# Find the longest path in a matrix with given constraints

Given a n*n matrix where all numbers are distinct, find the maximum length path (starting from any cell) such that all cells along the path are in increasing order with a difference of 1.

We can move in 4 directions from a given cell (i, j), i.e., we can move to (i+1, j) or (i, j+1) or (i-1, j) or (i, j-1) with the condition that the adjacent cells have a difference of 1.

Example:

```Input:  mat[][] = {{1, 2, 9}
{5, 3, 8}
{4, 6, 7}}
Output: 4
The longest path is 6-7-8-9.
```

## Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

The idea is simple, we calculate longest path beginning with every cell. Once we have computed longest for all cells, we return maximum of all longest paths. One important observation in this approach is many overlapping subproblems. Therefore this problem can be optimally solved using Dynamic Programming.

Below is Dynamic Programming based implementation that uses a lookup table dp[][] to check if a problem is already solved or not.

 `// C++ program to find the longest path in a matrix ` `// with given constraints ` `#include ` `#define n 3 ` `using` `namespace` `std; ` ` `  `// Returns length of the longest path beginning with mat[i][j]. ` `// This function mainly uses lookup table dp[n][n] ` `int` `findLongestFromACell(``int` `i, ``int` `j, ``int` `mat[n][n], ``int` `dp[n][n]) ` `{ ` `    ``if` `(i < 0 || i >= n || j < 0 || j >= n) ` `        ``return` `0; ` ` `  `    ``// If this subproblem is already solved ` `    ``if` `(dp[i][j] != -1) ` `        ``return` `dp[i][j]; ` ` `  `    ``// To store the path lengths in all the four directions ` `    ``int` `x = INT_MIN, y = INT_MIN, z = INT_MIN, w = INT_MIN; ` ` `  `    ``// Since all numbers are unique and in range from 1 to n*n, ` `    ``// there is atmost one possible direction from any cell ` `    ``if` `(j < n - 1 && ((mat[i][j] + 1) == mat[i][j + 1])) ` `        ``x = 1 + findLongestFromACell(i, j + 1, mat, dp); ` ` `  `    ``if` `(j > 0 && (mat[i][j] + 1 == mat[i][j - 1])) ` `        ``y = 1 + findLongestFromACell(i, j - 1, mat, dp); ` ` `  `    ``if` `(i > 0 && (mat[i][j] + 1 == mat[i - 1][j])) ` `        ``z = 1 + findLongestFromACell(i - 1, j, mat, dp); ` ` `  `    ``if` `(i < n - 1 && (mat[i][j] + 1 == mat[i + 1][j])) ` `        ``w = 1 + findLongestFromACell(i + 1, j, mat, dp); ` ` `  `    ``// If none of the adjacent fours is one greater we will take 1 ` `    ``// otherwise we will pick maximum from all the four directions ` `    ``return` `dp[i][j] = max(x, max(y, max(z, max(w, 1)))); ` `} ` ` `  `// Returns length of the longest path beginning with any cell ` `int` `finLongestOverAll(``int` `mat[n][n]) ` `{ ` `    ``int` `result = 1; ``// Initialize result ` ` `  `    ``// Create a lookup table and fill all entries in it as -1 ` `    ``int` `dp[n][n]; ` `    ``memset``(dp, -1, ``sizeof` `dp); ` ` `  `    ``// Compute longest path beginning from all cells ` `    ``for` `(``int` `i = 0; i < n; i++) { ` `        ``for` `(``int` `j = 0; j < n; j++) { ` `            ``if` `(dp[i][j] == -1) ` `                ``findLongestFromACell(i, j, mat, dp); ` ` `  `            ``// Update result if needed ` `            ``result = max(result, dp[i][j]); ` `        ``} ` `    ``} ` ` `  `    ``return` `result; ` `} ` ` `  `// Driver program ` `int` `main() ` `{ ` `    ``int` `mat[n][n] = { { 1, 2, 9 }, ` `                      ``{ 5, 3, 8 }, ` `                      ``{ 4, 6, 7 } }; ` `    ``cout << ``"Length of the longest path is "` `         ``<< finLongestOverAll(mat); ` `    ``return` `0; ` `} `

 `// Java program to find the longest path in a matrix ` `// with given constraints ` ` `  `class` `GFG { ` `    ``public` `static` `int` `n = ``3``; ` ` `  `    ``// Function that returns length of the longest path ` `    ``// beginning with mat[i][j] ` `    ``// This function mainly uses lookup table dp[n][n] ` `    ``static` `int` `findLongestFromACell(``int` `i, ``int` `j, ``int` `mat[][], ``int` `dp[][]) ` `    ``{ ` `        ``// Base case ` `        ``if` `(i < ``0` `|| i >= n || j < ``0` `|| j >= n) ` `            ``return` `0``; ` ` `  `        ``// If this subproblem is already solved ` `        ``if` `(dp[i][j] != -``1``) ` `            ``return` `dp[i][j]; ` ` `  `        ``// To store the path lengths in all the four directions ` `        ``int` `x = Integer.MIN_VALUE, y = Integer.MIN_VALUE, z = Integer.MIN_VALUE, w = Integer.MIN_VALUE; ` `        ``// Since all numbers are unique and in range from 1 to n*n, ` `        ``// there is atmost one possible direction from any cell ` `        ``if` `(j < n - ``1` `&& ((mat[i][j] + ``1``) == mat[i][j + ``1``])) ` `            ``x = dp[i][j] = ``1` `+ findLongestFromACell(i, j + ``1``, mat, dp); ` ` `  `        ``if` `(j > ``0` `&& (mat[i][j] + ``1` `== mat[i][j - ``1``])) ` `            ``y = dp[i][j] = ``1` `+ findLongestFromACell(i, j - ``1``, mat, dp); ` ` `  `        ``if` `(i > ``0` `&& (mat[i][j] + ``1` `== mat[i - ``1``][j])) ` `            ``z = dp[i][j] = ``1` `+ findLongestFromACell(i - ``1``, j, mat, dp); ` ` `  `        ``if` `(i < n - ``1` `&& (mat[i][j] + ``1` `== mat[i + ``1``][j])) ` `            ``w = dp[i][j] = ``1` `+ findLongestFromACell(i + ``1``, j, mat, dp); ` ` `  `        ``// If none of the adjacent fours is one greater we will take 1 ` `        ``// otherwise we will pick maximum from all the four directions ` `        ``return` `dp[i][j] = Math.max(x, Math.max(y, Math.max(z, Math.max(w, ``1``)))); ` `    ``} ` ` `  `    ``// Function that returns length of the longest path ` `    ``// beginning with any cell ` `    ``static` `int` `finLongestOverAll(``int` `mat[][]) ` `    ``{ ` `        ``// Initialize result ` `        ``int` `result = ``1``; ` ` `  `        ``// Create a lookup table and fill all entries in it as -1 ` `        ``int``[][] dp = ``new` `int``[n][n]; ` `        ``for` `(``int` `i = ``0``; i < n; i++) ` `            ``for` `(``int` `j = ``0``; j < n; j++) ` `                ``dp[i][j] = -``1``; ` ` `  `        ``// Compute longest path beginning from all cells ` `        ``for` `(``int` `i = ``0``; i < n; i++) { ` `            ``for` `(``int` `j = ``0``; j < n; j++) { ` `                ``if` `(dp[i][j] == -``1``) ` `                    ``findLongestFromACell(i, j, mat, dp); ` ` `  `                ``// Update result if needed ` `                ``result = Math.max(result, dp[i][j]); ` `            ``} ` `        ``} ` ` `  `        ``return` `result; ` `    ``} ` ` `  `    ``// driver program ` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``int` `mat[][] = { { ``1``, ``2``, ``9` `}, ` `                        ``{ ``5``, ``3``, ``8` `}, ` `                        ``{ ``4``, ``6``, ``7` `} }; ` `        ``System.out.println(``"Length of the longest path is "` `+ finLongestOverAll(mat)); ` `    ``} ` `} ` ` `  `// Contributed by Pramod Kumar `

 `# Python3 program to find the longest path in a matrix ` `# with given constraints ` ` `  `n ``=` `3` `# Returns length of the longest path beginning with mat[i][j].  ` `# This function mainly uses lookup table dp[n][n]  ` `def` `findLongestFromACell(i, j, mat, dp): ` `    ``# Base case  ` `    ``if` `(i<``0` `or` `i>``=` `n ``or` `j<``0` `or` `j>``=` `n): ` `        ``return` `0` ` `  `    ``# If this subproblem is already solved  ` `    ``if` `(dp[i][j] !``=` `-``1``):  ` `        ``return` `dp[i][j] ` ` `  `    ``# To store the path lengths in all the four directions ` `    ``x, y, z, w ``=` `-``1``, ``-``1``, ``-``1``, ``-``1` ` `  `    ``# Since all numbers are unique and in range from 1 to n * n,  ` `    ``# there is atmost one possible direction from any cell  ` `    ``if` `(j``0` `and` `(mat[i][j] ``+``1` `=``=` `mat[i][j``-``1``])):  ` `        ``y ``=` `1` `+` `findLongestFromACell(i, j``-``1``, mat, dp) ` ` `  `    ``if` `(i>``0` `and` `(mat[i][j] ``+``1` `=``=` `mat[i``-``1``][j])): ` `        ``z ``=` `1` `+` `findLongestFromACell(i``-``1``, j, mat, dp) ` ` `  `    ``if` `(i

 `// C# program to find the longest path ` `// in a matrix with given constraints ` `using` `System; ` ` `  `class` `GFG { ` `    ``public` `static` `int` `n = 3; ` ` `  `    ``// Function that returns length of ` `    ``// the longest path beginning with mat[i][j] ` `    ``// This function mainly uses lookup ` `    ``// table dp[n][n] ` `    ``public` `static` `int` `findLongestFromACell(``int` `i, ``int` `j, ` `                                           ``int``[][] mat, ` `                                           ``int``[][] dp) ` `    ``{ ` `        ``// Base case ` `        ``if` `(i < 0 || i >= n || j < 0 || j >= n) { ` `            ``return` `0; ` `        ``} ` ` `  `        ``// If this subproblem is ` `        ``// already solved ` `        ``if` `(dp[i][j] != -1) { ` `            ``return` `dp[i][j]; ` `        ``} ` ` `  `        ``// To store the path lengths in all the four directions ` `        ``int` `x = ``int``.MinValue, y = ``int``.MinValue, z = ``int``.MinValue, w = ``int``.MinValue; ` ` `  `        ``// Since all numbers are unique and ` `        ``// in range from 1 to n*n, there is ` `        ``// atmost one possible direction ` `        ``// from any cell ` `        ``if` `(j < n - 1 && ((mat[i][j] + 1) == mat[i][j + 1])) { ` `            ``x = dp[i][j] = 1 + findLongestFromACell(i, j + 1, mat, dp); ` `        ``} ` ` `  `        ``if` `(j > 0 && (mat[i][j] + 1 == mat[i][j - 1])) { ` `            ``y = dp[i][j] = 1 + findLongestFromACell(i, j - 1, mat, dp); ` `        ``} ` ` `  `        ``if` `(i > 0 && (mat[i][j] + 1 == mat[i - 1][j])) { ` `            ``z = dp[i][j] = 1 + findLongestFromACell(i - 1, j, mat, dp); ` `        ``} ` ` `  `        ``if` `(i < n - 1 && (mat[i][j] + 1 == mat[i + 1][j])) { ` `            ``w = dp[i][j] = 1 + findLongestFromACell(i + 1, j, mat, dp); ` `        ``} ` ` `  `        ``// If none of the adjacent fours is one greater we will take 1 ` `        ``// otherwise we will pick maximum from all the four directions ` `        ``dp[i][j] = Math.Max(x, Math.Max(y, Math.Max(z, Math.Max(w, 1)))); ` `        ``return` `dp[i][j]; ` `    ``} ` ` `  `    ``// Function that returns length of the ` `    ``// longest path beginning with any cell ` `    ``public` `static` `int` `finLongestOverAll(``int``[][] mat) ` `    ``{ ` `        ``// Initialize result ` `        ``int` `result = 1; ` ` `  `        ``// Create a lookup table and fill ` `        ``// all entries in it as -1 ` `        ``int``[][] dp = RectangularArrays.ReturnRectangularIntArray(n, n); ` `        ``for` `(``int` `i = 0; i < n; i++) { ` `            ``for` `(``int` `j = 0; j < n; j++) { ` `                ``dp[i][j] = -1; ` `            ``} ` `        ``} ` ` `  `        ``// Compute longest path beginning ` `        ``// from all cells ` `        ``for` `(``int` `i = 0; i < n; i++) { ` `            ``for` `(``int` `j = 0; j < n; j++) { ` `                ``if` `(dp[i][j] == -1) { ` `                    ``findLongestFromACell(i, j, mat, dp); ` `                ``} ` ` `  `                ``// Update result if needed ` `                ``result = Math.Max(result, dp[i][j]); ` `            ``} ` `        ``} ` ` `  `        ``return` `result; ` `    ``} ` ` `  `    ``public` `static` `class` `RectangularArrays { ` `        ``public` `static` `int``[][] ReturnRectangularIntArray(``int` `size1, ` `                                                        ``int` `size2) ` `        ``{ ` `            ``int``[][] newArray = ``new` `int``[size1][]; ` `            ``for` `(``int` `array1 = 0; ` `                 ``array1 < size1; array1++) { ` `                ``newArray[array1] = ``new` `int``[size2]; ` `            ``} ` ` `  `            ``return` `newArray; ` `        ``} ` `    ``} ` ` `  `    ``// Driver Code ` `    ``public` `static` `void` `Main(``string``[] args) ` `    ``{ ` `        ``int``[][] mat = ``new` `int``[][] { ` `            ``new` `int``[] { 1, 2, 9 }, ` `            ``new` `int``[] { 5, 3, 8 }, ` `            ``new` `int``[] { 4, 6, 7 } ` `        ``}; ` `        ``Console.WriteLine(``"Length of the longest path is "` `+ finLongestOverAll(mat)); ` `    ``} ` `} ` ` `  `// This code is contributed by Shrikant13 `

Output:
`Length of the longest path is 4`

Time complexity of the above solution is O(n2). It may seem more at first look. If we take a closer look, we can notice that all values of dp[i][j] are computed only once.