Project Euler 83：Path sum: four ways 路径和：4个方向

Path sum: four ways

NOTE: This problem is a significantly more challenging version of Problem 81.

In the 5 by 5 matrix below, the minimal path sum from the top left to the bottom right, by moving left, right, up, and down, is indicated in bold red and is equal to 2297.

131	673	234	103	18
201	96	342	965	150
630	803	746	422	111
537	699	497	121	956
805	732	524	37	331

Find the minimal path sum, in matrix.txt  (right click and “Save Link/Target As…”), a 31K text file containing a 80 by 80 matrix, from the top left to the bottom right by moving left, right, up, and down.

---

路径和：四个方向

注意：这是第81题的一个极具挑战性的版本。

在如下的5乘5矩阵中，从左上角到右下角任意地向上、向下、向左或向右移动的最小路径和为2297，由标注红色的路径给出。

131	673	234	103	18
201	96	342	965	150
630	803	746	422	111
537	699	497	121	956
805	732	524	37	331

在这个31K的文本文件matrix.txt （右击并选择“目标另存为……”）中包含了一个80乘80的矩阵，求出从左上角到右下角任意地向上、向下、向左或向右移动的最小路径和。

 解题

表示很复杂，这个应该用到图，dijkstra算法可解。

参考解题论坛中的程序，也就是dijkstra算法，只是用Python实现起来，比较简单。

实现思路：

1.当前节点(0,0)开始，在临近节点，寻找最短路径

2.是最短路径的节点位置保存

3,根据2中保存的节点，再找其到临近节点的最短路径

Python

import time 
def readData(filename):
    fl = open(filename)
    data =[]
    for row in fl:
        row = row.split(',')
        line = [int(i) for i in row]
        data.append(line)
    fl.close()
    return data

def next_steps(pos):
    (j,i) = pos
    if i+1<size:
        right = minnum[j,i] + data[j][i+1]
        if right< minnum[j,i+1]:
            minnum[j,i+1] = right 
            next_list.append((j,i+1))
    if j+1< size:
        down = minnum[j,i] + data[j+1][i]
        if down < minnum[j+1,i]:
            minnum[j+1,i] = down
            next_list.append((j+1,i))
    if i-1 > -1:
        left = minnum[j,i] + data[j][i-1]
        if left < minnum[j,i-1]:
            minnum[j,i-1] = left
            next_list.append((j,i-1))
    if j-1 > -1:
        up = minnum[j,i] + data[j-1][i]
        if up < minnum[j-1,i]:
            minnum[j-1,i] = up
            next_list.append((j-1,i))
   
t0 = time.time()         
filename = 'E:/java/projecteuler/src/Level3/p083_matrix.txt'
data = readData(filename)
size = 80
infinity = 10**10
minnum = {}
for i in range(0,size):
    for j in range(0,size):
        minnum[j,i] = infinity 

next_list = []
            
minnum[0,0] = data[0][0]
test = [(0,0)]
while test!=[]:
    next_list = []
    for el in test:
        next_steps(el)
    test = next_list
print minnum[size-1,size-1]
t1 = time.time()
print "running time=",(t1-t0),"s"

# 425185
# running time= 0.112999916077 s
            

Java

package Level3;

import java.io.BufferedReader;
import java.io.FileReader;
import java.io.IOException;
import java.util.ArrayList;


public class PE083{
    
    static int[][] grid;
    static void run() throws IOException{
        String filename = "src/Level3/p083_matrix.txt";
        String lineString = "";
        ArrayList<String> listData = new ArrayList<String>();
        BufferedReader data = new BufferedReader(new FileReader(filename));
        while((lineString = data.readLine())!= null){
            listData.add(lineString);
        }
        // 分配大小空间的 定义的grid 没有定义大小
        assignArray(listData.size());
        // 按照行添加到数组grid中 
        for(int index = 0,row_counter=0;index <=listData.size() - 1;++index,row_counter++){
            populateArray(listData.get(index),row_counter);
        }
        int result = minPath(grid,0,0,80-1,80-1);
        System.out.println(result);
        
    }
    // matrix[a][b] to matrix[c][d] 的最小值
    public static int minPath(int[][] matrix,int a,int b,int c,int d){
        int[][] D = new int[matrix.length][matrix[0].length];
        for(int i=0;i<D.length;i++)
            for(int j=0;j<D[0].length;j++)
                D[i][j] = Integer.MAX_VALUE;
        D[a][b] = matrix[a][b];
        int x=a,y=b;
        while(true){
            // 计算 x y 节点到上下左右四个方向的路径，若小则更新 
            // 下
            if(x < D.length -1)
                if(D[x+1][y] > 0)
                    D[x+1][y] = Math.min(matrix[x+1][y] + D[x][y], D[x+1][y]);
            // 右
            if( y<D[0].length -1)
                if(D[x][y+1] >0)
                    D[x][y+1] = Math.min(matrix[x][y+1] + D[x][y], D[x][y+1]);
            //上
            if(x>0)
                if(D[x-1][y] >0)
                    D[x-1][y] = Math.min(matrix[x-1][y] + D[x][y], D[x-1][y]);
            // 左
            if(y>0)
                if(D[x][y-1]>0)
                    D[x][y-1] = Math.min(matrix[x][y-1] + D[x][y], D[x][y-1]);
            if(x==c && y==d)
                return D[x][y];
            
            // 访问过的节点取其相反数
            D[x][y] =-D[x][y];
            // 选取下一个节点 
            // 在未被访问的节点中，选取路径值最小的
            int min = Integer.MAX_VALUE;
            for(int i=0;i< D.length;i++){
                for(int j=0;j<D[0].length;j++){
                    if(D[i][j]>0 && D[i][j] < min){
                        min = D[i][j];
                        x = i;
                        y = j;
                    }
                }
            }
        }
    }
    public static int Path_min(int[][] A){
        int size = A.length;
        int B[][] = new int[size][size];
        B[0][0] = A[0][0];
        B[0][1] = A[0][0] + A[0][1];
        B[1][0] = A[0][0] + A[1][0];
        for(int i = 1;i<size; i++){
            for(int j = 1;j<size ;j++){
                B[i][j] = A[i][j] + get4min(B[i-1][j],B[i+1][j],
                        B[i][j-1],B[i][j+1]);
            }
        }
        return B[size-1][size-1];
    }
    public static int get4min(int a,int b,int c,int d){
        int min1 = Math.min(a, b);
        int min2 = Math.min(c, d);
        return Math.min(min1, min2);
    }
    // 每行的数据添加到数组中
    public static void populateArray(String str,int row){
        int counter = 0;
        String[] data = str.split(",");
        for(int index = 0;index<=data.length -1;++index){
            grid[row][counter++] = Integer.parseInt(data[index]);
        }
    }
    public static void assignArray(int no_of_row){
        grid = new int[no_of_row][no_of_row];
    }
    
    

    public static void main(String[] args) throws IOException{
        long t0 = System.currentTimeMillis();
        run();
        long t1 = System.currentTimeMillis();
        long t = t1 - t0;
        System.out.println("running time="+t/1000+"s"+t%1000+"ms");
//        425185
//        running time=0s187ms
    }
}