Related Articles

# Lexicographically largest prime path from top-left to bottom-right in a matrix

• Difficulty Level : Hard
• Last Updated : 09 Dec, 2020

Given a m x n matrix of positive integers. The task is to find the number of paths from the top left of the matrix to the bottom right of the matrix such that each integer in the path is prime.
Also, print the lexicographical largest path among all the path. A cell (a, b) is lexicographical larger than cell (c, d) either a > b or if a == b then b > d. From cell (x, y), you are allowed to move (x + 1, y), (x, y + 1), (x + 1, y + 1).

Note: It is given that the top-left cell will always have a prime number.

Examples:

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

Approach: The idea is to use Dynamic Programming to solve the problem. First, observe, a non-prime number in the matrix can be treated as an obstacle and a prime number can be treated as a cell that can be used in the path. So, we can use a sieve to identify the obstacle and convert the given matrix into a binary matrix where 0 indicates the obstacle and 1 indicates the valid path.
So, we will define a 2D matrix, say dp[][], where d[i][j] indicate the number of path from cell (1, 1) to cell(i, j). Also, we can define dp[i][j] as,

`dp[i][j] = dp[i-1][j] + dp[i][j-1] + dp[i-1][j-1]`

i.e sum of path from left cell, right cell and upper left diagonal (moves allowed).
To find the lexicographical largest path, we can use DFS (Depth-first search). Consider each cell as a node which is having three outgoing edges, one to the cell adjacent right, cell adjacent down, and cell diagonal to lower left. Now, we will travel using a depth-first search in a manner so that we get the lexicographical largest path. So, to get the lexicographical largest path, from the cell (x, y) we first try to travel to cell (x + 1, y + 1) (if no path possible from that cell) then try travel to cell (x + 1, y) and finally to (x, y + 1).

Below is the implementation of this approach:

## 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<``int``, ``int``> 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<``int``, ``int``> 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;` `    ``// 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_56`

## C#

 `// 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`
Output:
```4
1 1
2 1
3 2
3 3```

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.  To complete your preparation from learning a language to DS Algo and many more,  please refer Complete Interview Preparation Course.

In case you wish to attend live classes with experts, please refer DSA Live Classes for Working Professionals and Competitive Programming Live for Students.

My Personal Notes arrow_drop_up