动态规划求二维网格中从左上角到右下角的最短路径( 每次只能向下、向右、向右下走 ) java 实现-编程知识

dp[i][j] 表示在以点(0,0)作为左上角，点(i,i) 作为右下角的二维网格中 左上角到右下角的最短路径，
动态转移方程为：dp[i][j] = min{ dp[i][j-1],dp[i-1][j],dp[i-1][j-1] }.distance + weight[i][j]

ImageUtils.java：

import java.awt.*;
import java.awt.image.BufferedImage;
import java.io.File;
import java.io.IOException;
import java.util.HashSet;
import java.util.List;
import java.util.Random;
import java.util.Set;
import javax.imageio.ImageIO;public class ImageUtils {/**** @param mat* @param outputPath* @param cellSize 格子的边长*/public static void printMat2Image(int[][] mat, String outputPath,int cellSize,PathVO path) {// 总行数int rowCount = mat.length;// 总列数int colCount = mat[0].length;int fontSize = cellSize / 2;BufferedImage image = new BufferedImage(colCount * cellSize, rowCount * cellSize, BufferedImage.TYPE_INT_RGB);Graphics g = image.getGraphics();Random random = new Random();// 设置背景色为白色g.setColor(Color.WHITE);g.fillRect(0, 0, colCount * cellSize, rowCount * cellSize);// 画网格g.setColor(Color.BLACK);for (int i = 0; i <= rowCount; i++) {g.drawLine(0, i * cellSize, colCount * cellSize, i * cellSize);}for (int i = 0; i <= colCount; i++) {g.drawLine(i * cellSize, 0, i * cellSize, rowCount * cellSize);}Set<String> codes = new HashSet<>();List<PointVO> points = path.getPoints();for( PointVO point:points ){codes.add( point.getRowNum() + "_" + point.getColNum() );}/*   Color color = g.getColor();// 画路径List<PointVO> points = path.getPoints();g.setColor( Color.GREEN );for( PointVO point:points ){int x = point.getRowNum() * cellSize;int y = point.getColNum() * cellSize;g.fillRect(x, y, cellSize, cellSize); // 填充矩形}g.setColor( color );*/// 写入文本g.setFont(new Font("Arial", Font.BOLD, fontSize));Color color_old = g.getColor();Color color_green = Color.GREEN;FontMetrics fontMetrics = g.getFontMetrics();for (int rowNum = 0; rowNum < rowCount; rowNum++) {int[] row = mat[rowNum];for (int colNum = 0; colNum < colCount; colNum++) {int num = row[colNum];String code = rowNum + "_" + colNum;if( codes.contains( code ) ){g.setColor( color_green );}else {g.setColor( color_old );}String text = String.valueOf( num );int x = colNum * cellSize + (cellSize - fontMetrics.stringWidth(text) ) / 2;int y = rowNum * cellSize + (cellSize - fontMetrics.getHeight()) / 2 + fontMetrics.getAscent();g.drawString(text, x, y);}}// 输出图片try {ImageIO.write(image, "png", new File(outputPath));System.out.println("图片已生成：" + outputPath);} catch (IOException e) {e.printStackTrace();}}
}

PathVO.java：

import lombok.Getter;
import lombok.Setter;import java.io.Serializable;
import java.util.List;@Getter
@Setter
public class PathVO implements Serializable {private List<PointVO> points;private Integer distance;}

PointVO.java：

import lombok.Getter;
import lombok.Setter;import java.io.Serializable;@Getter
@Setter
public class PointVO implements Serializable {private Integer rowNum;private Integer colNum;public PointVO(Integer rowNum, Integer colNum) {this.rowNum = rowNum;this.colNum = colNum;}
}

Test.java：

import lombok.Getter;
import lombok.Setter;import java.util.ArrayList;
import java.util.List;
import java.util.Random;@Getter
@Setter
public class Test   {public static void main(String[] args) {test();}private static void test(){/*   int rowCount = 10;int colCount = 10;*/// int[][] mat = initRandomWeightMat( rowCount,colCount );int[][] mat = new int[][]{{ 9, 9, 9, 9, 9, 9 },{ 9, 1, 9, 9, 9, 9 },{ 9, 1, 1, 9, 9, 9 },{ 9, 1, 9, 1, 9, 9 },{ 9, 1, 1, 1, 9, 1 },{ 9, 9, 9, 9, 1, 9 }};// dp[i][j] 表示在以点(0,0)作为左上角，点(i,i) 作为右下角的二维网格中 左上角到右下角的最短路径，// 动态转移方程为：dp[i][j] = min{ dp[i][j-1],dp[i-1][j] }.distance + weight[i][j]int rowCount = mat.length;int colCount = mat[0].length;PathVO[][] dp = new PathVO[rowCount][colCount];PathVO path = new PathVO();List<PointVO> points = new ArrayList<>();points.add( new PointVO( 0,0 ) );path.setPoints( points );path.setDistance( mat[0][0] );dp[0][0] = path;for (int colNum = 1; colNum < colCount; colNum++) {int rowNum = 0;int weight = mat[rowNum][colNum];PathVO path_prev = dp[rowNum][colNum - 1];path = new PathVO();points = new ArrayList<>();points.addAll( path_prev.getPoints() );points.add( new PointVO( rowNum,colNum ) );path.setPoints( points );path.setDistance( path_prev.getDistance() + weight );dp[rowNum][colNum] = path;}for (int rowNum = 1; rowNum < rowCount; rowNum++) {int colNum = 0;int weight = mat[rowNum][colNum];PathVO path_prev = dp[rowNum-1][colNum];path = new PathVO();points = new ArrayList<>();points.addAll( path_prev.getPoints() );points.add( new PointVO( rowNum,colNum ) );path.setPoints( points );path.setDistance( path_prev.getDistance() + weight );dp[rowNum][colNum] = path;}for (int rowNum = 1; rowNum < rowCount; rowNum++) {for (int colNum = 1; colNum < colCount; colNum++) {Integer weight = mat[ rowNum ][colNum];path = new PathVO();points = new ArrayList<>();// 当前点要么从上面的点到达，要么从 左边的点到达PathVO path_prev_up = dp[ rowNum -1 ][colNum];PathVO path_prev_left = dp[ rowNum ][colNum-1];PathVO path_prev_left_up = dp[ rowNum-1 ][colNum-1];PathVO path_prev_min = path_prev_up;if( path_prev_left.getDistance() < path_prev_min.getDistance() ){path_prev_min = path_prev_left;}if( path_prev_left_up.getDistance() < path_prev_min.getDistance() ){path_prev_min = path_prev_left_up;}points.addAll( path_prev_min.getPoints() );points.add( new PointVO( rowNum,colNum ) );path.setDistance( path_prev_min.getDistance() +  weight );path.setPoints( points );dp[ rowNum ][colNum] = path;}}path = dp[ rowCount -1 ][colCount -1];String outputPath = "C:\\E\\xxx.png";ImageUtils.printMat2Image( mat,outputPath,40,path );}private static int[][] initRandomWeightMat(int rowCount, int colCount) {Random random = new Random();// 初始化 二维网格矩阵int[][] mat = new int[rowCount][colCount];for (int rowNum = 0; rowNum < rowCount; rowNum++) {for (int colNum = 0; colNum < colCount; colNum++) {mat[ rowNum ][colNum] = random.nextInt( 2 ) + 1;}}return mat;}
}

测试输出：