# What's π(i) in a communication network?

I was reading a text about communication networks, and they define something called $π(i)$ like this:

We will specify the desired destinations of the packets by a permutation of $0, 1, . . . , N-1$. So a permutation, $π$, defines a routing problem: get a packet that starts at input $i$ to output $π(i)$ for $0 \leq i < N$. A routing $P$ that solves a routing problem $π(i)$ is a set of paths from each input to its specified output. That is, $P$ is a set of paths, $P_i$, for $i = 0, ..., N-1$, where $P_i$ goes from input $i$ to output $π(i)$.

I don't exactly get what they mean with $π(i)$. I only heard $π$ so far in context of geometry. So I want to send something from $i$ to $π(i)$, but is $π$ here some kind of function? Is there any reason why they chose $π$ and not, say, $j$ or something? It's also mentioned it's a permutation. Permutation of what?

A permutation of the numbers $0,\dots,N$ is (by definition) a bijective function from the set $\{ 0, \dots, N \}$ to itself.
So yes, $\pi$ is a function, and the choice of name is perhaps because $\pi$ is the Greek letter P as in permutation.
• Ah I see. So $\pi$ represents a routing, this part I got. But given a 2D-array network, what do the numbers represent? For example, in a 3x3 2D array, I have 3 inputs $0, 1, 2$ and 3 outputs $0, 1, 2$. A permutation $\pi$ would be $2, 1, 0$ for example? But what does it mean? Is it the id of the nodes I need to visit?
• As it says: “get a packet that starts at input $i$ to output $\pi(i)$”. In your example, you have $\pi(0)=2$, $\pi(1)=1$, $\pi(2)=0$. Nov 2, 2017 at 5:26