Apply Kruskal's Algorithm

4 A simplified map of an area of moorland is shown below. The vertices represent farmhouses and the arcs represent the paths between the farmhouses. The weights on the arcs show distances in km. \includegraphics[max width=\textwidth, alt={}, center]{dbefedb2-b398-45e8-92eb-eb510ff16def-4_618_1420_356_319} Ted wants to visit each farmhouse and then return to his starting point.

1. In your answer book the arcs have been sorted into increasing order of weight. Use Kruskal's algorithm to find a minimum spanning tree for the network, and give its total weight. Hence find a route visiting each farmhouse, and returning to the starting point, which has length 82 km .
2. Give the weight of the minimum spanning tree for the six vertices \(A , B , C , E , F , G\). Hence find a route visiting each of the seven farmhouses once, and returning to the starting point, which has length 81 km .
3. Show that the nearest neighbour method fails to produce a cycle through every vertex when started from \(A\) but that it succeeds when started from \(B\). Adapt this cycle to find a complete cycle of total weight less than 70 , and find the total length of the shorter cycle.

3 Answer this question on the insert provided. \includegraphics[max width=\textwidth, alt={}, center]{43fe5fd5-4b98-4c3a-90ca-a1bd5cf065fe-3_492_1006_356_568}

1. This diagram shows a network. The insert has a copy of this network together with a list of the arcs, sorted into increasing order of weight. Use Kruskal's algorithm on the insert to find a minimum spanning tree for this network. Draw your tree and give its total weight.
2. Use your answer to part (i) to find the weight of a minimum spanning tree for the network with vertex \(E\), and all the arcs joined to \(E\), removed. Hence find a lower bound for the travelling salesperson problem on the original network.
3. Show that the nearest neighbour method, starting from vertex \(A\), fails on the original network.
4. Apply the nearest neighbour method, starting from vertex \(B\), to find an upper bound for the travelling salesperson problem on the original network.
5. Apply Dijkstra's algorithm to the copy of the network in the insert to find the least weight path from \(A\) to \(G\). State the weight of the path and give its route.
6. The sum of the weights of all the arcs is 300 . Apply the route inspection algorithm, showing all your working, to find the weight of the least weight closed route that uses every arc at least once. The weights of least weight paths from vertex \(A\) should be found using your answer to part (v); the weights of other such paths should be determined by inspection.

Answer this question on the insert provided. \includegraphics{figure_2}

1. This diagram shows a network. The insert has a copy of this network together with a list of the arcs, sorted into increasing order of weight. Use Kruskal's algorithm on the insert to find a minimum spanning tree for this network. Draw your tree and give its total weight. [5]
2. Use your answer to part (i) to find the weight of a minimum spanning tree for the network with vertex \(G\), and all the arcs joined to \(G\), removed. Hence find a lower bound for the travelling salesperson problem on the original network. [3]
3. Apply the nearest neighbour method, starting from vertex \(A\), to find an upper bound for the travelling salesperson problem on the original network. [3]