PERMUTATION GROUPS, COMPLEXES, AND
REARRANGEABLE CONNECTING NETWORKS
V.E.BENEŠ
SUNITHA THODUPUNURI
Connecting Networks:
A connecting network is an arrangement of switches and
transmission links through which certain terminals can be
connected together in many combinations.
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Eg: Telephone central offices.
Performance of the connecting networks
Structure of the connecting networks
Combinatorial properties of Connecting Networks:
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Blocking
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NonBlocking
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Rearrangeable
Rearrangeability:
A connecting network is rearrangeable if its permitted
states realize every assignments of inlets to outlets or
alternatively if given any state of the network, any inlet
idle in x, and any outlet idle in x, there is a way of
assigning new routes (if necessary) to the calls in progress
in x so as to lead to new state of the network in which idle
inlet can be connected to the idle outlet.
Cross bar switch
1
1
1
1
1
n
n
Customer
lines
n
n
n
Rearrangeable Network
Structure of No. 5 crossbar network
Goal
What stages and link patterns are used to construct the
rearrangeable networks?
Overview
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General notion of a stage of switching in connecting network
Formulation of the problem in terms of partitions and
permutation groups using notation of stage
How a stage generates complexes?
Rearrangeability theorems for connecting
 First theorem gives sufficient conditions for connecting
terminals and link patterns which gives rise to
rearrangeable networks
Second theorem indicates a simple way of describing link
patterns and stages that satisfy the hypothesis of first
theorem
A Network in terms of Group Theory
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Stage - Column of switches
Link Pattern - Each link pattern i.e., the number of crosspoints
on a stage, N can be represented as the permutation on
{1,2,3…. N}
assume
I - The set of inlets
 - The set of outlets
Stage S is a subset of I
A stage S is made of square switches if and only if there is a
partition  of {1,2,3…. N} such that
S =  (A  A)
A
Group theory terms Contd…
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Group – Set of elements with an operation
Eg: Permutation group: permutations on {1,2,3…. N}
Complex – Subset of a Group is called a Complex
Eg: A subset KG is a complex
Imprimitivity –If Inlets and outlets are numbered {1,2,3…. N}
and a stage contains enough cross points it can be used to
connect every inlet to some outlet in one to one fashion with no
inlet connected to more than one outlet and vice-versa
Group theory terms Contd…
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A stage can generate a permutation
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Eg: If a stage S generates a permutation , if there is a
setting of N switches of S which connects each inlet to
one and only outlet in such a way that i is connected to
(i), i=1,2,…, N i.e., (i, (i)) S
The set of permutations is generated by a stage S is denoted by
P(S)
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Similarly, a network with ‘N’ inlets and ‘N’ outlets can
generate a permutation
Example: If two stages S1, S2 are connected by a link pattern
corresponding to a permutation  then it generates exactly the
permutations in the set
P(S1)  P(S2)
Generally,
If a network has ‘s’ stages and are connected by link patterns
corresponding to the permutations 1, 2, 3…….s-1
then the network generates the complex
P(S1) 1P(S2) 2 …….. s-1P(Ss)
The generation of complexes by stages
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A stage is made of square switches if there is a partition  on
{1,2,3……N} such that
S =  (A  A)
A
A stage can generate a subgroup only if it contains a substage
made of square switches
i.e., any stage made of square switches necessarily arise
in the generation of the symmetric group by products of
complexes some of which are subgroups
Group Theory Terminology
Suppose X, Y are sets
|X|=|Y|
: YX (one-one function)
BY
1, 2 – partitions on X,Y
 (B)={x  X:  -1 (x)  B}
  hits 1 from B if and only if A  1 implies (B)  A  
  covers 1 from 2 if and only if B  2 implies  hits 1
from B
Group theory terminology contd…
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(2) is the permutation of X induced by  acting on the
elements of 2 i.e.,
(2) = { (B) : B  2 }
is a restriction of  to B
A X
A1 is the partition of A induced by 1
A1 = {CA: C  1}
B covers 1 from 2 if and only if B covers (B)1 from B2
 B
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Generating the permutation group
P(S1) 1P(S2) 2 …….. s-1P(Ss)
Let s > 3 be an odd integer, let φ1, …, φs-1be permutations on X = { 1, …, N }, , let Пk,
k
k = 1, ..., s, and П , k = 1, ..., ½(s-1),be partitions of X, and let
={X}
k=0
-1
-1
k
= φk ... φ1/2(S-1) ( П )
k = 1, … , ½(s-1)
s-k
ψ
= φk … φ1/2(s+1) (П )
k = ½(s+1), … , s-1
={X}
k=s
Suppose that
i.
If s ≥ 3, then Пk < Пk+1, k = 0, … , ½(s-3).
ii.
П1/2(s+1) = П1/2(s-1)
iii.
For k = 1, … , ½(s-1), and every B  ψk-1, φk -1 B-covers Пk from φk from φk(ψk)
iv.
For k = ½(s+1), …, s-1, and every B  ψk+1, φk B-covers Пk+1 from φk-1(ψk)
v.
If A  Пk and B  ψk-1  ψk+1/2+(s+1) then
| BПk | = | BПs-k+1 | = | A |, k = 1, …, ½(s-1).
Let Hk, k = 1, …, s be the largest strictly imprimitive sub group of S(X) whose sets of
imprimitivity are exactly the elements of Пk (i.e. A  Пk implies { Aφ: φ  Hk } = S(A) ).
Then the complex K defined by
K = H1φ1H2 … Hs-1φs-1Hs
Assignments of Inlets and outlets of a rearrangeable network
V=S11…….s-1Ss
k =
-1
Sk
nk inlets
1
s-k
k
Sk+1
1
2
2
Sk=Ss-k+1
nk
nk+1
nk+2
2nk
Conclusion
A network is rearrangeable if the product of complexes
generated by the stages of the network is a symmetric group.
(following the conditions specified by BENEŠ)
References
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Benes V.E., Mathematical Theory of Connecting Networks
and Telephone Traffic, 1962, pp 82-135