1. 无向图和有向图
图是由结点和结点之间的连接构成。术语：图的结点就是顶点（v），结点之间的链接就是边（e）。
无向图是一种边为无序结点对的图。
有向图是一种边为有序结点对的图。
2. 网络
网络或者称为甲加权图，是一种每条边都带有权重或代价的图。
对于加权图的表示我们需要定义边为三元组。（起点，终点；权重）。
3. 常用的图算法
3.1 遍历
- 广度优先遍历BFS:使用队列来辅助实现。
@Override
public Iterator iteratorBFS(T startVertex) {
return iteratorBFS(getIndex(startVertex));
}
private Iterator iteratorBFS(int startIndex) {
Integer x;
Queue<Integer> tq = new LinkedList<Integer>();
List<T> rls = new ArrayList<T>();
if (!indexIsValid(startIndex)) {
return rls.iterator();
}
boolean[] visited = new boolean[numVertices];
for (int i = 0; i < numVertices; i++) {
visited[i] = false;
}
tq.add(new Integer(startIndex));
visited[startIndex] = true;
while (!tq.isEmpty()) {
// 出队列
x = tq.poll();
// 遍历记录表
rls.add(vertices[x.intValue()]);
for (int i = 0; i < numVertices; i++) {
if (adjMatrix[x.intValue()][i] && !visited[i]) {
tq.offer(new Integer(i));
visited[i] = true;
}
}
}
return new GraphIterator(rls.iterator());
} - 深度优先遍历DFS:使用栈来辅助实现。
@Override
public Iterator iteratorDFS(T startIndex) {
return iteratorDFS(getIndex(startIndex));
}
/**
* 深度优先遍历的实现
*
* @param index
* @return
*/
private Iterator iteratorDFS(int startIndex) {
Integer x;
Stack<Integer> ts = new Stack<Integer>();
List<T> rls = new ArrayList<T>();
boolean found;
if (!indexIsValid(startIndex)) {
return rls.iterator();
}
boolean[] visited = new boolean[numVertices];
for (int i = 0; i < numVertices; i++) {
visited[i] = false;
}
ts.push(new Integer(startIndex));
rls.add(vertices[startIndex]);
visited[startIndex] = true;
while (!ts.isEmpty()) {
x = ts.peek();
found = false;///用于作为栈顶是否是活顶点的标志变量
for (int i = 0; (i < numVertices) && !found; i++) {
if (adjMatrix[x.intValue()][i] && !visited[i]) {
ts.push(new Integer(i));
rls.add(vertices[i]);
visited[i] = true;
found = true;
}
}
//如果栈顶结点不是活结点并且栈不为空
if (!found && !ts.isEmpty())
ts.pop();
}
return new GraphIterator(rls.iterator());
}3.2 测试联通性
对图的连通性我们有一个简单的解法:在一个含有n个顶点的图中，当且仅当对每一个顶点v，从v开始的广度优先遍历的resultList的大小都是n，那么整个图就是连通的。
@Override
public boolean isConnected() {
boolean flag = true;
for(int i=0;i<vertices.length;i++){
int temp=0;
temp=getSizeOfIterator(this.iteratorBFS(vertices[i]));
if(temp!=vertices.length){
flag = false;
break;
}
}
return flag;
}3.3 最小生成树
生成树是一棵含有图中所有顶点和部分边（但可能不是所有边）的树。最小生成树(MST)是这样一棵生成树，其边的权重和小于或等于同一个图中其他任何一棵生成树的权重总和。
/**
* MST算法实现
*
* @return a minimum spanning tree of the graph
*/
public Graph getMST()
{
int x, y;
int[] edge = new int[2];
Stack<int[]> vertexStack = new Stack<int[]>();
Graph<T> resultGraph = new Graph<T>();
if (isEmpty() || !isConnected())
return resultGraph;
resultGraph.adjMatrix = new boolean[numVertices][numVertices];
for (int i = 0; i < numVertices; i++)
for (int j = 0; j < numVertices; j++)
resultGraph.adjMatrix[i][j] = false;
resultGraph.vertices = (T[])(new Object[numVertices]);
boolean[] visited = new boolean[numVertices];
for (int i = 0; i < numVertices; i++)
visited[i] = false;
edge[0] = 0;
resultGraph.vertices[0] = this.vertices[0];
resultGraph.numVertices++;
visited[0] = true;
// Add all edges that are adjacent to vertex 0 to the stack.
for (int i = 0; i < numVertices; i++)
{
if (!visited[i] && this.adjMatrix[0][i])
{
edge[1] = i;
vertexStack.push(edge.clone());
visited[i] = true;
}
}
while ((resultGraph.size() < this.size()) && !vertexStack.isEmpty())
{
// Pop an edge off the stack and add it to the resultGraph.
edge = vertexStack.pop();
x = edge[0];
y = edge[1];
resultGraph.vertices[y] = this.vertices[y];
resultGraph.numVertices++;
resultGraph.adjMatrix[x][y] = true;
resultGraph.adjMatrix[y][x] = true;
visited[y] = true;
// Add all unvisited edges that are adjacent to vertex y
// to the stack.
for (int i = 0; i < numVertices; i++)
{
if (!visited[i] && this.adjMatrix[i][y])
{
edge[0] = y;
edge[1] = i;
vertexStack.push(edge.clone());
visited[i] = true;
}
}
}
return resultGraph;
}3.4 判定最短路径
判定图的最短路径有两种情况。
- 第一种，也是最简单的一种，是判定起始节点与目标结点之间的字面意义上的最短路径，也就是两个顶点之间的最小边数，方法就是使用广度优先遍历的一个简单变体。
- 第二种，寻找加权图的最短路径。Dijkstra为此开发了一种类似于上述算法的算法。
鉴于篇幅关系我们这里实现第一种方法
@Override
public Iterator iteratorShortestPath(T startVertex, T targetVertex) {
return iteratorShortestPath(getIndex(startVertex),
getIndex(targetVertex));
}
/**
* 查找两个顶点之间的最近路径
* @param startIndex
* @param targetIndex
* @return
*/
private Iterator iteratorShortestPath(int startIndex, int targetIndex) {
List<T> resultList = new ArrayList<T>();
if (!indexIsValid(startIndex) || !indexIsValid(targetIndex))
return resultList.iterator();
//获取结点集合的迭代器对象
Iterator<Integer> it = iteratorShortestPathIndices(startIndex,
targetIndex);
while (it.hasNext())
resultList.add(vertices[((Integer)it.next()).intValue()]);
return new GraphIterator(resultList.iterator());
}
/**
* 构建从开始到结束的节点集合的迭代器对象
* @param startIndex
* @param targetIndex
* @return
*/
private Iterator<Integer> iteratorShortestPathIndices(int startIndex,
int targetIndex) {
int index = startIndex;
int[] pathLength = new int[numVertices];
int[] predecessor = new int[numVertices];
Queue<Integer> traversalQueue = new LinkedList<Integer>();
List<Integer> resultList = new ArrayList<Integer>();
if (!indexIsValid(startIndex) || !indexIsValid(targetIndex) ||
(startIndex == targetIndex))
return resultList.iterator();
boolean[] visited = new boolean[numVertices];
for (int i = 0; i < numVertices; i++)
visited[i] = false;
traversalQueue.offer(new Integer(startIndex));
visited[startIndex] = true;
pathLength[startIndex] = 0;
predecessor[startIndex] = -1;
while (!traversalQueue.isEmpty() && (index != targetIndex))
{
index = (traversalQueue.poll()).intValue();
//Update the pathLength for each unvisited vertex adjacent
// to the vertex at the current index.
for (int i = 0; i < numVertices; i++)
{
if (adjMatrix[index][i] && !visited[i])
{
pathLength[i] = pathLength[index] + 1;
predecessor[i] = index;
traversalQueue.offer(new Integer(i));
visited[i] = true;
}
}
}
if (index != targetIndex) // no path must have been found
return resultList.iterator();
Stack<Integer> stack = new Stack<Integer>();
index = targetIndex;
stack.push(new Integer(index));
do
{
index = predecessor[index];
stack.push(new Integer(index));
} while (index != startIndex);
while (!stack.isEmpty())
resultList.add(((Integer)stack.pop()));
return new GraphIndexIterator(resultList.iterator());
}
3.5 图扩大的算法
/*
* 扩大容量的函数
*/
private void expandCapacity() {
T[] largerVertices = (T[]) (new Object[vertices.length * 2]);
boolean[][] largerAdjMatrix = new boolean[vertices.length * 2][vertices.length * 2];
for (int i = 0; i < numVertices; i++) {
for (int j = 0; j < numVertices; j++) {
largerAdjMatrix[i][j] = adjMatrix[i][j];
}
largerVertices[i] = vertices[i];
}
vertices = largerVertices;
adjMatrix = largerAdjMatrix;
}
4. 邻接矩阵实现无向图
完整的链接矩阵实现图算法
package ds.java.ch15;
import java.util.Iterator;
/**
* @author LbZhang
* @version 创建时间：2015年12月2日 下午4:56:30
* @description 基本图的接口设计
*/
public interface GraphADT<T> {
public void addVertex(T vertex);
public void removeVertex(T vertex);
public void addEdge(T v1,T v2);
public void removeEdge(T v1,T v2);
/**
* 广度遍历
* @param startIndex
* @return
*/
public Iterator iteratorBFS(T startIndex);
/**
* 深度遍历
* @param startIndex
* @return
*/
public Iterator iteratorDFS(T startIndex);
/**
* 两个顶点之间最短路径的获取
* @param startVertex
* @param targetVertex
* @return
*/
public Iterator iteratorShortestPath(T startVertex,T targetVertex);
/**
* 判定是否为空
* @return
*/
public boolean isEmpty();
/**
* 测试连通性
* @return
*/
public boolean isConnected();
/**
* 返回顶点的规模
* @return
*/
public int size();
/**
* 输出图结构
* @return
*/
public String toString();
}
package ds.java.ch15;
import java.util.ArrayList;
import java.util.ConcurrentModificationException;
import java.util.Iterator;
import java.util.LinkedList;
import java.util.List;
import java.util.NoSuchElementException;
import java.util.Queue;
import java.util.Stack;
import ds.java.ch15.exceptions.EmptyCollectionException;
/**
* @author LbZhang
* @version 创建时间：2015年12月3日 上午10:29:09
* @description 类说明
*/
public class Graph<T> implements GraphADT<T> {
protected final int DEFAULT_CAPACITY = 5;
protected int numVertices; // 当前顶点个数
protected boolean[][] adjMatrix; // 邻接矩阵
protected T[] vertices; // 顶点的值
protected int modCount;// 修改标记数
/**
* 无参构造函数
*/
public Graph() {
numVertices = 0;
this.adjMatrix = new boolean[DEFAULT_CAPACITY][DEFAULT_CAPACITY];
this.vertices = (T[]) (new Object[DEFAULT_CAPACITY]);
}
@Override
public void addVertex(T vertex) {
// 如果顶点满了
if ((numVertices + 1) == adjMatrix.length)
expandCapacity();
// 添加结点
vertices[numVertices] = vertex;
// 添加的这个顶点和每一个顶点的连边默认的设置
for (int i = 0; i < numVertices; i++) {
adjMatrix[numVertices][i] = false;
adjMatrix[i][numVertices] = false;
}
numVertices++;
modCount++;
}
/**
* 扩大容量的函数
*/
private void expandCapacity() {
T[] largerVertices = (T[]) (new Object[vertices.length * 2]);
boolean[][] largerAdjMatrix = new boolean[vertices.length * 2][vertices.length * 2];
for (int i = 0; i < numVertices; i++) {
for (int j = 0; j < numVertices; j++) {
largerAdjMatrix[i][j] = adjMatrix[i][j];
}
largerVertices[i] = vertices[i];
}
vertices = largerVertices;
adjMatrix = largerAdjMatrix;
}
@Override
public void removeVertex(T vertex) {
if (isEmpty()) {
throw new EmptyCollectionException("Graph");
}
}
@Override
public void addEdge(T v1, T v2) {
addEdge(getIndex(v1), getIndex(v2));
}
/**
* 两个索引标号之间添加链接
*
* @param index
* @param index2
*/
private void addEdge(int index1, int index2) {
if (indexIsValid(index1) && indexIsValid(index2)) {
adjMatrix[index1][index2] = true;
adjMatrix[index2][index1] = true;
modCount++;
}
}
private boolean indexIsValid(int index) {
if (index < vertices.length) {
return true;
}
return false;
}
/**
* 获取结点的索引标号
*
* @param v1
* @return
*/
private int getIndex(T v) {
int index = vertices.length;
// boolean flag =
for (int i = 0; i < vertices.length; i++) {
if (v.equals(vertices[i])) {
index = i;
break;
}
}
return index;
}
@Override
public void removeEdge(T v1, T v2) {
removeEdge(getIndex(v1), getIndex(v2));
}
private void removeEdge(int index1, int index2) {
if (indexIsValid(index1) && indexIsValid(index2)) {
adjMatrix[index1][index2] = false;
adjMatrix[index2][index1] = false;
modCount++;
}
}
@Override
public Iterator iteratorBFS(T startVertex) {
return iteratorBFS(getIndex(startVertex));
}
private Iterator iteratorBFS(int startIndex) {
Integer x;
Queue<Integer> tq = new LinkedList<Integer>();
List<T> rls = new ArrayList<T>();
if (!indexIsValid(startIndex)) {
return rls.iterator();
}
boolean[] visited = new boolean[numVertices];
for (int i = 0; i < numVertices; i++) {
visited[i] = false;
}
tq.add(new Integer(startIndex));
visited[startIndex] = true;
while (!tq.isEmpty()) {
// 出队列
x = tq.poll();
// 遍历记录表
rls.add(vertices[x.intValue()]);
for (int i = 0; i < numVertices; i++) {
if (adjMatrix[x.intValue()][i] && !visited[i]) {
tq.offer(new Integer(i));
visited[i] = true;
}
}
}
return new GraphIterator(rls.iterator());
}
@Override
public Iterator iteratorDFS(T startIndex) {
return iteratorDFS(getIndex(startIndex));
}
/**
* 深度优先遍历的实现
*
* @param index
* @return
*/
private Iterator iteratorDFS(int startIndex) {
Integer x;
Stack<Integer> ts = new Stack<Integer>();
List<T> rls = new ArrayList<T>();
boolean found;
if (!indexIsValid(startIndex)) {
return rls.iterator();
}
boolean[] visited = new boolean[numVertices];
for (int i = 0; i < numVertices; i++) {
visited[i] = false;
}
ts.push(new Integer(startIndex));
rls.add(vertices[startIndex]);
visited[startIndex] = true;
while (!ts.isEmpty()) {
x = ts.peek();
found = false;///用于作为栈顶是否是活顶点的标志变量
for (int i = 0; (i < numVertices) && !found; i++) {
if (adjMatrix[x.intValue()][i] && !visited[i]) {
ts.push(new Integer(i));
rls.add(vertices[i]);
visited[i] = true;
found = true;
}
}
//如果栈顶结点不是活结点并且栈不为空
if (!found && !ts.isEmpty())
ts.pop();
}
return new GraphIterator(rls.iterator());
}
@Override
public Iterator iteratorShortestPath(T startVertex, T targetVertex) {
return iteratorShortestPath(getIndex(startVertex),
getIndex(targetVertex));
}
/**
* 查找两个顶点之间的最近路径
* @param startIndex
* @param targetIndex
* @return
*/
private Iterator iteratorShortestPath(int startIndex, int targetIndex) {
List<T> resultList = new ArrayList<T>();
if (!indexIsValid(startIndex) || !indexIsValid(targetIndex))
return resultList.iterator();
//获取结点集合的迭代器对象
Iterator<Integer> it = iteratorShortestPathIndices(startIndex,
targetIndex);
while (it.hasNext())
resultList.add(vertices[((Integer)it.next()).intValue()]);
return new GraphIterator(resultList.iterator());
}
/**
* 构建从开始到结束的节点集合的迭代器对象
* @param startIndex
* @param targetIndex
* @return
*/
private Iterator<Integer> iteratorShortestPathIndices(int startIndex,
int targetIndex) {
int index = startIndex;
int[] pathLength = new int[numVertices];
int[] predecessor = new int[numVertices];
Queue<Integer> traversalQueue = new LinkedList<Integer>();
List<Integer> resultList = new ArrayList<Integer>();
if (!indexIsValid(startIndex) || !indexIsValid(targetIndex) ||
(startIndex == targetIndex))
return resultList.iterator();
boolean[] visited = new boolean[numVertices];
for (int i = 0; i < numVertices; i++)
visited[i] = false;
traversalQueue.offer(new Integer(startIndex));
visited[startIndex] = true;
pathLength[startIndex] = 0;
predecessor[startIndex] = -1;
while (!traversalQueue.isEmpty() && (index != targetIndex))
{
index = (traversalQueue.poll()).intValue();
//Update the pathLength for each unvisited vertex adjacent
// to the vertex at the current index.
for (int i = 0; i < numVertices; i++)
{
if (adjMatrix[index][i] && !visited[i])
{
pathLength[i] = pathLength[index] + 1;
predecessor[i] = index;
traversalQueue.offer(new Integer(i));
visited[i] = true;
}
}
}
if (index != targetIndex) // no path must have been found
return resultList.iterator();
Stack<Integer> stack = new Stack<Integer>();
index = targetIndex;
stack.push(new Integer(index));
do
{
index = predecessor[index];
stack.push(new Integer(index));
} while (index != startIndex);
while (!stack.isEmpty())
resultList.add(((Integer)stack.pop()));
return new GraphIndexIterator(resultList.iterator());
}
/**
* Returns the weight of the least weight path in the network.
* Returns positive infinity if no path is found.
*
* @param startIndex the starting index
* @param targetIndex the target index
* @return the integer weight of the least weight path
* in the network
*/
public int shortestPathLength(int startIndex, int targetIndex)
{
int result = 0;
if (!indexIsValid(startIndex) || !indexIsValid(targetIndex))
return 0;
int index1, index2;
Iterator<Integer> it = iteratorShortestPathIndices(startIndex,
targetIndex);
if (it.hasNext())
index1 = ((Integer)it.next()).intValue();
else
return 0;
while (it.hasNext())
{
result++;
it.next();
}
return result;
}
/**
* Returns the weight of the least weight path in the network.
* Returns positive infinity if no path is found.
*
* @param startVertex the starting vertex
* @param targetVertex the target vertex
* @return the integer weight of teh least weight path
* in the network
*/
public int shortestPathLength(T startVertex, T targetVertex)
{
return shortestPathLength(getIndex(startVertex), getIndex(targetVertex));
}
/**
* Returns a minimum spanning tree of the graph.
*
* @return a minimum spanning tree of the graph
*/
public Graph getMST()
{
int x, y;
int[] edge = new int[2];
Stack<int[]> vertexStack = new Stack<int[]>();
Graph<T> resultGraph = new Graph<T>();
if (isEmpty() || !isConnected())
return resultGraph;
resultGraph.adjMatrix = new boolean[numVertices][numVertices];
for (int i = 0; i < numVertices; i++)
for (int j = 0; j < numVertices; j++)
resultGraph.adjMatrix[i][j] = false;
resultGraph.vertices = (T[])(new Object[numVertices]);
boolean[] visited = new boolean[numVertices];
for (int i = 0; i < numVertices; i++)
visited[i] = false;
edge[0] = 0;
resultGraph.vertices[0] = this.vertices[0];
resultGraph.numVertices++;
visited[0] = true;
// Add all edges that are adjacent to vertex 0 to the stack.
for (int i = 0; i < numVertices; i++)
{
if (!visited[i] && this.adjMatrix[0][i])
{
edge[1] = i;
vertexStack.push(edge.clone());
visited[i] = true;
}
}
while ((resultGraph.size() < this.size()) && !vertexStack.isEmpty())
{
// Pop an edge off the stack and add it to the resultGraph.
edge = vertexStack.pop();
x = edge[0];
y = edge[1];
resultGraph.vertices[y] = this.vertices[y];
resultGraph.numVertices++;
resultGraph.adjMatrix[x][y] = true;
resultGraph.adjMatrix[y][x] = true;
visited[y] = true;
// Add all unvisited edges that are adjacent to vertex y
// to the stack.
for (int i = 0; i < numVertices; i++)
{
if (!visited[i] && this.adjMatrix[i][y])
{
edge[0] = y;
edge[1] = i;
vertexStack.push(edge.clone());
visited[i] = true;
}
}
}
return resultGraph;
}
@Override
public boolean isEmpty() {
return (numVertices == 0);
}
@Override
public boolean isConnected() {
boolean flag = true;
for(int i=0;i<vertices.length;i++){
int temp=0;
temp=getSizeOfIterator(this.iteratorBFS(vertices[i]));
if(temp!=vertices.length){
flag = false;
break;
}
}
return flag;
}
/**
* 获取迭代器中顶点的数目
* @param it
* @return
*/
private int getSizeOfIterator(Iterator it) {
int size = 0;
while(it.hasNext()){
size++;
it.next();
}
return 0;
}
@Override
public int size() {
return numVertices;
}
@Override
public String toString() {
if (numVertices == 0)
return "Graph is empty";
String result = new String("");
result += "Adjacency Matrix
";
result += "----------------
";
result += "index ";
for (int i = 0; i < numVertices; i++) {
result += "" + i;
if (i < 10)
result += " ";
}
result += "
";
for (int i = 0; i < numVertices; i++) {
result += "" + i + " ";
for (int j = 0; j < numVertices; j++) {
if (adjMatrix[i][j])
result += "1 ";
else
result += "0 ";
}
result += "
";
}
result += "
Vertex Values";
result += "
-------------
";
result += "index value
";
for (int i = 0; i < numVertices; i++) {
result += "" + i + " ";
result += vertices[i].toString() + "
";
}
result += "
";
return result;
}
/**--------------------------------下面是自定义的内部迭代器的实现方式-----------------------------------***/
/**
* Inner class to represent an iterator over the elements of this graph
*/
protected class GraphIterator implements Iterator<T>
{
private int expectedModCount;
private Iterator<T> iter;
/**
* Sets up this iterator using the specified iterator.
*
* @param iter the list iterator created by a graph traversal
*/
public GraphIterator(Iterator<T> iter)
{
this.iter = iter;
expectedModCount = modCount;
}
/**
* Returns true if this iterator has at least one more element
* to deliver in the iteration.
*
* @return true if this iterator has at least one more element to deliver
* in the iteration
* @throws ConcurrentModificationException if the collection has changed
* while the iterator is in use
*/
public boolean hasNext() throws ConcurrentModificationException
{
if (!(modCount == expectedModCount))
throw new ConcurrentModificationException();
return (iter.hasNext());
}
/**
* Returns the next element in the iteration. If there are no
* more elements in this iteration, a NoSuchElementException is
* thrown.
*
* @return the next element in the iteration
* @throws NoSuchElementException if the iterator is empty
*/
public T next() throws NoSuchElementException
{
if (hasNext())
return (iter.next());
else
throw new NoSuchElementException();
}
/**
* The remove operation is not supported.
*
* @throws UnsupportedOperationException if the remove operation is called
*/
public void remove()
{
throw new UnsupportedOperationException();
}
}
/**
* Inner class to represent an iterator over the indexes of this graph
*/
protected class GraphIndexIterator implements Iterator<Integer>
{
private int expectedModCount;
private Iterator<Integer> iter;
/**
* Sets up this iterator using the specified iterator.
*
* @param iter the list iterator created by a graph traversal
*/
public GraphIndexIterator(Iterator<Integer> iter)
{
this.iter = iter;
expectedModCount = modCount;
}
/**
* Returns true if this iterator has at least one more element
* to deliver in the iteration.
*
* @return true if this iterator has at least one more element to deliver
* in the iteration
* @throws ConcurrentModificationException if the collection has changed
* while the iterator is in use
*/
public boolean hasNext() throws ConcurrentModificationException
{
if (!(modCount == expectedModCount))
throw new ConcurrentModificationException();
return (iter.hasNext());
}
/**
* Returns the next element in the iteration. If there are no
* more elements in this iteration, a NoSuchElementException is
* thrown.
*
* @return the next element in the iteration
* @throws NoSuchElementException if the iterator is empty
*/
public Integer next() throws NoSuchElementException
{
if (hasNext())
return (iter.next());
else
throw new NoSuchElementException();
}
/**
* The remove operation is not supported.
*
* @throws UnsupportedOperationException if the remove operation is called
*/
public void remove()
{
throw new UnsupportedOperationException();
}
}
}