Show basic concepts of permutation, Computer Engineering

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Q. Show Basic concepts of permutation?

Let us look at the basic concepts of permutation with respect to interconnection network.  Let us say the network has set of n input nodes and n output nodes.

Permutation P for a network of 5 nodes (i.e., n = 5) is written like this:

856_Show Basic concepts of permutation.png

It signifies node connections are 1↔5, 2↔4, 3↔1, 4↔3, 5↔2.

The connections are displayed in the Figure below.  

61_Show Basic concepts of permutation1.png

Figure: Node-Connections

The other permutation of the similar set of nodes can be 

423_Show Basic concepts of permutation2.png

That means connections are: 1↔2, 2↔3, 3↔5, 4↔1, and 5↔4 similarly other permutations are also possible.  The set of all permutations of a 3 node network would be 

2376_Show Basic concepts of permutation3.png

Connection, 790_Show Basic concepts of permutation4.png signifies connection from node 1 to node 1, node 2 to node 2, and node 3 to node 3 therefore it hasn't any meaning, so it's dropped. In these illustrations, only one set of links exist between output and input nodes and denotes it's a single stage network.  It might be probable that there exist many links between input and output (It means that multistage network).  Permutation of all these in a multistage network are known as permutation group and these stand for by a cycle for example permutation.

P= (1,2,3) (4,5) means that the network has two sets of input and output nodes, one group contains nodes 1,2,3 and another group contains nodes 4,5 and associations are 1→2, 2→3, 3→1, and 4→5. Here set (1, 2, 3) has period 3 and (4, 5) has period 2, Together these groups has periodicity 3×2=6. 

Interconnection from all the probable input nodes to all output nodes structures the permutation group.

1570_Show Basic concepts of permutation5.png

The permutations can be joined. This is known as composition operation. In composition operation two or more permutations are concerned in sequence, for example if P1 and P2 are two permutations stated like this:

2361_Show Basic concepts of permutation5.png

The composition of P1 and P2 will be

687_Show Basic concepts of permutation6.png

Similarly,

If P31498_Show Basic concepts of permutation7.png

And P4 = 1935_Show Basic concepts of permutation8.png

Then P3. P4 =

2297_Show Basic concepts of permutation9.png

1612_Show Basic concepts of permutation10.png

Compositions of those permutations P1 and P2 are represented in Figures a and b.


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