PERMUTATION GROUPS, COMPLEXES, AND REARRANGEABLE CONNECTING NETWORKS

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PERMUTATION GROUPS, COMPLEXES, AND REARRANGEABLE
CONNECTING NETWORKS
V.E.BENE
SUNITHA THODUPUNURI
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Connecting Networks

• A connecting network is an arrangement of
  switches and transmission links through which
  certain terminals can be connected together in
  many combinations.
•   Eg Telephone central offices.
• Performance of the connecting networks
• Structure of the connecting networks

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Combinatorial properties of Connecting Networks

• Blocking
• NonBlocking
• Rearrangeable

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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.
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Goal

• What stages and link patterns are used to
  construct the rearrangeable networks?

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Overview

• 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
• ü  First theorem gives sufficient conditions
  for connecting terminals and link
  patterns which gives rise to rearrangeable
  networks
• ü Second theorem indicates a simple way of
  describing link patterns and stages
  that satisfy the hypothesis of first theorem

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A Network in terms of Group Theory

• 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 ? ?

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Group theory terms Contd

• 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

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Group theory terms Contd

• A stage can generate a permutation
• 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), i1,2,,
  N i.e., (i, ?(i)) ?S
• The set of permutations is generated by a stage S
  is denoted by P(S)
• Similarly, a network with N inlets and N
  outlets can generate a permutation

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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)
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The generation of complexes by stages

• A stage is made of square switches if there is a
  partition ? on 1,2,3N 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

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Group Theory Terminology

• Suppose X, Y are sets
• XY
• ? 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

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Group theory terminology contd

• ?(?2) is the permutation of X induced by ? acting
  on the elements of ?2 i.e.,
• ?(?2) ?(B) B ? ?2
• ?B 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

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Generating the permutation group

• P(S1) ?1P(S2) ?2 .. ?s-1P(Ss)
• Let s gt 3 be an odd integer, let f1, , fs-1be
  permutations on X 1, , N , , let ?k,
• k 1, ..., s, and ?k, k 1, ..., ½(s-1),be
  partitions of X, and let
• X k 0
• fk -1 ... f1/2(S-1) -1 ( ?k ) k 1, ,
  ½(s-1)
• ? fk f1/2(s1) (?s-k ) k ½(s1),
  , s-1
• X k s
• Suppose that
• If s 3, then ?k lt ?k1, k 0, , ½(s-3).
• ?1/2(s1) ?1/2(s-1)
• For k 1, , ½(s-1), and every B ? ?k-1, fk -1
  B-covers ?k from fk from fk(?k)
• For k ½(s1), , s-1, and every B ? ?k1, fk
  B-covers ?k1 from fk-1(?k)
• If A ? ?k and B ? ?k-1 ? ?k1/2(s1) then
• B?k B?s-k1 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 ?k
  (i.e. A ? ?k implies Af f ? Hk S(A) ).
  Then the complex K defined by
• K H1f1H2 Hs-1fs-1Hs

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Assignments of Inlets and outlets of a
rearrangeable network
VS1?1.?s-1Ss ?k ?-1s-k SkSs-k1
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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 BENE)

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References

• Benes V.E., Mathematical Theory of Connecting
  Networks and Telephone Traffic, 1962, pp 82-135