给定一个mxn正整数矩阵。任务是找到从矩阵左上角到矩阵右下角的路径数，使得路径中的每个整数都是素数。
另外，打印所有路径中字典序最大的路径。单元格 (a, b) 在字典序上比单元格 (c, d) 大，要么 a > b，要么如果 a == b 则 b > d。从单元格 (x, y)，您可以移动 (x + 1, y), (x, y + 1), (x + 1, y + 1)。
注意：假设左上角的单元格总是有一个质数。
例子：
Input: n = 3, m = 3
m[][] = { { 2, 3, 7 },
{ 5, 4, 2 },
{ 3, 7, 11 } }
Output:
Number of paths: 4
Lexicographical largest path: (1, 1) -> (2, 1) -> (3, 2) -> (3, 3)
There are four ways to reach (3, 3) from (1, 1).
Path 1: (1, 1) (1, 2) (1, 3) (2, 3) (3, 3)
Path 2: (1, 1) (1, 2) (2, 3) (3, 3)
Path 3: (1, 1) (2, 1) (3, 1) (3, 2) (3, 3)
Path 4: (1, 1) (2, 1) (3, 2) (3, 3)
Lexicographical Order -> 4 > 3 > 2 > 1
方法：思路是使用动态规划来解决问题。首先，观察，矩阵中的非质数可以被视为障碍，质数可以被视为可以在路径中使用的单元格。因此，我们可以使用筛子来识别障碍物，并将给定的矩阵转换为二进制矩阵，其中 0 表示障碍物，1 表示有效路径。
因此，我们将定义一个二维矩阵，比如 dp[][]，其中 d[i][j] 表示从单元格 (1, 1) 到单元格 (i, j) 的路径数。此外，我们可以将 dp[i][j] 定义为，
dp[i][j] = dp[i-1][j] + dp[i][j-1] + dp[i-1][j-1]即从左单元格、右单元格和左上角对角线(允许移动)的路径总和。
为了找到字典序最大的路径，我们可以使用 DFS(深度优先搜索)。将每个单元格视为一个节点，该节点具有三个外向边，一条指向右邻单元格、下邻单元格和左下角单元格对角线。现在，我们将以某种方式使用深度优先搜索进行旅行，以便我们获得字典序最大的路径。因此，为了获得字典序最大的路径，我们首先尝试从单元格 (x, y) 前往单元格 (x + 1, y + 1)(如果该单元格没有可能的路径)，然后尝试前往单元格 (x + 1, y) 最后到 (x, y + 1)。
下面是这个方法的实现：
C++
// C++ implementation of above approach
#include
using namespace std;
#define MAX 105
void sieve(int prime[])
{
for (int i = 2; i * i <= MAX; i++) {
if (prime[i] == 0) {
for (int j = i * i; j <= MAX; j += i)
prime[j] = 1;
}
}
}
// Depth First Search
void dfs(int i, int j, int k, int* q, int n, int m,
int mappedMatrix[][MAX], int mark[][MAX],
pair ans[])
{
// Return if cell contain non prime number or obstacle,
// or going out of matrix or already visited the cell
// or already found the lexicographical larget path
if (mappedMatrix[i][j] == 0 || i > n
|| j > m || mark[i][j] || (*q))
return;
// marking cell is already visited
mark[i][j] = 1;
// storing the lexicographical largest path index
ans[k] = make_pair(i, j);
// if reached the end of the matrix
if (i == n && j == m) {
// updating the final number of
// steps in lexicographical largest path
(*q) = k;
return;
}
// moving diagonal (trying lexicographical largest path)
dfs(i + 1, j + 1, k + 1, q, n, m, mappedMatrix, mark, ans);
// moving cell right to current cell
dfs(i + 1, j, k + 1, q, n, m, mappedMatrix, mark, ans);
// moving cell down to current cell.
dfs(i, j + 1, k + 1, q, n, m, mappedMatrix, mark, ans);
}
// Print lexicographical largest prime path
void lexicographicalPath(int n, int m, int mappedMatrix[][MAX])
{
// to count the number of step in
// lexicographical largest prime path
int q = 0;
// to store the lexicographical
// largest prime path index
pair ans[MAX];
// to mark if the cell is already traversed or not
int mark[MAX][MAX];
// traversing by DFS
dfs(1, 1, 1, &q, n, m, mappedMatrix, mark, ans);
// printing the lexicographical largest prime path
for (int i = 1; i <= q; i++)
cout << ans[i].first << " " << ans[i].second << "\n";
}
// Return the number of prime path in ther matrix.
void countPrimePath(int mappedMatrix[][MAX], int n, int m)
{
int dp[MAX][MAX] = { 0 };
dp[1][1] = 1;
// for each cell
for (int i = 1; i <= n; i++) {
for (int j = 1; j <= m; j++) {
// If on the top row or leftmost column,
// there is no path there.
if (i == 1 && j == 1)
continue;
dp[i][j] = (dp[i - 1][j] + dp[i][j - 1]
+ dp[i - 1][j - 1]);
// If non prime number
if (mappedMatrix[i][j] == 0)
dp[i][j] = 0;
}
}
cout << dp[n][m] << "\n";
}
// Finding the matrix mapping by considering
// non prime number as obstacle and prime number be valid path.
void preprocessMatrix(int mappedMatrix[][MAX],
int a[][MAX], int n, int m)
{
int prime[MAX];
// Sieve
sieve(prime);
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) {
// If prime
if (prime[a[i][j]] == 0)
mappedMatrix[i + 1][j + 1] = 1;
// if non prime
else
mappedMatrix[i + 1][j + 1] = 0;
}
}
}
// Driver code
int main()
{
int n = 3;
int m = 3;
int a[MAX][MAX] = { { 2, 3, 7 },
{ 5, 4, 2 },
{ 3, 7, 11 } };
int mappedMatrix[MAX][MAX] = { 0 };
preprocessMatrix(mappedMatrix, a, n, m);
countPrimePath(mappedMatrix, n, m);
lexicographicalPath(n, m, mappedMatrix);
return 0;
} Python3
# Python3 implementation of above approach
MAX = 105
def sieve():
i = 2
while(i * i < MAX):
if (prime[i] == 0):
for j in range(i * i,
MAX, i):
prime[j] = 1;
i += 1
# Depth First Search
def dfs(i, j, k,
q, n, m):
# Return if cell contain non
# prime number or obstacle,
# or going out of matrix or
# already visited the cell
# or already found the
# lexicographical larget path
if (mappedMatrix[i][j] == 0 or
i > n or j > m or mark[i][j] or
q != 0):
return q;
# marking cell is already
# visited
mark[i][j] = 1;
# storing the lexicographical
# largest path index
ans[k] = [i, j]
# if reached the end of
# the matrix
if (i == n and j == m):
# updating the final number
# of steps in lexicographical
# largest path
q = k;
return q;
# moving diagonal (trying lexicographical
# largest path)
q = dfs(i + 1, j + 1, k + 1, q, n, m);
# moving cell right to current cell
q = dfs(i + 1, j, k + 1, q, n, m);
# moving cell down to current cell.
q = dfs(i, j + 1, k + 1, q, n, m);
return q
# Print lexicographical largest
# prime path
def lexicographicalPath(n, m):
# To count the number of step
# in lexicographical largest
# prime path
q = 0;
global ans, mark
# To store the lexicographical
# largest prime path index
ans = [[0, 0] for i in range(MAX)]
# To mark if the cell is already
# traversed or not
mark = [[0 for j in range(MAX)]
for i in range(MAX)]
# traversing by DFS
q = dfs(1, 1, 1, q, n, m);
# printing the lexicographical
# largest prime path
for i in range(1, q + 1):
print(str(ans[i][0]) + ' ' +
str(ans[i][1]))
# Return the number of prime
# path in ther matrix.
def countPrimePath(n, m):
global dp
dp = [[0 for j in range(MAX)]
for i in range(MAX)]
dp[1][1] = 1;
# for each cell
for i in range(1, n + 1):
for j in range(1, m + 1):
# If on the top row or
# leftmost column, there
# is no path there.
if (i == 1 and j == 1):
continue;
dp[i][j] = (dp[i - 1][j] +
dp[i][j - 1] +
dp[i - 1][j - 1]);
# If non prime number
if (mappedMatrix[i][j] == 0):
dp[i][j] = 0;
print(dp[n][m])
# Finding the matrix mapping by
# considering non prime number
# as obstacle and prime number
# be valid path.
def preprocessMatrix(a, n, m):
global prime
prime = [0 for i in range(MAX)]
# Sieve
sieve();
for i in range(n):
for j in range(m):
# If prime
if (prime[a[i][j]] == 0):
mappedMatrix[i + 1][j + 1] = 1;
# if non prime
else:
mappedMatrix[i + 1][j + 1] = 0;
# Driver code
if __name__ == "__main__":
n = 3;
m = 3;
a = [[ 2, 3, 7 ],
[ 5, 4, 2 ],
[ 3, 7, 11 ]];
mappedMatrix = [[0 for j in range(MAX)]
for i in range(MAX)]
preprocessMatrix(a, n, m);
countPrimePath(n, m);
lexicographicalPath(n, m);
# This code is contributed by Rutvik_56C#
// C# implementation of above approach
using System;
using System.Collections;
using System.Collections.Generic;
class GFG{
static int MAX = 105;
static void sieve(int []prime)
{
for(int i = 2; i * i < MAX; i++)
{
if (prime[i] == 0)
{
for (int j = i * i; j < MAX; j += i)
prime[j] = 1;
}
}
}
class pair
{
public int first,second;
public pair(int first, int second)
{
this.first = first;
this.second = second;
}
}
// Depth First Search
static void dfs(int i, int j, int k,
ref int q, int n, int m,
int [,]mappedMatrix,
int [,]mark, pair []ans)
{
// Return if cell contain non prime
// number or obstacle, or going out
// of matrix or already visited the
// cell or already found the
// lexicographical larget path
if ((mappedMatrix[i, j] == 0 ? true : false) ||
(i > n ? true : false) ||
(j > m ? true : false) ||
(mark[i, j] != 0 ? true : false) ||
(q != 0 ? true : false))
return;
// Marking cell is already visited
mark[i, j] = 1;
// Storing the lexicographical
// largest path index
ans[k] = new pair(i, j);
// If reached the end of the matrix
if (i == n && j == m)
{
// Updating the final number of
// steps in lexicographical
// largest path
(q) = k;
return;
}
// Moving diagonal (trying
// lexicographical largest path)
dfs(i + 1, j + 1, k + 1, ref q,
n, m, mappedMatrix, mark, ans);
// Moving cell right to current cell
dfs(i + 1, j, k + 1, ref q,
n, m, mappedMatrix, mark, ans);
// Moving cell down to current cell.
dfs(i, j + 1, k + 1, ref q,
n, m, mappedMatrix, mark, ans);
}
// Print lexicographical largest prime path
static void lexicographicalPath(int n, int m,
int [,]mappedMatrix)
{
// To count the number of step in
// lexicographical largest prime path
int q = 0;
// To store the lexicographical
// largest prime path index
pair []ans = new pair[MAX];
// To mark if the cell is already
// traversed or not
int [,]mark = new int[MAX, MAX];
// Traversing by DFS
dfs(1, 1, 1, ref q, n,
m, mappedMatrix, mark, ans);
// Printing the lexicographical
// largest prime path
for(int i = 1; i <= q; i++)
Console.WriteLine(ans[i].first + " " +
ans[i].second);
}
// Return the number of prime
// path in ther matrix.
static void countPrimePath(int [,]mappedMatrix,
int n, int m)
{
int [,]dp = new int[MAX, MAX];
for(int i = 0; i < MAX; i++)
{
for(int j = 0; j < MAX; j++)
{
dp[i, j] = 0;
}
}
dp[1, 1] = 1;
// For each cell
for(int i = 1; i <= n; i++)
{
for(int j = 1; j <= m; j++)
{
// If on the top row or leftmost
// column, there is no path there.
if (i == 1 && j == 1)
continue;
dp[i, j] = (dp[i - 1, j] + dp[i, j - 1] +
dp[i - 1, j - 1]);
// If non prime number
if (mappedMatrix[i, j] == 0)
dp[i, j] = 0;
}
}
Console.WriteLine(dp[n, m]);
}
// Finding the matrix mapping by considering
// non prime number as obstacle and prime
// number be valid path.
static void preprocessMatrix(int [,]mappedMatrix,
int [,]a, int n, int m)
{
int []prime = new int[MAX];
// Sieve
sieve(prime);
for(int i = 0; i < n; i++)
{
for(int j = 0; j < m; j++)
{
// If prime
if (prime[a[i, j]] == 0)
mappedMatrix[i + 1, j + 1] = 1;
// If non prime
else
mappedMatrix[i + 1, j + 1] = 0;
}
}
}
// Driver code
public static void Main(string []args)
{
int n = 3;
int m = 3;
int [,]a = new int[3, 3]{ { 2, 3, 7 },
{ 5, 4, 2 },
{ 3, 7, 11 } };
int [,]mappedMatrix = new int[MAX, MAX];
for(int i = 0; i < MAX; i++)
{
for(int j = 0; j < MAX; j++)
{
mappedMatrix[i, j] = 0;
}
}
preprocessMatrix(mappedMatrix, a, n, m);
countPrimePath(mappedMatrix, n, m);
lexicographicalPath(n, m, mappedMatrix);
}
}
// This code is contributed by pratham76输出：
4
1 1
2 1
3 2
3 3如果您希望与专家一起参加现场课程，请参阅DSA 现场工作专业课程和学生竞争性编程现场课程。