学习来源：日撸 Java 三百行（31-40天，图））_闵帆的博客-CSDN博客
32. 图的连通性检测
1.适用于有向图. 反正无向图是有向图的特殊形式.
2.0 次方的时候是单位矩阵.
3.为每一个方法写一个独立的测试方法. 测试代码有时比正常使用的代码更多.
4.第一个测试用例是无向图, 第二个是有向图. 可以看到, 后者从节点 1 不能到达节点 0.
5.Matrix 基础代码准备好之后, 其它的算法真的很方便. 后面会进一步体会到其威力.
package datastructure.graph;
import matrix.IntMatrix;
/**
* Directed graph. Note that undirected graphs are a special case of directed
* graphs.
*
* @author Rui Chen 1369097405@qq.com.
*/
public class Graph {
/**
* The connectivity matrix.
*/
IntMatrix connectivityMatrix;
/**
*********************
* The first constructor.
*
* @param paraNumNodes
* The number of nodes in the graph.
*********************
*/
public Graph(int paraNumNodes) {
connectivityMatrix = new IntMatrix(paraNumNodes, paraNumNodes);
}// Of the first constructor
/**
*********************
* The second constructor.
*
* @param paraMatrix
* The data matrix.
*********************
*/
public Graph(int[][] paraMatrix) {
connectivityMatrix = new IntMatrix(paraMatrix);
}// Of the second constructor
/**
*********************
* Overrides the method claimed in Object, the superclass of any class.
*********************
*/
public String toString() {
String resultString = "This is the connectivity matrix of the graph.\r\n"
+ connectivityMatrix;
return resultString;
}// Of toString
/**
*********************
* Get the connectivity of the graph.
*
* @throws Exception
* for internal error.
*********************
*/
public boolean getConnectivity() throws Exception {
// Step 1. Initialize accumulated matrix: M_a = I.
IntMatrix tempConnectivityMatrix = IntMatrix
.getIdentityMatrix(connectivityMatrix.getData().length);
// Step 2. Initialize M^1.
IntMatrix tempMultipliedMatrix = new IntMatrix(connectivityMatrix);
// Step 3. Determine the actual connectivity.
for (int i = 0; i < connectivityMatrix.getData().length - 1; i++) {
// M_a = M_a + M^k
tempConnectivityMatrix.add(tempMultipliedMatrix);
// M^k
tempMultipliedMatrix = IntMatrix.multiply(tempMultipliedMatrix, connectivityMatrix);
} // Of for i
// Step 4. Check the connectivity.
System.out.println("The connectivity matrix is: " + tempConnectivityMatrix);
int[][] tempData = tempConnectivityMatrix.getData();
for (int i = 0; i < tempData.length; i++) {
for (int j = 0; j < tempData.length; j++) {
if (tempData[i][j] == 0) {
System.out.println("Node " + i + " cannot reach " + j);
return false;
} // Of if
} // Of for j
} // Of for i
return true;
}// Of getConnectivity
/**
*********************
* Unit test for getConnectivity.
*********************
*/
public static void getConnectivityTest() {
// Test an undirected graph.
int[][] tempMatrix = { { 0, 1, 0 }, { 1, 0, 1 }, { 0, 1, 0 } };
Graph tempGraph2 = new Graph(tempMatrix);
System.out.println(tempGraph2);
boolean tempConnected = false;
try {
tempConnected = tempGraph2.getConnectivity();
} catch (Exception ee) {
System.out.println(ee);
} // Of try.
System.out.println("Is the graph connected? " + tempConnected);
// Test a directed graph.
// Remove one arc to form a directed graph.
tempGraph2.connectivityMatrix.setValue(1, 0, 0);
tempConnected = false;
try {
tempConnected = tempGraph2.getConnectivity();
} catch (Exception ee) {
System.out.println(ee);
} // Of try.
System.out.println("Is the graph connected? " + tempConnected);
}// Of getConnectivityTest
/**
*********************
* The entrance of the program.
*
* @param args
* Not used now.
*********************
*/
public static void main(String args[]) {
System.out.println("Hello!");
Graph tempGraph = new Graph(3);
System.out.println(tempGraph);
// Unit test.
getConnectivityTest();
}// Of main
}// Of class Graph
运行截图：
33.图的广度优先遍历
与树的广度优先遍历类似. 温故而知新. 应该看懂了自己写一遍.
为每个核心方法写一个测试方法吧. 这叫单元测试
package datastructure.graph;
import matrix.IntMatrix;
import datastructure.queue.*;
/**
* Directed graph. Note that undirected graphs are a special case of directed
* graphs.
*
* @author Rui Chen 1369097405@qq.com.
*/
public class Graph {
/**
* The connectivity matrix.
*/
IntMatrix connectivityMatrix;
/**
*********************
* The first constructor.
*
* @param paraNumNodes
* The number of nodes in the graph.
*********************
*/
public Graph(int paraNumNodes) {
connectivityMatrix = new IntMatrix(paraNumNodes, paraNumNodes);
}// Of the first constructor
/**
*********************
* The second constructor.
*
* @param paraMatrix
* The data matrix.
*********************
*/
public Graph(int[][] paraMatrix) {
connectivityMatrix = new IntMatrix(paraMatrix);
}// Of the second constructor
/**
*********************
* Overrides the method claimed in Object, the superclass of any class.
*********************
*/
public String toString() {
String resultString = "This is the connectivity matrix of the graph.\r\n"
+ connectivityMatrix;
return resultString;
}// Of toString
/**
*********************
* Get the connectivity of the graph.
*
* @throws Exception
* for internal error.
*********************
*/
public boolean getConnectivity() throws Exception {
// Step 1. Initialize accumulated matrix: M_a = I.
IntMatrix tempConnectivityMatrix = IntMatrix
.getIdentityMatrix(connectivityMatrix.getData().length);
// Step 2. Initialize M^1.
IntMatrix tempMultipliedMatrix = new IntMatrix(connectivityMatrix);
// Step 3. Determine the actual connectivity.
for (int i = 0; i < connectivityMatrix.getData().length - 1; i++) {
// M_a = M_a + M^k
tempConnectivityMatrix.add(tempMultipliedMatrix);
// M^k
tempMultipliedMatrix = IntMatrix.multiply(tempMultipliedMatrix, connectivityMatrix);
} // Of for i
// Step 4. Check the connectivity.
System.out.println("The connectivity matrix is: " + tempConnectivityMatrix);
int[][] tempData = tempConnectivityMatrix.getData();
for (int i = 0; i < tempData.length; i++) {
for (int j = 0; j < tempData.length; j++) {
if (tempData[i][j] == 0) {
System.out.println("Node " + i + " cannot reach " + j);
return false;
} // Of if
} // Of for j
} // Of for i
return true;
}// Of getConnectivity
/**
*********************
* Unit test for getConnectivity.
*********************
*/
public static void getConnectivityTest() {
// Test an undirected graph.
int[][] tempMatrix = { { 0, 1, 0 }, { 1, 0, 1 }, { 0, 1, 0 } };
Graph tempGraph2 = new Graph(tempMatrix);
System.out.println(tempGraph2);
boolean tempConnected = false;
try {
tempConnected = tempGraph2.getConnectivity();
} catch (Exception ee) {
System.out.println(ee);
} // Of try.
System.out.println("Is the graph connected? " + tempConnected);
// Test a directed graph.
// Remove one arc to form a directed graph.
tempGraph2.connectivityMatrix.setValue(1, 0, 0);
tempConnected = false;
try {
tempConnected = tempGraph2.getConnectivity();
} catch (Exception ee) {
System.out.println(ee);
} // Of try.
System.out.println("Is the graph connected? " + tempConnected);
}// Of getConnectivityTest
/**
*********************
* Breadth first traversal.
*
* @param paraStartIndex The start index.
* @return The sequence of the visit.
*********************
*/
public String breadthFirstTraversal(int paraStartIndex) {
CircleObjectQueue tempQueue = new CircleObjectQueue();
String resultString = "";
int tempNumNodes = connectivityMatrix.getRows();
boolean[] tempVisitedArray = new boolean[tempNumNodes];
//Initialize the queue.
//Visit before enqueue.
tempVisitedArray[paraStartIndex] = true;
resultString += paraStartIndex;
tempQueue.enqueue(new Integer(paraStartIndex));
//Now visit the rest of the graph.
int tempIndex;
Integer tempInteger = (Integer)tempQueue.dequeue();
while (tempInteger != null) {
tempIndex = tempInteger.intValue();
//Enqueue all its unvisited neighbors.
for (int i = 0; i < tempNumNodes; i ++) {
if (tempVisitedArray[i]) {
continue; //Already visited.
}//Of if
if (connectivityMatrix.getData()[tempIndex][i] == 0) {
continue; //Not directly connected.
}//Of if
//Visit before enqueue.
tempVisitedArray[i] = true;
resultString += i;
tempQueue.enqueue(new Integer(i));
}//Of for i
//Take out one from the head.
tempInteger = (Integer)tempQueue.dequeue();
}//Of while
return resultString;
}//Of breadthFirstTraversal
/**
*********************
* Unit test for breadthFirstTraversal.
*********************
*/
public static void breadthFirstTraversalTest() {
// Test an undirected graph.
int[][] tempMatrix = { { 0, 1, 1, 0 }, { 1, 0, 0, 1 }, { 1, 0, 0, 1}, { 0, 1, 1, 0} };
Graph tempGraph = new Graph(tempMatrix);
System.out.println(tempGraph);
String tempSequence = "";
try {
tempSequence = tempGraph.breadthFirstTraversal(2);
} catch (Exception ee) {
System.out.println(ee);
} // Of try.
System.out.println("The breadth first order of visit: " + tempSequence);
}//Of breadthFirstTraversalTest
/**
*********************
* The entrance of the program.
*
* @param args
* Not used now.
*********************
*/
public static void main(String args[]) {
System.out.println("Hello!");
Graph tempGraph = new Graph(3);
System.out.println(tempGraph);
// Unit test.
getConnectivityTest();
breadthFirstTraversalTest();
}// Of main
}// Of class Graph
运行截图：
34.图的深度优先遍历
与二叉树的深度优先遍历类似. 但要难一点.
又见 while (true), 循环的退出条件是栈为空.
需要在前面 import ObjectStack.
/**
*********************
* Depth first traversal.
*
* @param paraStartIndex The start index.
* @return The sequence of the visit.
*********************
*/
public String depthFirstTraversal(int paraStartIndex) {
ObjectStack tempStack = new ObjectStack();
String resultString = "";
int tempNumNodes = connectivityMatrix.getRows();
boolean[] tempVisitedArray = new boolean[tempNumNodes];
//Initialize the stack.
//Visit before push.
tempVisitedArray[paraStartIndex] = true;
resultString += paraStartIndex;
tempStack.push(new Integer(paraStartIndex));
System.out.println("Push " + paraStartIndex);
System.out.println("Visited " + resultString);
//Now visit the rest of the graph.
int tempIndex = paraStartIndex;
int tempNext;
Integer tempInteger;
while (true){
//Find an unvisited neighbor.
tempNext = -1;
for (int i = 0; i < tempNumNodes; i ++) {
if (tempVisitedArray[i]) {
continue; //Already visited.
}//Of if
if (connectivityMatrix.getData()[tempIndex][i] == 0) {
continue; //Not directly connected.
}//Of if
//Visit this one.
tempVisitedArray[i] = true;
resultString += i;
tempStack.push(new Integer(i));
System.out.println("Push " + i);
tempNext = i;
//One is enough.
break;
}//Of for i
if (tempNext == -1) {
tempInteger = (Integer) tempStack.pop();
System.out.println("Pop " + tempInteger);
if (tempStack.isEmpty()) {
// No unvisited neighbor. Backtracking to the last one
// stored in the stack.
// Attention: This is the terminate condition!
break;
} else {
// Back to the previous node, however do not remove it.
tempInteger = (Integer) tempStack.pop();
tempIndex = tempInteger.intValue();
tempStack.push(tempInteger);
} // Of if
} else {
tempIndex = tempNext;
} // Of if
}//Of while
return resultString;
}//Of depthFirstTraversal
/**
*********************
* Unit test for depthFirstTraversal.
*********************
*/
public static void depthFirstTraversalTest() {
// Test an undirected graph.
int[][] tempMatrix = { { 0, 1, 1, 0 }, { 1, 0, 0, 1 }, { 1, 0, 0, 0}, { 0, 1, 0, 0} };
Graph tempGraph = new Graph(tempMatrix);
System.out.println(tempGraph);
String tempSequence = "";
try {
tempSequence = tempGraph.depthFirstTraversal(0);
} catch (Exception ee) {
System.out.println(ee);
} // Of try.
System.out.println("The depth first order of visit: " + tempSequence);
}//Of depthFirstTraversalTest
/**
*********************
* The entrance of the program.
*
* @param args
* Not used now.
*********************
*/
public static void main(String args[]) {
System.out.println("Hello!");
Graph tempGraph = new Graph(3);
System.out.println(tempGraph);
// Unit test.
getConnectivityTest();
breadthFirstTraversalTest();
depthFirstTraversalTest();
}// Of main
运行截图：
35.图的 m 着色问题
经典的回溯算法. 万能的暴力解题法, 一定要掌握啊!
调拭时注意 +1, -1 之类的下标控制.
单独写一个冲突检测方法.
由于它仅使用了图的连接性, 所以放在这个部分.
public void coloring(int paraNumColors) {
// Step 1. Initialize.
int tempNumNodes = connectivityMatrix.getRows();
int[] tempColorScheme = new int[tempNumNodes];
Arrays.fill(tempColorScheme, -1);
coloring(paraNumColors, 0, tempColorScheme);
}// Of coloring
/**
*********************
* Coloring. Output all possible schemes.
*
* @param paraNumColors The number of colors.
* @param paraCurrentNumNodes The number of nodes that have been colored.
* @param paraCurrentColoring The array recording the coloring scheme.
*********************
*/
public void coloring(int paraNumColors, int paraCurrentNumNodes, int[] paraCurrentColoring) {
// Step 1. Initialize.
int tempNumNodes = connectivityMatrix.getRows();
System.out.println("coloring: paraNumColors = " + paraNumColors + ", paraCurrentNumNodes = "
+ paraCurrentNumNodes + ", paraCurrentColoring" + Arrays.toString(paraCurrentColoring));
// A complete scheme.
if (paraCurrentNumNodes >= tempNumNodes) {
System.out.println("Find one:" + Arrays.toString(paraCurrentColoring));
return;
} // Of if
// Try all possible colors.
for (int i = 0; i < paraNumColors; i++) {
paraCurrentColoring[paraCurrentNumNodes] = i;
if (!colorConflict(paraCurrentNumNodes + 1, paraCurrentColoring)) {
coloring(paraNumColors, paraCurrentNumNodes + 1, paraCurrentColoring);
} // Of if
} // Of for i
}// Of coloring
/**
*********************
* Coloring conflict or not. Only compare the current last node with previous
* ones.
*
* @param paraCurrentNumNodes The current number of nodes.
* @param paraColoring The current coloring scheme.
* @return Conflict or not.
*********************
*/
public boolean colorConflict(int paraCurrentNumNodes, int[] paraColoring) {
for (int i = 0; i < paraCurrentNumNodes - 1; i++) {
// No direct connection.
if (connectivityMatrix.getValue(paraCurrentNumNodes - 1, i) == 0) {
continue;
} // Of if
if (paraColoring[paraCurrentNumNodes - 1] == paraColoring[i]) {
return true;
} // Of if
} // Of for i
return false;
}// Of colorConflict
/**
*********************
* Coloring test.
*********************
*/
public static void coloringTest() {
int[][] tempMatrix = { { 0, 1, 1, 0 }, { 1, 0, 0, 1 }, { 1, 0, 0, 0 }, { 0, 1, 0, 0 } };
Graph tempGraph = new Graph(tempMatrix);
//tempGraph.coloring(2);
tempGraph.coloring(3);
}// Of coloringTest
/**
*********************
* The entrance of the program.
*
* @param args Not used now.
*********************
*/
public static void main(String args[]) {
System.out.println("Hello!");
Graph tempGraph = new Graph(3);
System.out.println(tempGraph);
// Unit test.
getConnectivityTest();
breadthFirstTraversalTest();
depthFirstTraversalTest();
coloringTest();
}// Of main
运行截图：
