Floyd's Algorithm Application

2 A government has just created a new ministry, the Ministry of Administrative Affairs. The ministry is to have four departments:
the Administration
the Bureaucracy
the Certification Service
the Duplication Section.
Each of these departments is to be established in a separate office on one of four existing sites. The diagram shows the direct journey times in minutes between these four sites. \includegraphics[max width=\textwidth, alt={}, center]{52b8153f-e655-4852-a0f8-6f1c1e9c9170-3_342_403_721_829}

1. Use Floyd's algorithm to find the shortest journey times between the four office sites.
2. Draw a network showing your shortest times.
3. Use appropriate algorithms to find upper and lower bounds for the optimum solution to the Travelling Salesperson Problem in the original network, briefly explaining the steps taken.
4. A van is to be organised to deliver bundles of paperwork between the departments. Why might the optimum solution to the TSP be useful in planning this, and why might it not be?
5. Journeys to locations 2 and 3, from locations 1 and 4, are subject to a congestion charge which is equivalent in costing terms to 15 minutes of journey time. What sort of network would be needed to model this?

3 The weights on the network represent distances. \includegraphics[max width=\textwidth, alt={}, center]{eb4e9c34-7d8f-4118-b7ec-edcd9567077f-4_451_544_324_740}

1. The answer book shows the initial tables and the results of iterations \(1,2,3\) and 5 when Floyd's algorithm is applied to the network.
   (A) Complete the two tables for iteration 4.
   (B) Use the final route table to give the shortest route from vertex \(\mathbf { 3 }\) to vertex \(\mathbf { 5 }\).
   (C) Use the final distance table to produce a complete network with weights representing the shortest distances between vertices.
2. Using the complete network of shortest distances, find a lower bound for the solution to the Travelling Salesperson Problem by deleting vertex 5 and its arcs, and by finding the length of a minimum connector for the remainder. (You may find the minimum connector by inspection.)
3. Use the nearest neighbour algorithm, starting at vertex \(\mathbf { 1 }\), to produce a Hamilton cycle in the complete network. Give the length of your cycle.
4. Interpret your Hamilton cycle in part (iii) in terms of the original network.
5. Give a walk of minimum length which traverses every arc on the original network at least once, and which returns to the start. Give the length of your walk.