1 In the network below, the arcs represent the roads in Ayton, the vertices represent roundabouts, and the arc weights show the number of traffic lights on each road. Sam is an evening class student at Ayton Academy ( \(A\) ). She wants to drive from the academy to her home ( \(H\) ). Sam hates waiting at traffic lights so she wants to find the route for which the number of traffic lights is a minimum.
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1. Apply Dijkstra's algorithm to find the route that Sam should use to travel from \(A\) to \(H\). At each vertex, show the temporary labels, the permanent label and the order of permanent labelling. In the daytime, Sam works for the highways department. After an electrical storm, the highways department wants to check that all the traffic lights are working. Sam is sent from the depot ( \(D\) ) to drive along every road and return to the depot. Sam needs to pass every traffic light, but wants to repeat as few as possible.
2. Find the minimum number of traffic lights that must be repeated. Show your working. Suppose, instead, that Sam wants to start at the depot, drive along every road and end at her home, passing every traffic light but repeating as few as possible.
3. Find a route on which the minimum number of traffic lights must be repeated. Explain your reasoning.