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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]


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++

// 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;
}

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)

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.



This article is attributed to GeeksforGeeks.org

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