MST with edges pre-included

2. \begin{figure}[h] \end{figure} Figure 3 represents nine computer terminals, A, B, C, D, E, F, G, H and J, at Pearsonby School. The school wishes to connect them to form a single computer network. The number on each arc represents the cost, in pounds, of connecting the corresponding computer terminals.

1. Use Prim's algorithm, starting at B , to find the minimum spanning tree for the computer network. You must clearly state the order in which you select the arcs of your tree.
   (3)
2. State the minimum cost of connecting the nine computer terminals.
   (1) It is discovered that some computer terminals are already connected. There are already direct connections along BD and FJ, as shown in bold in Diagram 1 in the answer book. It is decided to use these connections.
3. Use Kruskal's algorithm to find the minimum spanning tree that includes arcs BD and FJ. You must list the arcs in the order that you consider them. In each case, state whether or not you are adding the arcs to your spanning tree.
   (3)
   (Total 7 marks)

2. \begin{figure}[h] \end{figure} Figure 1 represents nine buildings, A, B, C, D, E, F, G, H and I, recently bought by Newberry Enterprises. The company wishes to connect the alarm systems between the buildings to form a single network. The number on each arc represents the cost, in pounds, of connecting the alarm systems between the buildings.

1. Use Prim's algorithm, starting at A , to find the minimum spanning tree for this network. You must list the arcs that form your tree in the order that you select them.
2. State the minimum cost of connecting the alarm systems in the nine buildings. It is discovered that some alarm systems are already connected. There are connections along BC and EF, as shown in bold in Diagram 1 in the answer book. Since these already exist, it is decided to use these arcs as part of the spanning tree.
3. 1. Use Kruskal's algorithm to find the minimum spanning tree that includes arcs BC and EF . You must list the arcs in the order that you consider them. In each case, state whether you are adding the arc to your spanning tree.
   2. Explain why Kruskal's algorithm is a better choice than Prim's algorithm in this case. Since arcs BC and EF already exist, there is no cost for these connections.
4. State the new minimum cost of connecting the nine buildings.

4
15
7

1. Use the bubble sort algorithm to perform ONE complete pass towards sorting these numbers into ascending order. The original list is now to be sorted into descending order.
2. Use a quick sort to obtain the sorted list, giving the state of the list after each complete pass. You must make your pivots clear. The numbers are to be packed into bins of size 26
3. Calculate a lower bound for the minimum number of bins required. You must show your working.
   2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e3ac0632-9560-4cb8-99dd-8f4bf28315f4-3_549_1175_260_443} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 represents nine buildings, A, B, C, D, E, F, G, H and I, recently bought by Newberry Enterprises. The company wishes to connect the alarm systems between the buildings to form a single network. The number on each arc represents the cost, in pounds, of connecting the alarm systems between the buildings.
   1. Use Prim's algorithm, starting at A , to find the minimum spanning tree for this network. You must list the arcs that form your tree in the order that you select them.
   2. State the minimum cost of connecting the alarm systems in the nine buildings. It is discovered that some alarm systems are already connected. There are connections along BC and EF, as shown in bold in Diagram 1 in the answer book. Since these already exist, it is decided to use these arcs as part of the spanning tree.
   3. 1. Use Kruskal's algorithm to find the minimum spanning tree that includes arcs BC and EF . You must list the arcs in the order that you consider them. In each case, state whether you are adding the arc to your spanning tree.
      2. Explain why Kruskal's algorithm is a better choice than Prim's algorithm in this case. Since arcs BC and EF already exist, there is no cost for these connections.
      3. State the new minimum cost of connecting the nine buildings.
         3. \begin{figure}[h]
         \includegraphics[alt={},max width=\textwidth]{e3ac0632-9560-4cb8-99dd-8f4bf28315f4-4_547_413_260_504} \captionsetup{labelformat=empty} \caption{Figure 2}
         \end{figure} \begin{figure}[h]
         \includegraphics[alt={},max width=\textwidth]{e3ac0632-9560-4cb8-99dd-8f4bf28315f4-4_549_412_258_1146} \captionsetup{labelformat=empty} \caption{Figure 3}
         \end{figure} Figure 2 shows the possible allocations of six people, Beth (B), Charlie (C), Harry (H), Karam (K), Sam (S) and Theresa (T), to six tasks 1, 2, 3, 4, 5 and 6. Figure 3 shows an initial matching.
   4. Define the term 'matching'.
      (2)
   5. Starting from the given initial matching, use the maximum matching algorithm to find an improved matching. You should list the alternating path that you use, and state the improved matching.
      (3) After training, a possible allocation for Harry is task 6, and an additional possible allocation for Karam is task 1.
   6. Starting from the matching found in (b), use the maximum matching algorithm to find a complete matching. You should list the alternating path that you use, and state your complete matching.
      (3)
      4. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e3ac0632-9560-4cb8-99dd-8f4bf28315f4-5_814_1303_251_390} \captionsetup{labelformat=empty} \caption{Figure 4
      [0pt] [The total weight of the network is 367 metres]}
      \end{figure} Figure 4 represents a network of water pipes. The number on each arc represents the length, in metres, of that water pipe. A robot will travel along each pipe to check that the pipe is in good repair.
      The robot will travel along each pipe at least once. It will start and finish at A and the total distance travelled must be minimised.
   7. Use the route inspection algorithm to find the pipes that will need to be traversed twice. You must make your method and working clear.
   8. Write down the length of a shortest inspection route. A new pipe, IJ, of length 35 m is added to the network. This pipe must now be included in a new minimum inspection route starting and finishing at A .
   9. Determine if the addition of this pipe will increase or decrease the distance the robot must travel. You must give a reason for your answer.

3 A theme park has 11 rides, \(A , B , \ldots K\). The network shows the distances, in metres, between pairs of rides. The rides are to be connected by cabling so that information can be collated. The manager of the theme park wishes to use the minimum amount of cabling. \includegraphics[max width=\textwidth, alt={}, center]{a1290c22-f28d-42aa-89d5-10d60ca4741c-03_988_1575_488_248}

1. Use Prim's algorithm, starting from \(A\), to find the minimum spanning tree for the network.
2. State the length of cabling required.
3. Draw your minimum spanning tree.
4. The manager decides that the edge \(A E\) must be included. Find the extra length of cabling now required to give the smallest spanning tree that includes \(A E\).

\includegraphics{figure_2} The network in Figure 2 shows the distances, in metres, between 10 wildlife observation points. The observation points are to be linked by footpaths, to form a network along the arcs indicated, using the least possible total length.

1. Find a minimum spanning tree for the network in Figure 2, showing clearly the order in which you selected the arcs for your tree, using
   1. Kruskal's algorithm, [3]
   2. Prim's algorithm, starting from \(A\). [3]
   Given that footpaths are already in place along \(AB\) and \(FI\) and so should be included in the spanning tree,
2. explain which algorithm you would choose to complete the tree, and how it should be adapted. (You do not need to find the tree.) [2]