How to find the longest path in a network?

To find the longest path in a network, we need to use the algorithm of Dijkstra.

Dijkstra's algorithm is a popular algorithm used to find the shortest path between two nodes in a graph. However, with a slight modification, we can use it to find the longest path as well. The algorithm works by starting at the source node and exploring all possible paths from that node. It then selects the path with the lowest cost and moves to the next node. This process is repeated until the destination node is reached.

To modify the algorithm to find the longest path, we simply need to change the cost function. Instead of adding the weights of the edges, we subtract them. This means that the algorithm will select the path with the highest cost at each step. The final result will be the longest path from the source node to the destination node.

Let's take an example to illustrate this. Consider the following network:

```
A -> B (5)
A -> C (2)
B -> D (4)
C -> D (3)
D -> E (6)
```

To find the longest path from A to E, we start by assigning a cost of 0 to A and infinity to all other nodes. We then explore all possible paths from A and update the costs of the nodes accordingly. The table below shows the progress of the algorithm:

```
Node | Cost
-----|-----
A | 0
B | -5
C | -2
D | -6
E | -12
```

The algorithm terminates when it reaches the destination node (E in this case). The longest path from A to E is -12, which corresponds to the path A -> C -> D -> E.