Exercise 5

  1. Reading: Chapter 16 (pp. 370-383).

  2. Pen-and-paper exercises:
    1. In each of the graph examples below, your graph should have at least 6 nodes.
      1. Draw an example of a graph such that if you remove any one edge, the resulting graph remains connected.
      2. Draw an example of a graph such that if you remove any edge, the resulting graph is not connected.
      3. Draw an example of a graph such that only one edge in the graph has the property that if you remove it, the resulting graph will not be connected.
    2. For the graph below, show the steps in executing Kruskal's algorithm and Prim's algorithm. Draw the graph with the MST edges identified at each step.

    3. Suppose a new degree curriculum is being offered with exactly N courses, all of which are required. Various courses have pre-requisites; a single course can have multiple pre-requisites. The curriculum is properly designed in that you can't go back a pre-requisite chain and loop back to the starting course. Suppose also that a student can take any number of courses in a semester. Describe in pseudocode an algorithm to find the minimum number of semesters needed to graduate? What is the running time of your algorithm?
    Note: Credit will not be given only for answers - show all your work: your reasoning, steps you took to get your answer etc.

  3. In this part, you will use Kruskal's algorithm (with adjacency matrices) to estimate the minimum-spanning tree weight of a randomly-created set of points.

    Compute the average weight of the minimum spanning tree for a collection of n random points in the unit square:

    • Consider a set of random points in the plane. For example, suppose these were pegs nailed to a board. Suppose you had to connect these pegs with the least amount amount of string. Note that this is just the minimum spanning tree problem. (You need to think about this for a few minutes). The goal is to compute the weight of the minimum spanning tree for a random set of points. Here, you can assume an "edge" between each pair of points with the distance between as the weight.
    • How do you generate n random points in the unit square? And what is the unit square?
      The unit square is the square with one corner at the origin (0,0) and opposite corner at (1,1). To generate a point randomly, we can use UniformRandom.uniform() to generate a value randomly between 0 and 1 for the X value, and call it once again for the Y value. So, to generate n such X and Y values, you can use this code: 
          for (int i=0; i < n; i++)
      X[i] = UniformRandom.uniform();
    for (int i=0; i < n; i++)
      Y[i] = UniformRandom.uniform();
    • Modify the code given to you in Module 8 for Kruskal's algorithm. Read through this carefully.
    • Name your class Kruskal.java.
    • Your class will need to implement the SpanningTreeAlgorithm interface. In particular, you will need to provide implementations for these methods: initialize(), minimumSpanningTree(double[][] adjMatrix) and getTreeWeight().
    • The classes and interfaces involved in this exercise are:
      The Algorithm interface
      The SpanningTreeAlgorithm interface
    • Also, Kruskal's algorithm will need an implementation of Union-Find. For this purpose, you should create a class called UnionFindInt. You can intuit what this class needs to have from its use in the Module 8 examples.
    • Use the test environment to test your minimum spanning tree algorithm. However, it does not compute the average tree weight. Simply submit your hardcopy and evidence that your code worked for the average-weight problem.
    • Use this properties file for testing your adjacency matrix implementation.

    Next, use your implementation for the following:

    • Suppose I have n points and a minimum-spanning tree already computed. Consider two options if another point is added to the data set:
      1. Recompute. In this option, you recompute the MST again using the n+1 points as input.
      2. Join-Heuristic. In this option, you find the closest existing point (among the first n points) to the new ((n+1)-st) point and put the edge between them in the new MST.
    • How much more efficient (in order-notation) is the Join-Heuristic?
    • How much worse, on average, is the MST produced by the Join-Heuristic?

Submission: