without affecting the existing connections.

• Same functionality as a crossbar.

• The Clos network is proposed to reduce the cost of crossbar by decreasing the number of switches (1953).

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History (telephone net.)

• Early telephone networks were built from elector-mechanical crossbars.

• Key developments in telephone switching include the Clos network in 1953 and the Benes network in 1962.

• Many large telephone switches today are still built from Clos or Clos-like networks.

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3-stage Clos network (m,n,r)

Nonblocking: m >= 2n-1

Rearrangeable: 2n-1 > m >= n

N

inputsN

outputs

1

2

r

1

n

1

n

1

n

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2

m

1

2

r

1

n

1

n

1

n

n×m r×r m×n

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Rearrangeable nonblocking MIN

• Any input port can be connected to any free output port with rearrangement of paths for the existing connections.

• Less stages or smaller switches than a nonblocking networks.

• The best known example is the Benes network.

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Benes 　 network

An 8x8 Benes network (Eσ ３－１ E σ ２

－ 1 Eσ ２ Eσ ３　 E)

N(2log2N - 1) / 2 switches, 2log2N - 1 hops, E: stage

0

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σ ３－１ σ ２

－１ σ ２ σ ３

010

001

(2,2,2) clos

(2,2,2) clos

(2,4,2) Clos

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2x2 switch (radix: 2)

straight exchange lowerbroadcast

upperbroadcast

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Permutation 　• Permutations are traffic patterns in which all traffic

from each source is directed to one destination.• It can be compactly represented by a permutation

function that maps source to destination.• Bit (digit) permutation are those in which each bit

(digit) of the destination address is a function of one bit (digit) of the source address.– di = sj or dx = f(sx)

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Permutation functions (1/2)Communication from node address x(an,an-1,an-2,…, a1) to

• Perfect shuffle: σ(x)=(an-1,an-2,…, a1,an)

• k sub-shuffule: σk(x)=(an,an-1,…, ak+1,ak-1, … , a1,ak)

• k supershuffule: σk(x)=(an-1,an-2,…, an-k+1, an, an-k, … , a1)

• Inverse perfect shuffle: σ-1(x)=(a1 , an , an-1,an-2,…, a2)

• k sub-inverse shuffule: σk-1

(x)=(an , …, ak+1 , a1,ak,…, a2)

• Butterfly: β(x)=(a1 , an-1,an-2,…, a2 , an)

• k sub-butterfly: βk (x)=(an,…, ak+1 , a1,ak-1, … , a2,ak)

• k superbutterfly: βk (x)=(an-k+1, an-1,…, an-k+2 , an,an-k, … , a1)

NC 論２ (No.2) 13

Permutation functions (2/2)Communication from node address x(an,an-1,an-2,…, a1) to

• Bit reversal: ρ(x)=(a1,a2,…, an-1, an)

• k sub-bit reversal : ρk(x)=(an,an-1,…, ak+1,a1, a2,… , ak-1,ak)

• k super-bit reversal: ρk(x)=(an-k+1,an-k+2,…, an-1, an, an-k, … , a1)

• shift: α(x)= | x+1| (2n), where modulo by 2n

• k sub-shift: αk(x)= | x+1| (2n) +floor(x/2k)2k

• k super-shift: αk (x)= | x+ 2n-k | (2n)

• Exchange: Ei(x)=(an,an-1, …, ~ai,…, a1)

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Perfect shuffle (1/2)000

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011

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011

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011

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001

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011

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Perfect shuffle

σ(x)=(an-1,an-2,…, a1,an)Inverse perfect shuffle

σ-1(x)=(a1,an an-1,an-2,…, a2)

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Perfect shuffle (2/2)000

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3 sub-inverse shuffule

σ3-1

(x)=(a1 , a3 , a2)2 sub-inverse shuffle

σ2-1

(x)=(a3, a1, a2)

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Butterfly000

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butterfly (bit reversal)

β3(x)=(a1 , a2 , a3)

　　＝ ρ ３ (x)

2 sub-butterfly

β2(x)=(a3, a1, a2)

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shift000

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shift

α(x)= | x+1 | (23)2 sub-shift

α 2(x)=| x+1 | (22)+floor(x/ 22) 22

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exchange000

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E3(x) = (~a3, a2, a1) E2(x) = (a3, ~a2, a1)

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Blocking MIN

• A free input/output pair is not always possible because of conflicts with the existing connections.

• Minimizing the number of switches and stages.

• Unidirectional MIN (Cenju-3, IBM PR3)

• Bidirectional MIN (IBM SP, TMC CM5)

NC 論２ (No.2) 20

Unidirectional MINs• Channels and switches are unidirectional.• At least ceiling(logkN) stages are required for a MIN

of N input and output ports with k×k switches.• Every path through the MIN crosses all the stages

(same length).• Examples: baseline, butterfly, indirect cube

logN stages for N inputs/outputs with 2x2 switches( binary logN-fly in the form of k-ary n-fly)the number of communication patterns are (√N)N

NC 論２ (No.2) 21

Baseline networks (1/2)(B = Eσn

－１ Eσn-1－１ E ・・・ Eσ ２

－ 1 E)

000

001

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011

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000

001 (UUL)

010 　 (ULU)

011

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abc cab cbaσ ２

－ 1σ ３－ 1

001

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Baseline networks (2/2)

• To route (an,an-1,an-2,…, a1) to (bn,bn-1,bn-2,…, b1),

select upper link when bi=0, otherwise select lower link on (n-i+1) stage (MSB to LSB).

(path is determined only by the destination address)

switch

upper link

lower linkinputs

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Omega networks (1/2) Ω= (σE)n

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σσσabc bca abccab

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Omega networks (2/2)

• To route (an,an-1,an-2,…, a1) to (bn,bn-1,bn-2,…, b1),

set the switch as straight when ai^bi=0, otherwise set it as exchange on (n-i+1) stage (MSB to LSB).

straight exchange

NC 論２ (No.2) 25

Indirect binary n-cube (1/2)(C = Eβ ２ Eβ ３ ・・・ E β ｎ Eσ － 1)

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σ-1β ３β ２

000001

100101

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abc acb abcbca

NC 論２ (No.2) 26

Indirect binary n-cube (2/2)

• To route (an,an-1,an-2,…, a1) to (bn,bn-1,bn-2,…, b1),

set the switch as straight when ai^bi=0, otherwise set it as exchange on i-th stage (LSB to MSB).

• A k-ary n-fly network consists of kn source/destination terminals, n stages of kn-1 k×k crossbar switches.– Degree of the switches: 2k

– Diameter: n+1 or logkN +1 where N = kn

NC 論２ (No.2) 27

Relation among blocking networks

• For omega, indirect n-cube and Banyan(Y):Ω-1=C=Yσ 　

So they are almost functionally equal (isomorphic).

• The communication patterns are different, but same number of combinations ((√N)N).

• Smaller number of communication patterns than nonblocking networks (N!).

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Bitonic sort 　

8 inputs sorter4 inputs sorter2 inputs sorter

A L

B H

NC 論２ (No.2) 29

Batcher-banyan network

Bitonic

Sorter

(Batcher stage)

Ω

(banyan

stage)

Non-blocking

5

∞

3

1

∞

･･∞

1

3

5

∞

∞

･･∞

1

3

5

receiver Ordered addresses

NC 論２ (No.2) 30

Tree saturation (e.g. on Omega)000

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Combining: combine two messages

hotspot

NC 論２ (No.2) 31

Fault-tolerant MINs

• Blocking MINs provide a single path between any input/output pair.

• If a link becomes congested or fails, the unique path property can easily disrupt the communication (no path diversity).

• A solution is to provide multiple routing paths by adding extra stages or channels.

NC 論２ (No.2) 32

Fault-tolerant MINs (Omega’s case)000

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σσσAdditional

stage

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Fault-tolerant MINs (Baseline’s case)000

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σ ２－ 1σ ３

－ 1

NC 論２ (No.2) 34

Bidirectional MINs (BMINs)

• Channels and switches are bidirectional.• Information can be transmitted simultaneously in

opposite directions between neighboring switches.• Paths are established in BMINs by crossing stages

in forward, then turnaround, and finally backward.

• Several paths are possible in forward direction, but a single path is available in backward direction.

NC 論２ (No.2) 35

Bidirectional Banyan networks000

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β ３β ２

000001

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000001

100101

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forward

backward

turnaround

NC 論２ (No.2) 36

A fat tree BMIN

forwardbackward

NC 論２ (No.2) 37

Hybrid networks

• Combine mechanisms from shared-medium networks and direct or indirect networks.

• Increase bandwidth and reduce the distance between nodes.

• Examples:– Multiple backplane buses– Hierarchical buses– Cluster-based networks

NC 論２ (No.2) 38

Multiple backplane buses

device device device

bus

bus

bus

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Hierarchical buses

Global bus

Cluster bus

bridge

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Cluster-based networks

Cluster bus

ClusterCluster Cluster

ClusterCluster Cluster

router

NC 論２ (No.2) 41

History (Clos networks, etc.)

• Clos introduced non-blocking networks in1953.• Benes discovered rearrangeable non-blocking

networks in 1965.• Bachter introduced the bitonic sort and the

Batcher sorting network in 1968.• Knuth’s book in 1998 treats sorting networks in

detail.• Multicast in Clos networks was first considered by

Masson and Jordan(1972), and has since been studied by many researchers.

NC 論２ (No.2) 42

History (indirect networks)

• Processor-memory networks emerged in the late 1960s in the form of indirect netwoks.

• The smallest machine employed crossbar, whereas larger sytems used butterfly, etc..

• Since the early 1990s, a variant of the Clos and Benes networks of telephony has emerged in SMP (MPP) networks in the form of the fat tree.

• Today, the high pin bandwidth of router chips relative to message length motivates the use of networks of higher node degree, such as Clos and fat free.