# Shortest Route

The second problem that we will consider for networks is that of finding the shortest route between any two vertices in the network. As the name suggests the shortest route problem is most commonly applied to transport networks. Other applications include, for example, cost minimisation. Although there are a number of algorithms that are designed to find minimum cost routes between vertices in a network the only algorithm with which you need to be familiar at A Level is Dijkstra’s algorithm.

# Dijkstra’s Algorithm

Dijkstra’s algorithm generates the shortest path tree from a given node to any (or every) other node in the network.

Although the problem that we will use as an example is fairly trivial and can be solved by inspection, the technique that we will use can be applied to much larger problems.

Dijkstra’s algorithm requires that each node in the network be assigned values (labels). There is a working label and a permanent label, as well as an ordering label. Whilst going through the steps of the algorithm you will assign a working label to each vertex. The smallest working label at each iteration will become permanent.

The steps of Dijkstra’s algorithm are:

1. Give the start point the permanent label of 0, and the ordering label 1.
2. Any vertex directly connected to the last vertex given a permanent label is assigned a working label equal to the weight of the connecting edge added to the permanent label you are coming from. If it already has a working label replace it only if the new working label is lower.
3. Select the minimum current working value in the network and make it the permanent label for that node.
4. If the destination node has a permanent label go to step 5, otherwise go to step 2.
5. Connect the destination to the start, working backwards. Select any edge for which the difference between the permanent labels at each end is equal to the weight of the edge.

It is a good idea to use a system for keeping track of the current working labels and the ordering and permanent labels of each of the nodes in the network. A standard system found on A Level exams is that a small grid is drawn near each of the nodes, working labels are written in the lower box, ordering labels in the upper left box, and permanent labels in the upper right box, as shown in figure 1.

## Example

In this example we will consider the network in figure 2. We would like to find the shortest path from node A to node H.

As you can see at the end of the video, the destination node (node H) has a permanent label. This means that a shortest route from A to H has been found.

The shortest route is found by tracing back from the destination node (node H) to the start (node A), selecting each edge for which the difference between permanent values at its terminating nodes is equal to the weight of the edge. The shortest routes for this network are shown in figure 3 below. The edges shown in green (AB and BC) are an alternative to the edge AC shown in red. In this case there are two equivalent shortest routes from A to H.

The solution to the shortest route problem, from A to H, in this network is therefore A-C-G-H, or the equivalent distance A-B-C-G-H. Both have a total distance of 11 units, given by the permanent label of the terminal node.

# Dijkstra’s Complexity

Dijkstra’s algorithm inspects each of the nodes that have not yet been permanently labelled, so the work per step is approximately proportional to the number of remaining nodes. For the entire process this number is:
$(n - 1) + (n - 2) + (n - 3) + \dotsc + 1 = \frac{1}{2}n(n - 1)$.