indirect (dynamic) networks
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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 - PowerPoint PPT PresentationTRANSCRIPT
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
NC 論２ (No.2) 5
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).
NC 論２ (No.2) 6
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.
NC 論２ (No.2) 7
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
1
2
m
1
2
r
1
n
1
n
1
n
n×m r×r m×n
NC 論２ (No.2) 8
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.
NC 論２ (No.2) 9
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
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
σ ３－１ σ ２
－１ σ ２ σ ３
010
001
(2,2,2) clos
(2,2,2) clos
(2,4,2) Clos
NC 論２ (No.2) 10
2x2 switch (radix: 2)
straight exchange lowerbroadcast
upperbroadcast
NC 論２ (No.2) 11
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)
NC 論２ (No.2) 12
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)
NC 論２ (No.2) 14
Perfect shuffle (1/2)000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
Perfect shuffle
σ(x)=(an-1,an-2,…, a1,an)Inverse perfect shuffle
σ-1(x)=(a1,an an-1,an-2,…, a2)
NC 論２ (No.2) 15
Perfect shuffle (2/2)000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
3 sub-inverse shuffule
σ3-1
(x)=(a1 , a3 , a2)2 sub-inverse shuffle
σ2-1
(x)=(a3, a1, a2)
NC 論２ (No.2) 16
Butterfly000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
butterfly (bit reversal)
β3(x)=(a1 , a2 , a3)
＝ ρ ３ (x)
2 sub-butterfly
β2(x)=(a3, a1, a2)
NC 論２ (No.2) 17
shift000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
shift
α(x)= | x+1 | (23)2 sub-shift
α 2(x)=| x+1 | (22)+floor(x/ 22) 22
NC 論２ (No.2) 18
exchange000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
E3(x) = (~a3, a2, a1) E2(x) = (a3, ~a2, a1)
NC 論２ (No.2) 19
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
010
011
100
101
110
111
000
001 (UUL)
010 (ULU)
011
100
101
110
111
abc cab cbaσ ２
－ 1σ ３－ 1
001
NC 論２ (No.2) 22
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
NC 論２ (No.2) 23
Omega networks (1/2) Ω= (σE)n
000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
σσσabc bca abccab
NC 論２ (No.2) 24
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)
000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
σ-1β ３β ２
000001
100101
010
110
011
111
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!).
NC 論２ (No.2) 28
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
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
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
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
σσσAdditional
stage
NC 論２ (No.2) 33
Fault-tolerant MINs (Baseline’s case)000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
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
001
010
011
100
101
110
111
β ３β ２
000001
100101
010
110
011
111
000001
100101
010
110
011
111
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
NC 論２ (No.2) 39
Hierarchical buses
Global bus
Cluster bus
bridge
NC 論２ (No.2) 40
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.