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Problem Solving with Algorithms and Data Structures using Java The Interactive Edition

Bradley N. Miller, David L. Ranum, Roman Yasinovskyy, J. David Eisenberg

Section 7.25 Graphs and Graphing Algorithms: Exercises

Exercises Exercises

1.

Draw the graph corresponding to the following adjacency matrix.
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Figure 7.25.1. 6 x 6 Adjacency Matrix
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2.

Draw the graph corresponding to the following list of edges.
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Table 7.25.2.
From To Cost
1 2 10
1 3 15
1 6 5
2 3 7
3 4 7
3 6 10
4 5 7
6 4 5
5 6 13
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3.

Ignoring the weights, perform a breadth-first search on the graph drawn for question 1 or 2.
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4.

What is the Big-O running time of the build_graph function?
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  • O(n)
  • O(n) would suggest that there is no nesting. There are several nested for loops.
  • O(n2)
  • Correct. The two consecutively nested for loops would dictate that this is in the realm of O(n2)
  • O(1)
  • O(1) would suggest that the function is constant. Since there are multiple for loops intertwined, it is not in constant time.
  • O(n3)
  • O(n3) would suggest that there are three consecutively nested for loops. There are only two.
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5.

Derive the Big-O running time for the topological sort algorithm.
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6.

Derive the Big-O running time for the strongly connected components algorithm.
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7.

Show each step in applying Dijkstra’s algorithm to the graph drawn for question 1 or 2.
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8.

Using Prim’s algorithm, find the minimum weight spanning tree for the graph drawn for question 1 or 2.
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9.

Draw a dependency graph illustrating the steps needed to send an email. Perform a topological sort on your graph.
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10.

Express branching factor k as a function of the board size >n.
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11.

Derive an expression for the base of the exponent used in expressing the running time of the knights tour.
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12.

Explain why the general DFS algorithm is not suitable for solving the knight’s tour problem.
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13.

What is the Big-O running time for Prim’s minimum spanning tree algorithm?
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  • O(1)
  • O(1) would mean that the algorithm runs in constant time. This isn’t true because there are several comparisons happening in the algorithm.
  • O(n3)
  • O(n3) suggests that there are three consecutively nested loops. If you look at the example algorithm, you can see that there are not three nested loops.
  • O(n)
  • O(n) is linear time. The time it takes for this program to run doesn’t grow linearly.
  • O(n2)
  • Correct. Since you are not only comparing the weight of a branch but also if the branch has already been connected to, this would make the Big-O of the algorithm O(n2).
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14.

Modify the depth-first search function to produce a topological sort.
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15.

Write the transpose method for the Graph class.
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16.

Modify the depth-first search to produce strongly connected components. Hint: produce an ArrayList<ArrayList<T>> (where T is your data type). The main ArrayList contains one ArrayList per forest, with each forest containing the keys for that forest.
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17.

Using breadth-first search, write an algorithm that can determine the shortest path from each vertex to every other vertex. This is called the “all pairs shortest path problem.”
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18.

Using breadth-first search revise the maze program from Chapter 4 to find the shortest path out of a maze.
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