Numbering path

Intersections in a city are connected by one-way streets. Write a program to calculate the number of different routes between each pair of intersections. A route is a sequence of one-way streets connecting two intersections.

Intersections are labeled with non-negative integers. A one-way street is specified by the pair of intersections it connects. For example, $\text{j k}$ indicates that there is a one-way street from intersection $j$ to intersection $k$. Note that a two-way street can be modeled by two one-way streets: $\text{j k}$ and $\text{k j}$.

Consider a city with four intersections connected as follows:

$\text{0 1}$

$\text{0 2}$

$\text{1 2}$

$\text{2 3}$

There is one route from intersection $0$ to $1$, two routes from $0$ to $2$ (these are 0 \to 1 \to 2 and 0 \to 2), two routes from $0$ to $3$, one route from $1$ to $2$, one route from $1$ to $3$, one route from $2$ to $3$, and no other routes.

It is possible for an infinite number of different routes to exist between some intersections. For example, if the street $\text{3 2}$ is added to the above, there is still only one route from $0$ to $1$, but now there are infinitely many different routes from $0$ to $2$. This is because after reaching $2$ and $3$, you can repeat traveling from $2$ to $3$ and back any number of times, yielding distinct routes. For example, 0 \to 2 \to 3 \to 2 \to 3 \to 2 and 0 \to 2 \to 3 \to 2 are both different routes.