Chapter 29 Exercise 18, Introduction to Java Programming, Tenth Edition Y. Daniel LiangY.

29.18 (Alternative version of Dijkstra algorithm) An alternative version of the Dijkstra algorithm can be described as follows:
Input: a weighted graph G = (V, E) with non-negative weights
Output: A shortest path tree from a source vertex s
1 ShortestPathTree getShortestPath(s) {
2 Let T be a set that contains the vertices whose
3 paths to s are known;
4 Initially T contains source vertex s with cost[s] = 0;
5 for (each u in V – T)
6 cost[u] = infinity;
7
8 while (size of T < n) {
9 Find v in V – T with the smallest cost[u] + w(u, v) value
10 among all u in T;
11 Add v to T and set cost[v] = cost[u] + w(u, v);
12 parent[v] = u;
13 }
14 }

The algorithm uses cost[v] to store the cost of a shortest path from vertex v
to the source vertex s. cost[s] is 0. Initially assign infinity to cost[v] to
indicate that no path is found from v to s. Let V denote all vertices in the graph
and T denote the set of the vertices whose costs are known. Initially, the source
vertex s is in T. The algorithm repeatedly finds a vertex u in T and a vertex v
in V – T such that cost[u] + w(u, v) is the smallest, and moves v to T.
The shortest path algorithm given in the text coninously update the cost and
parent for a vertex in V – T. This algorithm initializes the cost to infinity for
each vertex and then changes the cost for a vertex only once when the vertex is
added into T. Implement this algorithm and use Listing 29.7, TestShortestPath.
java, to test your new algorithm.

import java.util.*;

public class Exercise18 {
  public static void main(String[] args) {
    String[] vertices = { "Seattle", "San Francisco", "Los Angeles", "Denver",
        "Kansas City", "Chicago", "Boston", "New York", "Atlanta", "Miami",
        "Dallas", "Houston" };

    int[][] edges = { { 0, 1, 807 }, { 0, 3, 1331 }, { 0, 5, 2097 },
        { 1, 0, 807 }, { 1, 2, 381 }, { 1, 3, 1267 }, { 2, 1, 381 },
        { 2, 3, 1015 }, { 2, 4, 1663 }, { 2, 10, 1435 }, { 3, 0, 1331 },
        { 3, 1, 1267 }, { 3, 2, 1015 }, { 3, 4, 599 }, { 3, 5, 1003 },
        { 4, 2, 1663 }, { 4, 3, 599 }, { 4, 5, 533 }, { 4, 7, 1260 },
        { 4, 8, 864 }, { 4, 10, 496 }, { 5, 0, 2097 }, { 5, 3, 1003 },
        { 5, 4, 533 }, { 5, 6, 983 }, { 5, 7, 787 }, { 6, 5, 983 },
        { 6, 7, 214 }, { 7, 4, 1260 }, { 7, 5, 787 }, { 7, 6, 214 },
        { 7, 8, 888 }, { 8, 4, 864 }, { 8, 7, 888 }, { 8, 9, 661 },
        { 8, 10, 781 }, { 8, 11, 810 }, { 9, 8, 661 }, { 9, 11, 1187 },
        { 10, 2, 1435 }, { 10, 4, 496 }, { 10, 8, 781 }, { 10, 11, 239 },
        { 11, 8, 810 }, { 11, 9, 1187 }, { 11, 10, 239 } };

    WeightedGraph<String> graph1 = new WeightedGraph<>(edges, vertices);
    WeightedGraph<String>.ShortestPathTree tree1 = graph1
        .getShortestPath(graph1.getIndex("Chicago"));
    tree1.printAllPaths();

    // Display shortest paths from Houston to Chicago
    System.out.print("Shortest path from Houston to Chicago: ");
    java.util.List<String> path = tree1.getPath(graph1.getIndex("Houston"));
    for (String s : path) {
      System.out.print(s + " ");
    }

    edges = new int[][] { { 0, 1, 2 }, { 0, 3, 8 }, { 1, 0, 2 }, { 1, 2, 7 },
        { 1, 3, 3 }, { 2, 1, 7 }, { 2, 3, 4 }, { 2, 4, 5 }, { 3, 0, 8 },
        { 3, 1, 3 }, { 3, 2, 4 }, { 3, 4, 6 }, { 4, 2, 5 }, { 4, 3, 6 } };
    WeightedGraph<Integer> graph2 = new WeightedGraph<Integer>(edges, 5);
    WeightedGraph<Integer>.ShortestPathTree tree2 = graph2.getShortestPath(3);
    System.out.println("\n");
    tree2.printAllPaths();
  }

  public static interface Graph<V> {
    /** Return the number of vertices in the graph */
    public int getSize();

    /** Return the vertices in the graph */
    public java.util.List<V> getVertices();

    /** Return the object for the specified vertex index */
    public V getVertex(int index);

    /** Return the index for the specified vertex object */
    public int getIndex(V v);

    /** Return the neighbors of vertex with the specified index */
    public java.util.List<Integer> getNeighbors(int index);

    /** Return the degree for a specified vertex */
    public int getDegree(int v);

    /** Print the edges */
    public void printEdges();

    /** Clear graph */
    public void clear();

    /** Add a vertex to the graph */
    public boolean addVertex(V vertex);

    /** Add an edge to the graph */
    public boolean addEdge(int u, int v);

    /** Obtain a depth-first search tree */
    public AbstractGraph<V>.Tree dfs(int v);

    /** Obtain a breadth-first search tree */
    public AbstractGraph<V>.Tree bfs(int v);
  }

  public static abstract class AbstractGraph<V> implements Graph<V> {
    protected List<V> vertices = new ArrayList<V>(); // Store vertices
    protected List<List<Integer>> neighbors 
      = new ArrayList<List<Integer>>(); // Adjacency lists

    /** Construct an empty graph */
    protected AbstractGraph() {
    }
    
    /** Construct a graph from edges and vertices stored in arrays */
    protected AbstractGraph(int[][] edges, V[] vertices) {
      for (int i = 0; i < vertices.length; i++)
        addVertex(vertices[i]);
      
      createAdjacencyLists(edges, vertices.length);
    }

    /** Construct a graph from edges and vertices stored in List */
    protected AbstractGraph(List<Edge> edges, List<V> vertices) {
      for (int i = 0; i < vertices.size(); i++)
        addVertex(vertices.get(i));
          
      createAdjacencyLists(edges, vertices.size());
    }

    /** Construct a graph for integer vertices 0, 1, 2 and edge list */
    protected AbstractGraph(List<Edge> edges, int numberOfVertices) {
      for (int i = 0; i < numberOfVertices; i++) 
        addVertex((V)(new Integer(i))); // vertices is {0, 1, ...}
      
      createAdjacencyLists(edges, numberOfVertices);
    }

    /** Construct a graph from integer vertices 0, 1, and edge array */
    protected AbstractGraph(int[][] edges, int numberOfVertices) {
      for (int i = 0; i < numberOfVertices; i++) 
        addVertex((V)(new Integer(i))); // vertices is {0, 1, ...}
      
      createAdjacencyLists(edges, numberOfVertices);
    }

    /** Create adjacency lists for each vertex */
    private void createAdjacencyLists(
        int[][] edges, int numberOfVertices) {
      for (int i = 0; i < edges.length; i++) {
        int u = edges[i][0];
        int v = edges[i][1];
        addEdge(u, v);
      }
    }

    /** Create adjacency lists for each vertex */
    private void createAdjacencyLists(
        List<Edge> edges, int numberOfVertices) {
      for (Edge edge: edges) {
        addEdge(edge.u, edge.v);
      }
    }

    @Override /** Return the number of vertices in the graph */
    public int getSize() {
      return vertices.size();
    }

    @Override /** Return the vertices in the graph */
    public List<V> getVertices() {
      return vertices;
    }

    @Override /** Return the object for the specified vertex */
    public V getVertex(int index) {
      return vertices.get(index);
    }

    @Override /** Return the index for the specified vertex object */
    public int getIndex(V v) {
      return vertices.indexOf(v);
    }

    @Override /** Return the neighbors of the specified vertex */
    public List<Integer> getNeighbors(int index) {
      return neighbors.get(index);
    }

    @Override /** Return the degree for a specified vertex */
    public int getDegree(int v) {
      return neighbors.get(v).size();
    }

    @Override /** Print the edges */
    public void printEdges() {
      for (int u = 0; u < neighbors.size(); u++) {
        System.out.print(getVertex(u) + " (" + u + "): ");
        for (int j = 0; j < neighbors.get(u).size(); j++) {
          System.out.print("(" + u + ", " +
            neighbors.get(u).get(j) + ") ");
        }
        System.out.println();
      }
    }

    @Override /** Clear graph */
    public void clear() {
      vertices.clear();
      neighbors.clear();
    }
    
    @Override /** Add a vertex to the graph */  
    public boolean addVertex(V vertex) {
      if (!vertices.contains(vertex)) {
        vertices.add(vertex);
        neighbors.add(new ArrayList<Integer>());
        return true;
      }
      else {
        return false;
      }
    }

    @Override /** Add an edge to the graph */  
    public boolean addEdge(int u, int v) {
      if (u < 0 || u > getSize() - 1)
        throw new IllegalArgumentException("No such index: " + u);

      if (v < 0 || v > getSize() - 1)
        throw new IllegalArgumentException("No such index: " + v);
      
      if (!neighbors.get(u).contains(v)) {
        neighbors.get(u).add(v);
        return true;
      }
      else {
        return false;
      }
    }
    
    /** Edge inner class inside the AbstractGraph class */
    public static class Edge {
      public int u; // Starting vertex of the edge
      public int v; // Ending vertex of the edge

      /** Construct an edge for (u, v) */
      public Edge(int u, int v) {
        this.u = u;
        this.v = v;
      }
    }
    
    @Override /** Obtain a DFS tree starting from vertex v */
    /** To be discussed in Section 30.6 */
    public Tree dfs(int v) {
      List<Integer> searchOrder = new ArrayList<Integer>();
      int[] parent = new int[vertices.size()];
      for (int i = 0; i < parent.length; i++)
        parent[i] = -1; // Initialize parent[i] to -1

      // Mark visited vertices
      boolean[] isVisited = new boolean[vertices.size()];

      // Recursively search
      dfs(v, parent, searchOrder, isVisited);

      // Return a search tree
      return new Tree(v, parent, searchOrder);
    }

    /** Recursive method for DFS search */
    private void dfs(int v, int[] parent, List<Integer> searchOrder,
        boolean[] isVisited) {
      // Store the visited vertex
      searchOrder.add(v);
      isVisited[v] = true; // Vertex v visited

      for (int w : neighbors.get(v)) {
        if (!isVisited[w]) {
          parent[w] = v; // The parent of vertex i is v
          dfs(w, parent, searchOrder, isVisited); // Recursive search
        }
      }
    }

    @Override /** Starting bfs search from vertex v */
    /** To be discussed in Section 27.7 */
    public Tree bfs(int v) {
      List<Integer> searchOrder = new ArrayList<Integer>();
      int[] parent = new int[vertices.size()];
      for (int i = 0; i < parent.length; i++)
        parent[i] = -1; // Initialize parent[i] to -1

      java.util.LinkedList<Integer> queue =
        new java.util.LinkedList<Integer>(); // list used as a queue
      boolean[] isVisited = new boolean[vertices.size()];
      queue.offer(v); // Enqueue v
      isVisited[v] = true; // Mark it visited

      while (!queue.isEmpty()) {
        int u = queue.poll(); // Dequeue to u
        searchOrder.add(u); // u searched
        for (int w : neighbors.get(u)) {
          if (!isVisited[w]) {
            queue.offer(w); // Enqueue w
            parent[w] = u; // The parent of w is u
            isVisited[w] = true; // Mark it visited
          }
        }
      }

      return new Tree(v, parent, searchOrder);
    }

    /** Tree inner class inside the AbstractGraph class */
    /** To be discussed in Section 27.5 */
    public class Tree {
      private int root; // The root of the tree
      private int[] parent; // Store the parent of each vertex
      private List<Integer> searchOrder; // Store the search order

      /** Construct a tree with root, parent, and searchOrder */
      public Tree(int root, int[] parent, List<Integer> searchOrder) {
        this.root = root;
        this.parent = parent;
        this.searchOrder = searchOrder;
      }

      /** Return the root of the tree */
      public int getRoot() {
        return root;
      }

      /** Return the parent of vertex v */
      public int getParent(int v) {
        return parent[v];
      }

      /** Return an array representing search order */
      public List<Integer> getSearchOrder() {
        return searchOrder;
      }

      /** Return number of vertices found */
      public int getNumberOfVerticesFound() {
        return searchOrder.size();
      }
      
      /** Return the path of vertices from a vertex to the root */
      public List<V> getPath(int index) {
        ArrayList<V> path = new ArrayList<V>();

        do {
          path.add(vertices.get(index));
          index = parent[index];
        }
        while (index != -1);

        return path;
      }

      /** Print a path from the root to vertex v */
      public void printPath(int index) {
        List<V> path = getPath(index);
        System.out.print("A path from " + vertices.get(root) + " to " +
          vertices.get(index) + ": ");
        for (int i = path.size() - 1; i >= 0; i--)
          System.out.print(path.get(i) + " ");
      }

      /** Print the whole tree */
      public void printTree() {
        System.out.println("Root is: " + vertices.get(root));
        System.out.print("Edges: ");
        for (int i = 0; i < parent.length; i++) {
          if (parent[i] != -1) {
            // Display an edge
            System.out.print("(" + vertices.get(parent[i]) + ", " +
              vertices.get(i) + ") ");
          }
        }
        System.out.println();
      }
    }
  }

  public static class WeightedGraph<V> extends AbstractGraph<V> {
    // Priority adjacency lists
    private List<PriorityQueue<WeightedEdge>> queues
      = new ArrayList<PriorityQueue<WeightedEdge>>();

    /** Construct a WeightedGraph from edges and vertices in arrays */
    public WeightedGraph() {
    }
    
    /** Construct a WeightedGraph from edges and vertices in arrays */
    public WeightedGraph(int[][] edges, V[] vertices) {
      super(edges, vertices);
      createQueues(edges, vertices.length);
    }

    /** Construct a WeightedGraph from edges and vertices in List */
    public WeightedGraph(int[][] edges, int numberOfVertices) {
      super(edges, numberOfVertices);
      createQueues(edges, numberOfVertices);
    }

    /** Construct a WeightedGraph for vertices 0, 1, 2 and edge list */
    public WeightedGraph(List<WeightedEdge> edges, List<V> vertices) {
      super((List)edges, vertices);
      createQueues(edges, vertices.size());
    }

    /** Construct a WeightedGraph from vertices 0, 1, and edge array */
    public WeightedGraph(List<WeightedEdge> edges,
        int numberOfVertices) {
      super((List)edges, numberOfVertices);
      createQueues(edges, numberOfVertices);
    }

    /** Create priority adjacency lists from edge arrays */
    private void createQueues(int[][] edges, int numberOfVertices) {
      for (int i = 0; i < numberOfVertices; i++) {
        queues.add(new PriorityQueue<WeightedEdge>()); // Create a queue
      }

      for (int i = 0; i < edges.length; i++) {
        int u = edges[i][0];
        int v = edges[i][1];
        int weight = edges[i][2];
        // Insert an edge into the queue
        queues.get(u).offer(new WeightedEdge(u, v, weight));
      }
    }

    /** Create priority adjacency lists from edge lists */
    private void createQueues(List<WeightedEdge> edges,
        int numberOfVertices) {
      for (int i = 0; i < numberOfVertices; i++) {
        queues.add(new PriorityQueue<WeightedEdge>()); // Create a queue
      }

      for (WeightedEdge edge: edges) {
        queues.get(edge.u).offer(edge); // Insert an edge into the queue
      }
    }

    /** Display edges with weights */
    public void printWeightedEdges() {
      for (int i = 0; i < queues.size(); i++) {
        System.out.print(getVertex(i) + " (" + i + "): ");
        for (WeightedEdge edge : queues.get(i)) {
          System.out.print("(" + edge.u +
            ", " + edge.v + ", " + edge.weight + ") ");
        }
        System.out.println();
      }
    }

    /** Get the edges from the weighted graph */  
    public List<PriorityQueue<WeightedEdge>> getWeightedEdges() {
      return queues;
    }
    
    /** Clears the weighted graph */
    public void clear() {
      vertices.clear();
      neighbors.clear();
      queues.clear();
    }
    
    /** Add vertices to the weighted graph */  
    public boolean addVertex(V vertex) {
      if (super.addVertex(vertex)) {
        if (queues == null) 
          queues = new ArrayList<PriorityQueue<WeightedEdge>>();
        queues.add(new PriorityQueue<WeightedEdge>());
        return true;
      }
      else {
        return false;
      }
    }

    /** Add edges to the weighted graph */  
    public void addEdge(int u, int v, double weight) {
      if (super.addEdge(u, v)) {
        queues.get(u).add(new WeightedEdge(u, v, weight));
      }
    }

    /** Get a minimum spanning tree rooted at vertex 0 */
    public MST getMinimumSpanningTree() {
      return getMinimumSpanningTree(0);
    }
    
    /** Get a minimum spanning tree rooted at a specified vertex */
    public MST getMinimumSpanningTree(int startingVertex) {
      List<Integer> T = new ArrayList<Integer>();
      // T initially contains the startingVertex;
      T.add(startingVertex);

      int numberOfVertices = vertices.size(); // Number of vertices
      int[] parent = new int[numberOfVertices]; // Parent of a vertex
      // Initially set the parent of all vertices to -1
      for (int i = 0; i < parent.length; i++)
        parent[i] = -1;
      double totalWeight = 0; // Total weight of the tree thus far

      // Clone the priority queue, so to keep the original queue intact
      List<PriorityQueue<WeightedEdge>> queues = deepClone(this.queues);

      // All vertices are found?
      while (T.size() < numberOfVertices) {
        // Search for the vertex with the smallest edge adjacent to
        // a vertex in T
        int v = -1;
        double smallestWeight = Double.MAX_VALUE;
        for (int u: T) { 
          while (!queues.get(u).isEmpty() &&
              T.contains(queues.get(u).peek().v)) {
            // Remove the edge from queues[u] if the adjacent
            // vertex of u is already in T
            queues.get(u).remove();
          }

          if (!queues.get(u).isEmpty()) {
            // Current smallest weight on an edge adjacent to u
            WeightedEdge edge = queues.get(u).peek();
            if (edge.weight < smallestWeight) {
              v = edge.v;
              smallestWeight = edge.weight;
              // If v is added to the tree, u will be its parent
              parent[v] = u;
            }
          }
        } // End of for

        if (v != -1) {
          T.add(v); // Add a new vertex to the tree
          totalWeight += smallestWeight;
        }
        else 
          break; // The tree is not connected, a partial MST is found
      } // End of while

      return new MST(startingVertex, parent, T, totalWeight);
    }

    /** Clone an array of queues */
    private List<PriorityQueue<WeightedEdge>> deepClone(
      List<PriorityQueue<WeightedEdge>> queues) {
      List<PriorityQueue<WeightedEdge>> copiedQueues =
        new ArrayList<PriorityQueue<WeightedEdge>>();

      for (int i = 0; i < queues.size(); i++) {
        copiedQueues.add(new PriorityQueue<WeightedEdge>());
        for (WeightedEdge e : queues.get(i)) {
          copiedQueues.get(i).add(e);
        }
      }

      return copiedQueues;
    }

    /** MST is an inner class in WeightedGraph */
    public class MST extends Tree {
      private double totalWeight; // Total weight of all edges in the tree

      public MST(int root, int[] parent, List<Integer> searchOrder,
          double totalWeight) {
        super(root, parent, searchOrder);
        this.totalWeight = totalWeight;
      }

      public double getTotalWeight() {
        return totalWeight;
      }
    }

    /** Find single source shortest paths */
    public ShortestPathTree getShortestPath(int sourceVertex) {
      // T stores the vertices whose path found so far
      List<Integer> T = new ArrayList<Integer>();
      // T initially contains the sourceVertex;
      T.add(sourceVertex);

      // vertices is defined in AbstractGraph
      int numberOfVertices = vertices.size();

      // parent[v] stores the previous vertex of v in the path
      int[] parent = new int[numberOfVertices];
      parent[sourceVertex] = -1; // The parent of source is set to -1

      // cost[v] stores the cost of the path from v to the source
      double[] cost = new double[numberOfVertices];
      for (int i = 0; i < cost.length; i++) {
        cost[i] = Double.MAX_VALUE; // Initial cost set to infinity
      }
      cost[sourceVertex] = 0; // Cost of source is 0

      // Get a copy of queues
      List<PriorityQueue<WeightedEdge>> queues = deepClone(this.queues);

      // Expand T
      while (T.size() < numberOfVertices) {
        int v = -1; // Vertex to be determined
        double smallestCost = Double.MAX_VALUE; // Set to infinity
        for (int u : T) {
          while (!queues.get(u).isEmpty() &&
              T.contains(queues.get(u).peek().v)) {
            queues.get(u).remove(); // Remove the vertex in queue for u
          }

          if (!queues.get(u).isEmpty()) {
            WeightedEdge e = queues.get(u).peek();
            if (cost[u] + e.weight < smallestCost) {
              v = e.v;
              smallestCost = cost[u] + e.weight;
              // If v is added to the tree, u will be its parent
              parent[v] = u;
            }
          }
        } // End of for

        if (v != -1) {
          T.add(v); // Add a new vertex to T
          cost[v] = smallestCost;
        }
        else
          break; // Graph is not connected, s cannot reach all vertices
      } // End of while

      // Create a ShortestPathTree
      return new ShortestPathTree(sourceVertex, parent, T, cost);
    }

    /** ShortestPathTree is an inner class in WeightedGraph */
    public class ShortestPathTree extends Tree {
      private double[] cost; // cost[v] is the cost from v to source

      /** Construct a path */
      public ShortestPathTree(int source, int[] parent, 
          List<Integer> searchOrder, double[] cost) {
        super(source, parent, searchOrder);
        this.cost = cost;
      }

      /** Return the cost for a path from the root to vertex v */
      public double getCost(int v) {
        return cost[v];
      }

      /** Print paths from all vertices to the source */
      public void printAllPaths() {
        System.out.println("All shortest paths from " +
          vertices.get(getRoot()) + " are:");
        for (int i = 0; i < cost.length; i++) {
          printPath(i); // Print a path from i to the source
          System.out.println("(cost: " + cost[i] + ")"); // Path cost
        }
      }
    }
    
    /** Find single source shortest paths, implementation in the text */
    public ShortestPathTree getShortestPath1(int sourceVertex) {
      // T stores the vertices whose path found so far
      List<Integer> T = new ArrayList<Integer>();
      
      // T initially contains the sourceVertex;
      T.add(sourceVertex);

      // vertices is defined in AbstractGraph
      int numberOfVertices = vertices.size();

      // parent[v] stores the previous vertex of v in the path
      int[] parent = new int[numberOfVertices];
      parent[sourceVertex] = -1; // The parent of source is set to -1

      // cost[v] stores the cost of the path from v to the source
      double[] cost = new double[numberOfVertices];
      for (int i = 0; i < cost.length; i++) {
        cost[i] = Double.MAX_VALUE; // Initial cost set to infinity
      }
      cost[sourceVertex] = 0; // Cost of source is 0

      // Get a copy of queues
      List<PriorityQueue<WeightedEdge>> queues = deepClone(this.queues);

      // Set cost for the neighbors of sourceVertex
      while (!queues.get(sourceVertex).isEmpty()) {
        WeightedEdge e = queues.get(sourceVertex).poll();
        cost[e.v] = e.weight;
        parent[e.v] = sourceVertex;
      }
      
      // Expand T
      while (T.size() < numberOfVertices) {
        // Find smallest cost v in V - T 
        int v = -1; // Vertex to be determined
        double currentMinCost = Double.MAX_VALUE;
        for (int i = 0; i < getSize(); i++) {
          if (!T.contains(i) && cost[i] < currentMinCost) {
            currentMinCost = cost[i];
            v = i;
          }
        }
        
        if (v != -1) {
          T.add(v); // Add a new vertex to T
    
          // Adjust cost[u] for u that is adjacent to v and u in V - T
          while (!queues.get(v).isEmpty()) {
            WeightedEdge e = queues.get(v).poll();
            if (!T.contains(e.v) && cost[e.v] > cost[v] + e.weight) {
              cost[e.v] = cost[v] + e.weight;
              parent[e.v] = v; 
            }
          }
        }
        else
          break; // s cannot reach to all vertices
      } // End of while

      // Create a ShortestPathTree
      return new ShortestPathTree(sourceVertex, parent, T, cost);
    }
  }
  
 static class WeightedEdge extends AbstractGraph.Edge implements Comparable<WeightedEdge> {
  public double weight; // The weight on edge (u, v)

  /** Create a weighted edge on (u, v) */
  public WeightedEdge(int u, int v, double weight) {
   super(u, v);
   this.weight = weight;
  }

  /** Compare two edges on weights */
  public int compareTo(WeightedEdge edge) {
   if (weight > edge.weight) {
    return 1;
   } else if (weight == edge.weight) {
    return 0;
   } else {
    return -1;
   }
  }
  
  @Override
  public boolean equals(Object obj) {
   if (obj instanceof WeightedEdge) {
    WeightedEdge that = (WeightedEdge)obj;
    return (u == that.u) && (v == that.v) && (weight == that.weight);   
   } else {
    return false;
   }
  }
  
  @Override
  public String toString() {
   return "(" + u + ", " + v + ", " + weight + ")";
  }
 }
}