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Minimum Sum Path In 3-D Array
• Difficulty Level : Hard
• Last Updated : 27 Jul, 2018

Given a 3-D array arr[l][m][n], the task is to find the minimum path sum from the first cell of array to the last cell of array. We can only traverse to adjacent element, i.e., from a given cell (i, j, k), cells (i+1, j, k), (i, j+1, k) and (i, j, k+1) can be traversed, diagonal traversing is not allowed, We may assume that all costs are positive integers.

Examples:

```Input : arr[][][]= { {{1, 2}, {3, 4}},
{{4, 8}, {5, 2}} };
Output : 9
Explanation : arr + arr +
arr + arr

Input : { { {1, 2}, {4, 3}},
{ {3, 4}, {2, 1}} };
Output : 7
Explanation : arr + arr +
arr + arr
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Let us consider a 3-D array arr represented by a cuboid having values as:

```arr[][][] = {{{1, 2}, {3, 4}},
{ {4, 8}, {5, 2}}};
Result = 9 is calculated as: ```

This problem is similar to Min cost path. and can be solved using Dynamic Programming/

```// Array for storing result
int tSum[l][m][n];

tSum = arr;

/* Initialize first row of tSum array */
for (i = 1; i < l; i++)
tSum[i] = tSum[i-1] + arr[i];

/* Initialize first column of tSum array */
for (j = 1; j < m; j++)
tSum[j] = tSum[j-1] + arr[j];

/* Initialize first width of tSum array */
for (k = 1; k < n; k++)
tSum[k] = tSum[k-1] + arr[k];

/* Initialize first row- First column of tSum
array */
for (i = 1; i < l; i++)
for (j = 1; j < m; j++)
tSum[i][j] = min(tSum[i-1][j],
tSum[i][j-1],
INT_MAX)
+ arr[i][j];

/* Initialize first row- First width of tSum
array */
for (i = 1; i < l; i++)
for (k = 1; k < n; k++)
tSum[i][k] = min(tSum[i-1][k],
tSum[i][k-1],
INT_MAX)
+ arr[i][k];

/* Initialize first width- First column of
tSum array */
for (k = 1; k < n; k++)
for (j = 1; j < m; j++)
tSum[j][k] = min(tSum[j][k-1],
tSum[j-1][k],
INT_MAX)
+ arr[j][k];

/* Construct rest of the tSum array */
for (i = 1; i < l; i++)
for (j = 1; j < m; j++)
for (k = 1; k < n; k++)
tSum[i][j][k] = min(tSum[i-1][j][k],
tSum[i][j-1][k],
tSum[i][j][k-1])
+ arr[i][j][k];

return tSum[l-1][m-1][n-1];
```

## C++

 `// C++ program for Min path sum of 3D-array``#include``using` `namespace` `std;``#define l 3``#define m 3``#define n 3`` ` `// A utility function that returns minimum``// of 3 integers``int` `min(``int` `x, ``int` `y, ``int` `z)``{``  ``return` `(x < y)? ((x < z)? x : z) :``          ``((y < z)? y : z);``}`` ` `// function to calculate MIN path sum of 3D array``int` `minPathSum(``int` `arr[][m][n])``{``  ``int` `i, j, k;``  ``int` `tSum[l][m][n];`` ` `  ``tSum = arr;`` ` `  ``/* Initialize first row of tSum array */``  ``for` `(i = 1; i < l; i++)``    ``tSum[i] = tSum[i-1] + arr[i];`` ` `  ``/* Initialize first column of tSum array */``  ``for` `(j = 1; j < m; j++)``    ``tSum[j] = tSum[j-1] + arr[j];`` ` `  ``/* Initialize first width of tSum array */``  ``for` `(k = 1; k < n; k++)``    ``tSum[k] = tSum[k-1] + arr[k];`` ` `  ``/* Initialize first row- First column of``     ``tSum array */``  ``for` `(i = 1; i < l; i++)``    ``for` `(j = 1; j < m; j++)``      ``tSum[i][j] = min(tSum[i-1][j],``                          ``tSum[i][j-1],``                          ``INT_MAX)``                    ``+ arr[i][j];`` ` ` ` `  ``/* Initialize first row- First width of``     ``tSum array */``  ``for` `(i = 1; i < l; i++)``    ``for` `(k = 1; k < n; k++)``      ``tSum[i][k] = min(tSum[i-1][k],``                          ``tSum[i][k-1],``                          ``INT_MAX)``                    ``+ arr[i][k];`` ` ` ` `  ``/* Initialize first width- First column of``     ``tSum array */``  ``for` `(k = 1; k < n; k++)``    ``for` `(j = 1; j < m; j++)``      ``tSum[j][k] = min(tSum[j][k-1],``                          ``tSum[j-1][k],``                          ``INT_MAX)``                    ``+ arr[j][k];`` ` `  ``/* Construct rest of the tSum array */``  ``for` `(i = 1; i < l; i++)``    ``for` `(j = 1; j < m; j++)``      ``for` `(k = 1; k < n; k++)``        ``tSum[i][j][k] = min(tSum[i-1][j][k],``                            ``tSum[i][j-1][k],``                            ``tSum[i][j][k-1])``                        ``+ arr[i][j][k];`` ` `  ``return` `tSum[l-1][m-1][n-1];`` ` `}`` ` `// Driver program``int` `main()``{``  ``int` `arr[l][m][n] = { { {1, 2, 4}, {3, 4, 5}, {5, 2, 1}},``    ``{ {4, 8, 3}, {5, 2, 1}, {3, 4, 2}},``    ``{ {2, 4, 1}, {3, 1, 4}, {6, 3, 8}}``  ``};``  ``cout << minPathSum(arr);``  ``return` `0;``}`

## Java

 `// Java program for Min path sum of 3D-array``import` `java.io.*;`` ` `class` `GFG {``     ` `    ``static` `int` `l =``3``;``    ``static` `int` `m =``3``;``    ``static` `int` `n =``3``;``     ` `    ``// A utility function that returns minimum``    ``// of 3 integers``    ``static` `int` `min(``int` `x, ``int` `y, ``int` `z)``    ``{``         ``return` `(x < y)? ((x < z)? x : z) :``                ``((y < z)? y : z);``    ``}``     ` `    ``// function to calculate MIN path sum of 3D array``    ``static` `int` `minPathSum(``int` `arr[][][])``    ``{``        ``int` `i, j, k;``        ``int` `tSum[][][] =``new` `int``[l][m][n];``         ` `        ``tSum[``0``][``0``][``0``] = arr[``0``][``0``][``0``];``         ` `        ``/* Initialize first row of tSum array */``        ``for` `(i = ``1``; i < l; i++)``            ``tSum[i][``0``][``0``] = tSum[i-``1``][``0``][``0``] + arr[i][``0``][``0``];``         ` `        ``/* Initialize first column of tSum array */``        ``for` `(j = ``1``; j < m; j++)``            ``tSum[``0``][j][``0``] = tSum[``0``][j-``1``][``0``] + arr[``0``][j][``0``];``         ` `        ``/* Initialize first width of tSum array */``        ``for` `(k = ``1``; k < n; k++)``            ``tSum[``0``][``0``][k] = tSum[``0``][``0``][k-``1``] + arr[``0``][``0``][k];``         ` `        ``/* Initialize first row- First column of``            ``tSum array */``        ``for` `(i = ``1``; i < l; i++)``            ``for` `(j = ``1``; j < m; j++)``            ``tSum[i][j][``0``] = min(tSum[i-``1``][j][``0``],``                                ``tSum[i][j-``1``][``0``],``                                ``Integer.MAX_VALUE)``                            ``+ arr[i][j][``0``];``         ` `         ` `        ``/* Initialize first row- First width of``            ``tSum array */``        ``for` `(i = ``1``; i < l; i++)``            ``for` `(k = ``1``; k < n; k++)``            ``tSum[i][``0``][k] = min(tSum[i-``1``][``0``][k],``                                ``tSum[i][``0``][k-``1``],``                                ``Integer.MAX_VALUE)``                            ``+ arr[i][``0``][k];``         ` `         ` `        ``/* Initialize first width- First column of``            ``tSum array */``        ``for` `(k = ``1``; k < n; k++)``            ``for` `(j = ``1``; j < m; j++)``            ``tSum[``0``][j][k] = min(tSum[``0``][j][k-``1``],``                                ``tSum[``0``][j-``1``][k],``                                ``Integer.MAX_VALUE)``                            ``+ arr[``0``][j][k];``         ` `        ``/* Construct rest of the tSum array */``        ``for` `(i = ``1``; i < l; i++)``            ``for` `(j = ``1``; j < m; j++)``            ``for` `(k = ``1``; k < n; k++)``                ``tSum[i][j][k] = min(tSum[i-``1``][j][k],``                                    ``tSum[i][j-``1``][k],``                                    ``tSum[i][j][k-``1``])``                                ``+ arr[i][j][k];``         ` `        ``return` `tSum[l-``1``][m-``1``][n-``1``];``         ` `    ``}``     ` `    ``// Driver program``    ``public` `static` `void` `main (String[] args)``    ``{``        ``int` `arr[][][] = { { {``1``, ``2``, ``4``}, {``3``, ``4``, ``5``}, {``5``, ``2``, ``1``}},``                          ``{ {``4``, ``8``, ``3``}, {``5``, ``2``, ``1``}, {``3``, ``4``, ``2``}},``                          ``{ {``2``, ``4``, ``1``}, {``3``, ``1``, ``4``}, {``6``, ``3``, ``8``}}``                        ``};``        ``System.out.println ( minPathSum(arr));``             ` `    ``}``}`` ` `// This code is contributed by vt_m`

## C#

 `// C# program for Min ``// path sum of 3D-array``using` `System;`` ` `class` `GFG``{``     ` `    ``static` `int` `l = 3;``    ``static` `int` `m = 3;``    ``static` `int` `n = 3;``     ` `    ``// A utility function ``    ``// that returns minimum``    ``// of 3 integers``    ``static` `int` `min(``int` `x, ``int` `y, ``int` `z)``    ``{``        ``return` `(x < y) ? ((x < z) ? x : z) :``              ``((y < z) ? y : z);``    ``}``     ` `    ``// function to calculate MIN ``    ``// path sum of 3D array``    ``static` `int` `minPathSum(``int` `[,,]arr)``    ``{``        ``int` `i, j, k;``        ``int` `[ , , ]tSum = ``new` `int``[l, m, n];``         ` `        ``tSum[0, 0, 0] = arr[0, 0, 0];``         ` `        ``/* Initialize first``        ``row of tSum array */``        ``for` `(i = 1; i < l; i++)``            ``tSum[i, 0, 0] = tSum[i - 1, 0, 0] + ``                             ``arr[i, 0, 0];``         ` `        ``/* Initialize first column ``        ``of tSum array */``        ``for` `(j = 1; j < m; j++)``            ``tSum[0, j, 0] = tSum[0, j - 1, 0] + ``                             ``arr[0, j, 0];``         ` `        ``/* Initialize first``        ``width of tSum array */``        ``for` `(k = 1; k < n; k++)``            ``tSum[0, 0, k] = tSum[0, 0, k - 1] + ``                             ``arr[0, 0, k];``         ` `        ``/* Initialize first ``        ``row- First column of``        ``tSum array */``        ``for` `(i = 1; i < l; i++)``            ``for` `(j = 1; j < m; j++)``            ``tSum[i, j, 0] = min(tSum[i - 1, j, 0],``                                ``tSum[i, j - 1, 0],``                                ``int``.MaxValue) +``                                ``arr[i, j, 0];``         ` `         ` `        ``/* Initialize first ``        ``row- First width of``        ``tSum array */``        ``for` `(i = 1; i < l; i++)``            ``for` `(k = 1; k < n; k++)``            ``tSum[i, 0, k] = min(tSum[i - 1, 0, k],``                                ``tSum[i, 0, k - 1],``                                ``int``.MaxValue) + ``                                ``arr[i, 0, k];``         ` `         ` `        ``/* Initialize first ``        ``width- First column of``        ``tSum array */``        ``for` `(k = 1; k < n; k++)``            ``for` `(j = 1; j < m; j++)``            ``tSum[0, j, k] = min(tSum[0, j, k - 1],``                                ``tSum[0, j - 1, k],``                                ``int``.MaxValue) + ``                                ``arr[0, j, k];``         ` `        ``/* Construct rest of``        ``the tSum array */``        ``for` `(i = 1; i < l; i++)``            ``for` `(j = 1; j < m; j++)``            ``for` `(k = 1; k < n; k++)``                ``tSum[i, j, k] = min(tSum[i - 1, j, k],``                                    ``tSum[i, j - 1, k],``                                    ``tSum[i, j, k - 1]) +``                                    ``arr[i, j, k];``         ` `        ``return` `tSum[l-1,m-1,n-1];``         ` `    ``}``     ` `    ``// Driver Code``    ``static` `public` `void` `Main ()``    ``{``        ``int` `[, , ]arr= {{{1, 2, 4}, {3, 4, 5}, {5, 2, 1}},``                        ``{{4, 8, 3}, {5, 2, 1}, {3, 4, 2}},``                        ``{{2, 4, 1}, {3, 1, 4}, {6, 3, 8}}};``        ``Console.WriteLine(minPathSum(arr));``             ` `    ``}``}`` ` `// This code is contributed by ajit`

Output :
```20
```

Time Complexity : O(l*m*n)
Auxiliary Space : O(l*m*n)

This article is contributed by Shivam Pradhan (anuj_charm). If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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