问题是在矩阵中从左上角到右下角找到路径？

Question

我有一个用随机数20x30填充的[ 0, 1, 2 ]矩阵。我需要找到仅由1组成的路径，该路径从左上方开始，在右下方结束。我需要帮助找到1s的路径。另外，如何打印步进的每个数字的坐标？我可以显示步进的数字，但是在显示其坐标时遇到问题。这是我当前的代码

#include <iostream>
#include <vector>
#include <time.h>
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
using namespace std;

const int N = 3;
const int M = 3;

void mtxFeltolt(int (&mat)[N][M]);

void mtxPrint(int (&mat)[N][M]);

void printPaths(int mat[M][N], vector<int> &route, int i, int j)
{
    // if last cell is reached
    if (i == M - 1 && j == N - 1)
    {
        // print the current route

        for (int i: route) {
            cout << i << " - ";
        }

        cout << mat[i][j] << endl;
        return;
    }

    // include current cell in route
    route.push_back(mat[i][j]);


    // move right
    if (j + 1 < N){
        printPaths(mat, route, i, j + 1);
    }
    // move down
    if (i + 1 < M){
        printPaths(mat, route, i + 1, j);
    }
    // move diagonally
    if (i + 1 < M && j + 1 < N){
        printPaths(mat, route, i + 1, j + 1);
    }
    // backtrack
    route.pop_back();
}

// Print all shortest routes in a rectangular grid
void printPaths(int mat[][N])
{
    // vector to store current route
    vector<int> route;

    // start from the first cell (0, 0)
    printPaths(mat, route, 0, 0);
}

// main function
int main()
{
    int mat[N][M];

    srand (time(NULL));

    mtxFeltolt(mat);


    cout << "A matrix: " <<endl;
    mtxPrint(mat);

    cout << endl;
    cout << "---- A megfelelo utak ----" << endl;
    printPaths(mat);

    return 0;
}
void mtxFeltolt(int (&mat)[N][M]){
    for(int i=0; i < N; i++){
        for(int j=0; j < M; j++)
            mat[i][j] = rand() % 3;
    }

}

void mtxPrint(int (&mat)[N][M]){
    for(int i=0; i < N; i++){
        for(int j = 0; j < M; j++){
            cout << mat[i][j] << " ";
        }
        cout << endl;
    }
}

Answer 1

希望您可以遵循此代码。我没有使用std::pair，而是制作了一个简单的结构Coord来包含路径中坐标的行和列。用这种方式更容易阅读。我还提供了一种生成随机数的更好方法。

我使用深度优先搜索来找到路径。它不能保证最短的路径，但是会找到从左上方到右下方的路径。

#include <algorithm>
#include <iostream>
#include <random>
#include <vector>

struct Coord {unsigned long row, col;};

template<typename T>
using Matrix = std::vector<std::vector<T>>;
using Path = std::vector<Coord>;

/**
 * Generate a random number from [low, high]
 *
 * @param low  The lower bound
 * @param high The upper bound
 * @return     A random number on the range [low, high]
 */
int random_int(int low, int high)
{
  static std::random_device rd;

  // static std::mt19937 mt(rd()); // different random numbers each time
  static std::mt19937 mt(30); // seed that generates a matrix with a path

  std::uniform_int_distribution<> dist(low, high);
  return dist(mt);
}

Matrix<int> generateMatrix(const int m, const int n)
{
  Matrix<int> mat;
  for(int row = 0; row < m; ++row)
  {
    mat.push_back({});
    for(int col = 0; col < n; ++col)
    {
      mat[row].push_back(random_int(0,2));
    }
  }
  return mat;
}

void print(const Matrix<int>& mat)
{
  for(const auto & row : mat)
  {
    for(const auto & col : row)
      std::cout << col << " ";
    std::cout << std::endl;
  }

}

Path findPath(const Matrix<int>& mat,
              Matrix<bool>& visited,
              const Coord cur,
              const Coord end) 
{
  // out of range -> no path
  if(cur.row < 0
     || cur.row >= mat.size()
     || cur.col < 0
     || cur.col >= mat[0].size())
  {
    return {};
  }

  // visited current location -> no path
  if(visited[cur.row][cur.col])
  {
    return {};
  }
    visited[cur.row][cur.col] = true;

  // current location is not a 1 -> no path
  if(mat[cur.row][cur.col] != 1)
  {
    return {};
  }

  // if at the end, the path is trivial
  if(cur.row == end.row && cur.col == end.col)
  {
    return {cur};
  }

  Path p {cur};
  std::vector<Path> paths;

  // try to go in each direction
  // right
  paths.push_back(findPath(mat, visited, {cur.row, cur.col+1}, end));
  // left
  paths.push_back(findPath(mat, visited, {cur.row, cur.col-1}, end));
  // up
  paths.push_back(findPath(mat, visited, {cur.row-1, cur.col}, end));
  // down
  paths.push_back(findPath(mat, visited, {cur.row+1, cur.col}, end));
  // up-right
  paths.push_back(findPath(mat, visited, {cur.row-1, cur.col+1}, end));
  // down-right
  paths.push_back(findPath(mat, visited, {cur.row+1, cur.col+1}, end));
  // down-left
  paths.push_back(findPath(mat, visited, {cur.row+1, cur.col-1}, end));
  // up-left
  paths.push_back(findPath(mat, visited, {cur.row-1, cur.col-1}, end));

  Path longest = *std::max_element(paths.begin(), paths.end(),
    [](const auto a, const auto b){
      return a.size() < b.size();
  });

  p.insert(p.end(), longest.begin(), longest.end());


  return p;
}

Path findPath(const Matrix<int>& mat,
              const Coord cur,
              const Coord end) 
{
  Matrix<bool> visited;
  for(int row = 0; row < mat.size(); ++row)
  {
    visited.push_back({});
    for(int col = 0; col < mat[0].size(); ++col)
    {
      visited[row].push_back(false);
    }
  }

  return findPath(mat, visited, cur, end);
}

int main()
{
  auto mat = generateMatrix(5, 5);
  print(mat);
  auto path = findPath(mat, {0, 0}, {mat.size()-1, mat[0].size()-1});

  if(path.size() > 0
     && path.back().row == mat.size()-1 && path.back().col == mat[0].size()-1)
  {
    std::cout << "path length: " << path.size() << std::endl;
    for(const auto c : path)
    {
      std::cout << "(" << c.row << ", " << c.col << ")" << std::endl;
    }
  }
  else
    std::cout << "no path" << std::endl;
}