Lab: Shortest paths

Summary

We explore techniques for finding the shortest path in a graph.

Repository

https://github.com/Grinnell-CSC207/graphs-2019S-revised

Preparation

Fork and clone the repository. Then import it into Eclipse.

Exercises

Exercise 0: Code review

We have updated the code for Graphs.

a. Review the code to quickly explore each of the following additions.

• The readGraph procedure, which reads a graph from a file.
• A variety of new ways to print graphs, most importantly, a write method.
• Some mechanisms for marking graphs.
• The UndirectedGraph class, which has a simple approach for implementing undirected graphs. (Essentially: When we add an edge to the undirected graph, we add the edge in both directions to the underlying directed graph.)
• A path method that finds the path between two vertices.

b. Re-review the path method and be prepared to explain how it works.

Exercise 1: Prim’s algorithm

If you have not done so already, implement Prim’s algorithm. See the prior lab for details.

Exercise 2: Planning for Dijkstra’s algorithm

As you may recall, Dijsktra’s algorithm works something like the following:

To find the shortest path from source to sink,
  Indicate that all vertices as having infinite distance from the source
  Indicate that the source has a distance of 0 from itself
  While unmarked(sink) and there exists an unmarked node with finite distance
    Find the nearest unmarked vertex, u
    Mark u
    For each unmarked neighbor, v, of u
      If distanceTo(u) + edgeWeight(u,v) < distanceTo(v)
        Note that the best path to v is the path to u plus the
        edge from u to v.
  Report the path to sink, if there is one

a. How will you represent the path to each vertex?

b. How will you represent the distance (sum of edge weights) from the source to each vertex?

c. How will you “find the nearest unknown vertex”, particularly given that the distance of vertices can change?

d. How will you mark vertices?

Exercise 3: Implementing Dijkstra’s algorithm

Implement a method, shortestPath(int source, int sink), that finds the shortest path from source to sink.

If you’re unsure about how to deal with various aspects of the algorithm, you can find some notes at the end of this lab.

Exercise 4: Experiments

Write a few experiments or tests to ensure that your implementation does, indeed, find the shortest path. You can likely find some good graphs by doing a Web search for “shortest path example”. (Make sure to cite!)

For those with extra time

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

Some notes on implementing Dijkstra’s algorithm

Finding “the nearest unknown vertex”

While it is tempting to use Java’s built-in PriorityQueue for our collection of unmarked neighbors, there are some subtleties involved. As we noted, the priority of a vertex can change as we find new paths. How do we handle that? If we were using a heap and had a link to the node in the heap, we could “heap up” from that location. Of course, that changes the positions of other nodes, which would be problematic.

One strategy is to do without the PriorityQueue, keep a list of unmarked vertices, and do a linear search on that list.

Another strategy is to use the PriorityQueue, but re-insert the vertex with its new priority each time we update the priority. Doing so means that when we remove vertices from the queue, we will need to recheck their priority.

Paths

You will likely find it easiest to represent the paths with an array of incoming edges, as we did in the path method in Graph.java.

Other data structures

You will likely find it easiest to represent the distance as an array of integers.

To determine whether a vertex is known or unknown, you can either use the mark method or check whether there is an incoming edge to the vertex (using the array just mentioned).