29.20 (Test if a vertex u is in T efficiently) Since T is implemented using a list
T.contains(u) takes O(n) time. Modify these two methods by introducing
an array named isInT. Set isInT[u] to true when a vertex u is added to T.
Testing whether a vertex u is in T can now be done in O(1) time. Write a test
program using the following code, where graph1 is created from Figure 29.1.
import java.util.*;
public class Exercise20 {
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>.MST tree1 = graph1.getMinimumSpanningTree();
System.out.println("Total weight is " + tree1.getTotalWeight());
tree1.printTree();
System.out.println();
WeightedGraph<String>.ShortestPathTree tree2 =
graph1.getShortestPath(graph1.getIndex("Chicago"));
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);
}
/* NEW IMPLEMENTATION */
/** 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);
// Track if a vertex is in T
boolean[] isInT = new boolean[vertices.size()];
isInT[startingVertex] = true;
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() &&
isInT[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
isInT[v] = true;
totalWeight += smallestWeight;
}
else
return null; // 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);
// Track if a vertex is in T
boolean[] isInT = new boolean[vertices.size()];
isInT[sourceVertex] = true;
// parent[v] stores the previous vertex of v in the path
int[] parent = new int[vertices.size()];
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[vertices.size()];
for (int i = 0; i < cost.length; i++) {
cost[i] = Double.POSITIVE_INFINITY; // 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() < vertices.size()) {
// Find smallest cost v in V - T
int u = -1; // Vertex to be determined
double currentMinCost = Double.POSITIVE_INFINITY;
for (int i = 0; i < getSize(); i++) {
if (!isInT[i] && cost[i] < currentMinCost) {
currentMinCost = cost[i];
u = i;
}
}
if (u != -1) {
T.add(u); // Add a new vertex to T
isInT[u] = true;
// Adjust cost[v] for v that is adjacent to u and v in V - T
while (!queues.get(u).isEmpty()) {
WeightedEdge e = queues.get(u).poll();
if (!isInT[e.v] && cost[e.v] > cost[u] + e.weight) {
cost[e.v] = cost[u] + e.weight;
parent[e.v] = u;
}
}
}
else
return null; // s cannot reach to 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
}
}
}
}
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 + ")";
}
}
}