Lab: Minimum spanning trees

Summary
We explore techniques for building minimum spanning trees
Repository
https://github.com/Grinnell-CSC207/graphs-2019S

Preparation

If you have not done so already, fork and clone the repository. Import it into Eclipse.

Exercises

Exercise 1: Priority queues

As you may recall, all of the MST algorithms rely on some sort of priority queue that allows you to find the smallest edge in a set of edges (the whole set of edges or the edges in Kruskal’s; those adjacent to the partial MST in Prim’s).

a. Identify an appropriate implementation of priority queues in Java.

b. Sketch how you will use that implementation to order edges by weight.

Exercise 2: From directed to undirected

As you may recall, Prim’s algorithm is intended to work with undirected graphs, rather than directed graphs.

How will you accommodate that issue in your code?

Exercise 3: Parts of Prim’s algorithm

As you may recall, Prim’s algorithm relies on two structures (beyond the graph): a priority queue of edges left to process and a collection of the edges already determined to be in the MST. We’ll call the first thing remaining and the second mst.

The algorithm goes something like the following.

Pick a random vertex
Add all of the edges from that vertex to remaining
While edges remain
Grab the remaining edge with the lowest weight
If either vertex is not in the MST
Add all the edges from that vertex to remaining
(arguably, you should only add those that don't lead back to the MST)


How will you implement each of the following steps?

a. Represent the MST. (Remember it’s a collection of edges.)

b. Pick a random vertex.

c. Grab the remaining edge with lowest weight.

d. Determine if a vertex is in the MST.

e. Print out the MST.

Exercise 4: Implementing Prim’s algorithm

Implement Prim’s algorithm. If you are unsure about any of the steps suggested above, you can discuss them with your instructor or mentor, review our suggestions at the end of this lab, or both.

For those with extra time

If you find that you have extra time, implement Kruskal’s MST algorithm.

Suggested strategies

• To deal with non-directional edges, we can just make sure that we always add pairs of edges (both directions). Once an edge is in the MST, it doesn’t matter what it’s direction is.
• Option 1: Add an addUndirectedEdge method.
• Option 2: Subclass the Graph class and override addEdge.
• You can use a PriorityQueue of Edge objects to keep track of which edges remain. You’ll need to supply an edge comparator, which will look something like the following: (e1,e2) -> e1.weight().compareTo(e2.weight())
• You can use an ArrayList to keep track of the edges in the MST.
• You can determine whether or not a vertex is in the MST by marking vertices in the MST. (Note: You will need to clear the marks before you begin.)
• If you are implementing Prim’s algorithm within the Graph class (or one of its descendants), you can randomly select a non-null element from vertices. Alternately, you can choose the first (or last) non-null element.
• Since ArrayLists have a toString method, you don’t need to do anything special to print out the MST.