Graph 接口定义了图的常见操作。 Java 集合框架是设计复杂数据结构的一个很好的例子。数据结构的共同特征在接口中定义（例如，Collection、Set、List、Queue），如下所示图 20.1。抽象类（例如 AbstractCollection、AbstractSet、AbstractList）部分实现了接口。具体类（例如，HashSet、LinkedHashSet、TreeSet、ArrayList、LinkedList、PriorityQueue）提供具体的实现。这种设计模式对于图形建模很有用。我们将定义一个名为 Graph 的接口，其中包含图的所有常见操作，以及一个名为 AbstractGraph 的抽象类，它部分实现 Graph 接口。许多具体的图表可以添加到设计中。例如，我们将定义名为 UnweightedGraph 和 WeightedGraph 的图。这些接口和类的关系如下图所示。
图的常见操作有哪些？一般来说，您需要获取图中的顶点数量，获取图中的所有顶点，获取指定索引的顶点对象，获取指定名称的顶点的索引，获取顶点的邻居，获取顶点的度数，清除图，添加新顶点，添加新边，执行深度优先搜索，并执行广度优先搜索。深度优先搜索和广度优先搜索将在下一节中介绍。下图在 UML 图中说明了这些方法。
AbstractGraph 没有引入任何新方法。顶点列表和边邻接列表在 AbstractGraph 类中定义。有了这些数据字段，就足以实现 Graph 接口中定义的所有方法。为了方便起见，我们假设该图是一个简单图，即一个顶点本身没有边，并且从顶点 u 到 v 没有平行边。
AbstractGraph 实现了 Graph 中的所有方法，并且除了一个方便的 addEdge(edge) 添加 方法之外，它没有引入任何新方法>Edge 对象到邻接边列表。 UnweightedGraph 只是用五个构造函数扩展 AbstractGraph，用于创建具体的 Graph 实例。
您可以创建具有任何类型顶点的图形。每个顶点都与一个索引相关联，该索引与顶点列表中该顶点的索引相同。如果您创建图形时未指定顶点，则顶点与其索引相同。
AbstractGraph 类实现了 Graph 接口中的所有方法。那么为什么它被定义为抽象呢？将来，您可能需要向 Graph 接口添加在 AbstractGraph 中无法实现的新方法。为了使类易于维护，最好将 AbstractGraph 类定义为抽象类。
假设所有这些接口和类都可用。下面的代码给出了一个测试程序，该程序创建上图所示的图形以及下图 (a) 中的另一个图形。
public class TestGraph {
public static void main(String[] args) {
String[] vertices = {"Seattle", "San Francisco", "Los Angeles", "Denver", "Kansas City", "Chicago", "Boston", "New York", "Atlanta", "Miami", "Dallas", "Houston"};
// Edge array for graph
int[][] edges = {
{0, 1}, {0, 3}, {0, 5},
{1, 0}, {1, 2}, {1, 3},
{2, 1}, {2, 3}, {2, 4}, {2, 10},
{3, 0}, {3, 1}, {3, 2}, {3, 4}, {3, 5},
{4, 2}, {4, 3}, {4, 5}, {4, 7}, {4, 8}, {4, 10},
{5, 0}, {5, 3}, {5, 4}, {5, 6}, {5, 7},
{6, 5}, {6, 7},
{7, 4}, {7, 5}, {7, 6}, {7, 8},
{8, 4}, {8, 7}, {8, 9}, {8, 10}, {8, 11},
{9, 8}, {9, 11},
{10, 2}, {10, 4}, {10, 8}, {10, 11},
{11, 8}, {11, 9}, {11, 10}
};
Graph<String> graph1 = new UnweightedGraph<>(vertices, edges);
System.out.println("The number of vertices in graph1: " + graph1.getSize());
System.out.println("The vertex with index 1 is " + graph1.getVertex(1));
System.out.println("The index for Miami is " + graph1.getIndex("Miami"));
System.out.println("The edges for graph1:");
graph1.printEdges();
// List of Edge objects for graph
String[] names = {"Peter", "Jane", "Mark", "Cindy", "Wendy"};
java.util.ArrayList<AbstractGraph.Edge> edgeList = new java.util.ArrayList<>();
edgeList.add(new AbstractGraph.Edge(0, 2));
edgeList.add(new AbstractGraph.Edge(1, 2));
edgeList.add(new AbstractGraph.Edge(2, 4));
edgeList.add(new AbstractGraph.Edge(3, 4));
// Create a graph with 5 vertices
Graph<String> graph2 = new UnweightedGraph<>(java.util.Arrays.asList(names), edgeList);
System.out.println("\nThe number of vertices in graph2: " + graph2.getSize());
System.out.println("Te edges for graph2:");
graph2.printEdges();
}
}
graph1 中的顶点数量：12
索引为 1 的顶点是旧金山
迈阿密的指数是 9
graph1 的边：
西雅图 (0): (0, 1) (0, 3) (0, 5)
旧金山 (1): (1, 0) (1, 2) (1, 3)
洛杉矶 (2): (2, 1) (2, 3) (2, 4) (2, 10)
丹佛 (3): (3, 0) (3, 1) (3, 2) (3, 4) (3, 5)
堪萨斯城 (4): (4, 2) (4, 3) (4, 5) (4, 7) (4, 8) (4, 10)
芝加哥 (5): (5, 0) (5, 3) (5, 4) (5, 6) (5, 7)
波士顿 (6): (6, 5) (6, 7)
纽约 (7): (7, 4) (7, 5) (7, 6) (7, 8)
亚特兰大 (8): (8, 4) (8, 7) (8, 9) (8, 10) (8, 11)
迈阿密 (9): (9, 8) (9, 11)
达拉斯 (10): (10, 2) (10, 4) (10, 8) (10, 11)
休斯顿 (11): (11, 8) (11, 9) (11, 10)
The number of vertices in graph2: 5
The edges for graph2:
Peter (0): (0, 2)
Jane (1): (1, 2)
Mark (2): (2, 4)
Cindy (3): (3, 4)
Wendy (4):
The program creates graph1 for the graph in Figure 28.1 in lines 3–23. The vertices for graph1 are defined in lines 3–5. The edges for graph1 are defined in 8–21. The edges are represented using a two-dimensional array. For each row i in the array, edges[i][0] and edges[i][1] indicate that there is an edge from vertex edges[i][0] to vertex edges[i][1]. For example, the first row, {0, 1}, represents the edge from vertex 0 (edges[0][0]) to vertex 1 (edges[0][1]). The row {0, 5} represents the edge from vertex 0 (edges[2][0]) to vertex 5 (edges[2][1]). The graph is created in line 23. Line 31 invokes the printEdges() method on graph1 to display all edges in graph1.
The program creates graph2 for the graph in Figure 28.3a in lines 34–43. The edges for graph2 are defined in lines 37–40. graph2 is created using a list of Edge objects in line 43. Line 47 invokes the printEdges() method on graph2 to display all edges in graph2.
Note that both graph1 and graph2 contain the vertices of strings. The vertices are associated with indices 0, 1, . . . , n-1. The index is the location of the vertex in vertices. For example, the index of vertex Miami is 9.
Now we turn our attention to implementing the interface and classes. The codes below give the Graph interface, the AbstractGraph class, and the UnweightedGraph class, respectively.
public 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 the graph */
public void clear();
/** Add a vertex to the graph */
public void addVertex(V vertex);
/** Add an edge to the graph */
public void addEdge(int u, int v);
/** Obtain a depth-first search tree starting from v */
public AbstractGraph<V>.Tree dfs(int v);
/** Obtain a breadth-first search tree starting from v */
public AbstractGraph<V>.Tree bfs(int v);
}
import java.util.*;
public abstract class AbstractGraph<V> implements Graph<V> {
protected List<V> vertices = new ArrayList<>(); // Store vertices
protected List<List<Edge>> neighbors = new ArrayList<>(); // Adjacency lists
/** Construct an empty graph */
protected AbstractGraph() {}
/** Construct a graph from vertices and edges stored in arrays */
protected AbstractGraph(V[] vertices, int[][] edges) {
for(int i = 0; i < vertices.length; i++)
addVertex(vertices[i]);
createAdjacencyLists(edges, vertices.length);
}
/** Construct a graph from vertices and edges stored in List */
protected AbstractGraph(List<V> vertices, List<Edge> edges) {
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++) {
addEdge(edges[i][0], edges[i][1]);
}
}
/** 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) {
List<Integer> result = new ArrayList<>();
for(Edge e: neighbors.get(index))
result.add(e.v);
return result;
}
@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(Edge e: neighbors.get(u)) {
System.out.print("(" + getVertex(e.u) + ", " + getVertex(e.v) + ") ");
}
System.out.println();
}
}
@Override /** Clear the graph */
public void clear() {
vertices.clear();
neighbors.clear();
}
@Override /** Add a vertex to the graph */
public void addVertex(V vertex) {
if(!vertices.contains(vertex)) {
vertices.add(vertex);
neighbors.add(new ArrayList<Edge>());
}
}
/** Add an edge to the graph */
protected boolean addEdge(Edge e) {
if(e.u < 0 || e.u > getSize() - 1)
throw new IllegalArgumentException("No such index: " + e.u);
if(e.v < 0 || e.v > getSize() - 1)
throw new IllegalArgumentException("No such index: " + e.v);
if(!neighbors.get(e.u).contains(e)) {
neighbors.get(e.u).add(e);
return true;
}
else {
return false;
}
}
@Override /** Add an edge to the graph */
public void addEdge(int u, int v) {
addEdge(new Edge(u, v));
}
/** 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;
}
public boolean equals(Object o) {
return u == ((Edge)o).u && v == ((Edge)o).v;
}
}
@Override /** Obtain a DFS tree starting from vertex v */
public Tree dfs(int v) {
List<Integer> searchOrder = new ArrayList<>();
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 u, int[] parent, List<Integer> searchOrder, boolean[] isVisited) {
// Store the visited vertex
searchOrder.add(u);
isVisited[u] = true; // Vertex v visited
for(Edge e: neighbors.get(u)) {
if(!isVisited[e.v]) {
parent[e.v] = u; // The parent of vertex e.v is u
dfs(e.v, parent, searchOrder, isVisited); // Recursive search
}
}
}
@Override /** Starting bfs search from vertex v */
public Tree bfs(int v) {
List<Integer> searchOrder = new ArrayList<>();
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<>(); // list used as 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(Edge e: neighbors.get(u)) {
if(!isVisited[e.v]) {
queue.offer(e.v); // Enqueue w
parent[e.v] = u; // The parent of w is u
isVisited[e.v] = true; // Mark it visited
}
}
}
return new Tree(v, parent, searchOrder);
}
/** Tree inner class inside the AbstractGraph class */
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<>();
do {
path.add(vertices.get(index));
index = parent[index];
}
while(index != -1);
return path;
}
/** Print a path from the root 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();
}
}
}
import java.util.*;
public class UnweightedGraph<V> extends AbstractGraph<V> {
/** Construct an empty graph */
public UnweightedGraph() {}
/** Construct a graph from vertices and edges stored in arrays */
public UnweightedGraph(V[] vertices, int[][] edges) {
super(vertices, edges);
}
/** Construct a graph from vertices and edges stored in List */
public UnweightedGraph(List<V> vertices, List<Edge> edges) {
super(vertices, edges);
}
/** Construct a graph for integer vertices 0, 1, 2, and edge list */
public UnweightedGraph(List<Edge> edges, int numberOfVertices) {
super(edges, numberOfVertices);
}
/** Construct a graph from integer vertices 0, 1, and edge array */
public UnweightedGraph(int[][] edges, int numberOfVertices) {
super(edges, numberOfVertices);
}
}
The code in the Graph interface and the UnweightedGraph class are straightforward. Let us digest the code in the AbstractGraph class.
The AbstractGraph class defines the data field vertices (line 4) to store vertices and neighbors (line 5) to store edges in adjacency lists. neighbors.get(i) stores all edges adjacent to vertex i. Four overloaded constructors are defined in lines 9–42 to create a default graph, or a graph from arrays or lists of edges and vertices. The createAdjacencyLists(int[][] edges, int numberOfVertices) method creates adjacency lists from edges in an array (lines 45–50). The createAdjacencyLists(List edges, int numberOfVertices) method creates adjacency lists from edges in a list (lines 53–58).
The getNeighbors(u) method (lines 81–87) returns a list of vertices adjacent to vertex u. The clear() method (lines 106–110) removes all vertices and edges from the graph. The addVertex(u) method (lines 112–122) adds a new vertex to vertices and returns true. It returns false if the vertex is already in the graph (line 120).
The addEdge(e) method (lines 124–139) adds a new edge the adjacency edge list and returns true. It returns false if the edge is already in the graph. This method may throw IllegalArgumentException if the edge is invalid (lines 126–130).
The printEdges() method (lines 95–104) displays all vertices and edges adjacent to each vertex.
The code in lines 164–293 gives the methods for finding a depth-first search tree and a breadth-first search tree, which will be introduced in Depth-First Search (DFS) and Breadth-First Search (BFS), respectively.
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