# Minimum Sum Path In 3-D Array

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[0][0][0] + arr[0][0][1] +
arr[0][1][1] + arr[1][1][1]

Input : { { {1, 2}, {4, 3}},
{ {3, 4}, {2, 1}} };
Output : 7
Explanation : arr[0][0][0] + arr[0][0][1] +
arr[0][1][1] + arr[1][1][1]
```

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

Let us consider a 3-D array arr[2][2][2] 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[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_MAX)
+ 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_MAX)
+ 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_MAX)
+ 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];
```

```// C++ program for Min path sum of 3D-array
#include<bits/stdc++.h>
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[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_MAX)
+ 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_MAX)
+ 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_MAX)
+ 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
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;
}
```

Output :

```20
```

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

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