OCR D1 (Decision Mathematics 1) 2010 June

1 Owen and Hari each want to sort the following list of marks into decreasing order. $$\begin{array} { l l l l l l l l l l } 31 & 28 & 75 & 87 & 42 & 43 & 70 & 56 & 61 & 95 \end{array}$$

1. Owen uses bubble sort, starting from the left-hand end of the list.
   1. Show the result of the first pass through the list. Record the number of comparisons and the number of swaps used in this first pass. Which marks, if any, are guaranteed to be in their correct final positions after the first pass?
   2. Write down the list at the end of the second pass of bubble sort.
   3. How many more passes are needed to get the value 95 to the start of the list?
   4. Hari uses shuttle sort, starting from the left-hand end of the list. Show the results of the first and the second pass through the list. Record the number of comparisons and the number of swaps used in each of these passes.
   5. Explain why, for this particular list, the total number of comparisons will be greater using bubble sort than using shuttle sort. Shuttle sort is a quadratic order algorithm.
   6. If it takes Hari 20 seconds to sort a list of ten marks using shuttle sort, approximately how long will it take Hari to sort a list of fifty marks?

2 A simple graph is one in which any two vertices are directly connected by at most one arc and no vertex is directly connected to itself. A connected graph is one in which every vertex is joined, directly or indirectly, to every other vertex.
A simply connected graph is one that is both simple and connected.

1. Explain why it is impossible to draw a graph with exactly three vertices in which the vertex orders are 2, 3 and 4.
2. Draw a graph with exactly four vertices of orders 1, 2, 3 and 4 that is neither simple nor connected.
3. Explain why there is no simply connected graph with exactly four vertices of orders \(1,2,3\) and 4. State which of the properties 'simple' and 'connected' cannot be achieved.
4. A simply connected Eulerian graph has exactly five vertices.
   1. Explain why there cannot be exactly three vertices of order 4.
   2. By considering the vertex orders, explain why there are only four such graphs. Draw an example of each.

3 The constraints of a linear programming problem are represented by the graph below. The feasible region is the unshaded region, including its boundaries. \includegraphics[max width=\textwidth, alt={}, center]{7ca6d572-d776-4ad7-a0ed-9ec43c975585-03_908_915_392_614}

1. Write down the inequalities that define the feasible region. The objective is to maximise \(P _ { 1 } = x + 6 y\).
2. Find the values of \(x\) and \(y\) at the optimal point, and the corresponding value of \(P _ { 1 }\). The objective is changed to maximise \(P _ { k } = k x + 6 y\), where \(k\) is positive.
3. Calculate the coordinates of the optimal point, and the corresponding value of \(P _ { k }\) when the optimal point is not the same as in part (ii).
4. Find the range of values of \(k\) for which the point identified in part (ii) is still optimal.

4 The network below represents a small village. The arcs represent the streets and the weights on the arcs represent distances in km . \includegraphics[max width=\textwidth, alt={}, center]{7ca6d572-d776-4ad7-a0ed-9ec43c975585-04_503_1179_324_482}

1. Use Dijkstra's algorithm to find the shortest path from \(A\) to \(G\). You must show your working, including temporary labels, permanent labels and the order in which permanent labels are assigned. Write down the route of the shortest path from \(A\) to \(G\). Hannah wants to deliver newsletters along every street; she will start and end at \(A\).
2. Which standard network problem does Hannah need to solve to find the shortest route that uses every arc? The total weight of all the arcs is 3.7 km .
3. Hannah knows that she will need to travel \(A B\) and \(E F\) twice, once in each direction. With this information, use an appropriate algorithm to find the length of the shortest route that Hannah can use. Show all your working. (You may find the lengths of shortest paths between vertices by inspection.) There are street name signs at each vertex except for \(A\) and \(E\). Hannah's friend Peter wants to check that the signs have not been vandalised. He will start and end at \(B\). The table below shows the complete set of shortest distances between vertices \(B , C , D , F\) and \(G\).

   	\(B\)	\(C\)	\(D\)	\(F\)	\(G\)
   \(B\)	-	0.2	0.1	0.3	0.75
   \(C\)	0.2	-	0.3	0.5	0.95
   \(D\)	0.1	0.3	-	0.2	0.65
   \(F\)	0.3	0.5	0.2	-	0.45
   \(G\)	0.75	0.95	0.65	0.45	-

4. Apply the nearest neighbour method to this table, starting from \(B\), to find an upper bound for the distance that Peter must travel.
5. Apply Prim's algorithm to the matrix formed by deleting the row and column for vertex \(G\) from the table. Start building your tree at vertex \(B\). Draw your tree. Give the order in which vertices are built into your tree and calculate the total weight of your tree. Hence find a lower bound for the distance that Peter must travel.

8 \section*{PLEASE DO NOT WRITE ON THIS PAGE} RECOGNISING ACHIEVEMENT