Min Cost Path | DP-6

• Difficulty Level : Easy
• Last Updated : 21 Oct, 2021

Given a cost matrix cost[][] and a position (m, n) in cost[][], write a function that returns cost of minimum cost path to reach (m, n) from (0, 0). Each cell of the matrix represents a cost to traverse through that cell. The total cost of a path to reach (m, n) is the sum of all the costs on that path (including both source and destination). You can only traverse down, right and diagonally lower cells from a given cell, i.e., from a given cell (i, j), cells (i+1, j), (i, j+1), and (i+1, j+1) can be traversed. You may assume that all costs are positive integers.

For example, in the following figure, what is the minimum cost path to (2, 2)? The path with minimum cost is highlighted in the following figure. The path is (0, 0) –> (0, 1) –> (1, 2) –> (2, 2). The cost of the path is 8 (1 + 2 + 2 + 3). 1) Optimal Substructure
The path to reach (m, n) must be through one of the 3 cells: (m-1, n-1) or (m-1, n) or (m, n-1). So minimum cost to reach (m, n) can be written as “minimum of the 3 cells plus cost[m][n]”.
minCost(m, n) = min (minCost(m-1, n-1), minCost(m-1, n), minCost(m, n-1)) + cost[m][n]

2) Overlapping Subproblems
Following is a simple recursive implementation of the MCP (Minimum Cost Path) problem. The implementation simply follows the recursive structure mentioned above.

C++

 // A Naive recursive implementation // of MCP(Minimum Cost Path) problem #include using namespace std;   #define R 3 #define C 3   int min(int x, int y, int z);   // Returns cost of minimum cost path // from (0,0) to (m, n) in mat[R][C] int minCost(int cost[R][C], int m, int n) {     if (n < 0 || m < 0)         return INT_MAX;     else if (m == 0 && n == 0)         return cost[m][n];     else         return cost[m][n] +          min(minCost(cost, m - 1, n - 1),              minCost(cost, m - 1, n),              minCost(cost, m, n - 1)); }   // A utility function that returns // minimum of 3 integers int min(int x, int y, int z) {     if (x < y)         return (x < z) ? x : z;     else         return (y < z) ? y : z; }   // Driver code int main() {     int cost[R][C] = { { 1, 2, 3 },                        { 4, 8, 2 },                        { 1, 5, 3 } };                              cout << minCost(cost, 2, 2) << endl;           return 0; }   // This code is contributed by nikhilchhipa9

C

 /* A Naive recursive implementation of MCP(Minimum Cost Path) problem */ #include #include #define R 3 #define C 3   int min(int x, int y, int z);   /* Returns cost of minimum cost path from (0,0) to (m, n) in mat[R][C]*/ int minCost(int cost[R][C], int m, int n) {    if (n < 0 || m < 0)       return INT_MAX;    else if (m == 0 && n == 0)       return cost[m][n];    else       return cost[m][n] + min( minCost(cost, m-1, n-1),                                minCost(cost, m-1, n),                                minCost(cost, m, n-1) ); }   /* A utility function that returns minimum of 3 integers */ int min(int x, int y, int z) {    if (x < y)       return (x < z)? x : z;    else       return (y < z)? y : z; }   /* Driver program to test above functions */ int main() {    int cost[R][C] = { {1, 2, 3},                       {4, 8, 2},                       {1, 5, 3} };    printf(" %d ", minCost(cost, 2, 2));    return 0; }

Java

 /* A Naive recursive implementation of MCP(Minimum Cost Path) problem */ public class GFG {       /* A utility function that returns     minimum of 3 integers */     static int min(int x, int y, int z)     {         if (x < y)             return (x < z) ? x : z;         else             return (y < z) ? y : z;     }           /* Returns cost of minimum cost path     from (0,0) to (m, n) in mat[R][C]*/     static int minCost(int cost[][], int m,                                      int n)     {         if (n < 0 || m < 0)             return Integer.MAX_VALUE;         else if (m == 0 && n == 0)             return cost[m][n];         else             return cost[m][n] +                 min( minCost(cost, m-1, n-1),                      minCost(cost, m-1, n),                      minCost(cost, m, n-1) );     }       // Driver code     public static void main(String args[])     {                   int cost[][] = { {1, 2, 3},                          {4, 8, 2},                          {1, 5, 3} };                                    System.out.print(minCost(cost, 2, 2));     } }   // This code is contributed by Sam007

Python3

 # A Naive recursive implementation of MCP(Minimum Cost Path) problem R = 3 C = 3 import sys   # Returns cost of minimum cost path from (0,0) to (m, n) in mat[R][C] def minCost(cost, m, n):     if (n < 0 or m < 0):         return sys.maxsize     elif (m == 0 and n == 0):         return cost[m][n]     else:         return cost[m][n] + min( minCost(cost, m-1, n-1),                                 minCost(cost, m-1, n),                                 minCost(cost, m, n-1) )   #A utility function that returns minimum of 3 integers */ def min(x, y, z):     if (x < y):         return x if (x < z) else z     else:         return y if (y < z) else z     # Driver program to test above functions cost= [ [1, 2, 3],         [4, 8, 2],         [1, 5, 3] ] print(minCost(cost, 2, 2))   # This code is contributed by # Smitha Dinesh Semwal

C#

 /* A Naive recursive implementation of MCP(Minimum Cost Path) problem */ using System;   class GFG {       /* A utility function that     returns minimum of 3 integers */     static int min(int x,                    int y, int z)     {         if (x < y)             return ((x < z) ? x : z);         else             return ((y < z) ? y : z);     }           /* Returns cost of minimum     cost path from (0,0) to     (m, n) in mat[R][C]*/     static int minCost(int [,]cost,                        int m , int n)     {         if (n < 0 || m < 0)             return int.MaxValue;         else if (m == 0 && n == 0)             return cost[m, n];         else             return cost[m, n] +                    min(minCost(cost, m - 1, n - 1),                    minCost(cost, m - 1, n),                    minCost(cost, m, n - 1) );     }       // Driver code     public static void Main()     {                   int [,]cost = {{1, 2, 3},                        {4, 8, 2},                        {1, 5, 3}};                                   Console.Write(minCost(cost, 2, 2));     } }   // This code is contributed // by shiv_bhakt.



Javascript



Output

8

Time Complexity: O(m * n)

It should be noted that the above function computes the same subproblems again and again. See the following recursion tree, there are many nodes which appear more than once. The time complexity of this naive recursive solution is exponential and it is terribly slow.

mC refers to minCost()
mC(2, 2)
/            |           \
/             |            \
mC(1, 1)           mC(1, 2)             mC(2, 1)
/     |     \       /     |     \           /     |     \
/      |      \     /      |      \         /      |       \
mC(0,0) mC(0,1) mC(1,0) mC(0,1) mC(0,2) mC(1,1) mC(1,0) mC(1,1) mC(2,0)

So the MCP problem has both properties (see this and this) of a dynamic programming problem. Like other typical Dynamic Programming(DP) problems, recomputations of the same subproblems can be avoided by constructing a temporary array tc[][] in a bottom-up manner.

C++

 /* Dynamic Programming implementation of MCP problem */ #include #include #define R 3 #define C 3 using namespace std; int min(int x, int y, int z);   int minCost(int cost[R][C], int m, int n) {      int i, j;        // Instead of following line, we can use int tc[m+1][n+1] or      // dynamically allocate memory to save space. The following line is      // used to keep the program simple and make it working on all compilers.      int tc[R][C];         tc = cost;        /* Initialize first column of total cost(tc) array */      for (i = 1; i <= m; i++)         tc[i] = tc[i - 1] + cost[i];        /* Initialize first row of tc array */      for (j = 1; j <= n; j++)         tc[j] = tc[j - 1] + cost[j];        /* Construct rest of the tc array */      for (i = 1; i <= m; i++)         for (j = 1; j <= n; j++)             tc[i][j] = min(tc[i - 1][j - 1],                            tc[i - 1][j],                            tc[i][j - 1]) + cost[i][j];        return tc[m][n]; }   /* A utility function that returns minimum of 3 integers */ int min(int x, int y, int z) {    if (x < y)       return (x < z)? x : z;    else       return (y < z)? y : z; }   /* Driver code*/ int main() {    int cost[R][C] = { {1, 2, 3},                       {4, 8, 2},                       {1, 5, 3} };    cout << "  " << minCost(cost, 2, 2);    return 0; }   // This code is contributed by shivanisinghss2110

C

 /* Dynamic Programming implementation of MCP problem */ #include #include #define R 3 #define C 3   int min(int x, int y, int z);   int minCost(int cost[R][C], int m, int n) {      int i, j;        // Instead of following line, we can use int tc[m+1][n+1] or      // dynamically allocate memory to save space. The following line is      // used to keep the program simple and make it working on all compilers.      int tc[R][C];         tc = cost;        /* Initialize first column of total cost(tc) array */      for (i = 1; i <= m; i++)         tc[i] = tc[i-1] + cost[i];        /* Initialize first row of tc array */      for (j = 1; j <= n; j++)         tc[j] = tc[j-1] + cost[j];        /* Construct rest of the tc array */      for (i = 1; i <= m; i++)         for (j = 1; j <= n; j++)             tc[i][j] = min(tc[i-1][j-1],                            tc[i-1][j],                            tc[i][j-1]) + cost[i][j];        return tc[m][n]; }   /* A utility function that returns minimum of 3 integers */ int min(int x, int y, int z) {    if (x < y)       return (x < z)? x : z;    else       return (y < z)? y : z; }   /* Driver program to test above functions */ int main() {    int cost[R][C] = { {1, 2, 3},                       {4, 8, 2},                       {1, 5, 3} };    printf(" %d ", minCost(cost, 2, 2));    return 0; }

Java

 /* Java program for Dynamic Programming implementation    of Min Cost Path problem */ import java.util.*;   class MinimumCostPath {     /* A utility function that returns minimum of 3 integers */     private static int min(int x, int y, int z)     {         if (x < y)             return (x < z)? x : z;         else             return (y < z)? y : z;     }       private static int minCost(int cost[][], int m, int n)     {         int i, j;         int tc[][]=new int[m+1][n+1];           tc = cost;           /* Initialize first column of total cost(tc) array */         for (i = 1; i <= m; i++)             tc[i] = tc[i-1] + cost[i];           /* Initialize first row of tc array */         for (j = 1; j <= n; j++)             tc[j] = tc[j-1] + cost[j];           /* Construct rest of the tc array */         for (i = 1; i <= m; i++)             for (j = 1; j <= n; j++)                 tc[i][j] = min(tc[i-1][j-1],                                tc[i-1][j],                                tc[i][j-1]) + cost[i][j];           return tc[m][n];     }       /* Driver program to test above functions */     public static void main(String args[])     {         int cost[][]= {{1, 2, 3},                        {4, 8, 2},                        {1, 5, 3}};         System.out.println(minCost(cost,2,2));     } } // This code is contributed by Pankaj Kumar

Python

 # Dynamic Programming Python implementation of Min Cost Path # problem R = 3 C = 3   def minCost(cost, m, n):       # Instead of following line, we can use int tc[m+1][n+1] or     # dynamically allocate memoery to save space. The following     # line is used to keep te program simple and make it working     # on all compilers.     tc = [[0 for x in range(C)] for x in range(R)]       tc = cost       # Initialize first column of total cost(tc) array     for i in range(1, m+1):         tc[i] = tc[i-1] + cost[i]       # Initialize first row of tc array     for j in range(1, n+1):         tc[j] = tc[j-1] + cost[j]       # Construct rest of the tc array     for i in range(1, m+1):         for j in range(1, n+1):             tc[i][j] = min(tc[i-1][j-1], tc[i-1][j], tc[i][j-1]) + cost[i][j]       return tc[m][n]   # Driver program to test above functions cost = [[1, 2, 3],         [4, 8, 2],         [1, 5, 3]] print(minCost(cost, 2, 2))   # This code is contributed by Bhavya Jain

C#

 // C# program for Dynamic Programming implementation // of Min Cost Path problem using System;   class GFG {     // A utility function that     // returns minimum of 3 integers     private static int min(int x, int y, int z)     {         if (x < y)             return (x < z)? x : z;         else             return (y < z)? y : z;     }       private static int minCost(int [,]cost, int m, int n)     {         int i, j;         int [,]tc=new int[m+1,n+1];           tc[0,0] = cost[0,0];           /* Initialize first column of total cost(tc) array */         for (i = 1; i <= m; i++)             tc[i, 0] = tc[i - 1, 0] + cost[i, 0];           /* Initialize first row of tc array */         for (j = 1; j <= n; j++)             tc[0, j] = tc[0, j - 1] + cost[0, j];           /* Construct rest of the tc array */         for (i = 1; i <= m; i++)             for (j = 1; j <= n; j++)                 tc[i, j] = min(tc[i - 1, j - 1],                             tc[i - 1, j],                             tc[i, j - 1]) + cost[i, j];           return tc[m, n];     }       // Driver program     public static void Main()     {         int [,]cost= {{1, 2, 3},                     {4, 8, 2},                     {1, 5, 3}};         Console.Write(minCost(cost,2,2));     } }   // This code is contributed by Sam007.



Javascript



Output

8

Time Complexity of the DP implementation is O(mn) which is much better than Naive Recursive implementation.

Auxiliary Space: O(m * n)

Space Optimization: The idea is to use the same given array to store the solutions of subproblems.

C++

 #include #define endl "\n" using namespace std;   const int row = 3; const int col = 3;   int minCost(int cost[row][col]) {       // for 1st column     for (int i=1 ; i

Java

 // Java program for the // above approach import java.util.*; class GFG{   static int row = 3; static int col = 3;   static int minCost(int cost[][]) {   // for 1st column   for (int i = 1; i < row; i++)   {     cost[i] += cost[i - 1];   }     // for 1st row   for (int j = 1; j < col; j++)   {     cost[j] += cost[j - 1];   }     // for rest of the 2d matrix   for (int i = 1; i < row; i++)   {     for (int j = 1; j < col; j++)     {       cost[i][j] += Math.min(cost[i - 1][j - 1],                     Math.min(cost[i - 1][j],                              cost[i][j - 1]));     }   }     // Returning the value in   // last cell   return cost[row - 1][col - 1]; }     // Driver code  public static void  main(String[] args) {       int cost[][] = {{1, 2, 3},                   {4, 8, 2},                   {1, 5, 3} };   System.out.print(minCost(cost) + "\n"); } }   // This code is contributed by Amit Katiyar

Python3

 # Python3 program for the # above approach   def minCost(cost, row, col):           # For 1st column     for i in range(1, row):         cost[i] += cost[i - 1]       # For 1st row     for j in range(1, col):         cost[j] += cost[j - 1]       # For rest of the 2d matrix     for i in range(1, row):         for j in range(1, col):             cost[i][j] += (min(cost[i - 1][j - 1],                            min(cost[i - 1][j],                                cost[i][j - 1])))       # Returning the value in     # last cell     return cost[row - 1][col - 1]   # Driver code if __name__ == '__main__':           row = 3     col = 3           cost = [ [ 1, 2, 3 ],              [ 4, 8, 2 ],              [ 1, 5, 3 ] ]                    print(minCost(cost, row, col));   # This code is contributed by Amit Katiyar

C#

 // C# program for the // above approach using System; class GFG{   static int row = 3; static int col = 3;   static int minCost(int [,]cost) {   // for 1st column   for (int i = 1; i < row; i++)   {     cost[i, 0] += cost[i - 1, 0];   }     // for 1st row   for (int j = 1; j < col; j++)   {     cost[0, j] += cost[0, j - 1];   }     // for rest of the 2d matrix   for (int i = 1; i < row; i++)   {     for (int j = 1; j < col; j++)     {       cost[i,j] += Math.Min(cost[i - 1,                                  j - 1],                    Math.Min(cost[i - 1, j],                             cost[i, j - 1]));     }   }     // Returning the value in   // last cell   return cost[row - 1, col - 1]; }     // Driver code  public static void  Main(String[] args) {       int [,]cost = {{1, 2, 3},                  {4, 8, 2},                  {1, 5, 3} };   Console.Write(minCost(cost) + "\n"); } }   // This code is contributed by Rajput-Ji

Javascript



Output

8

Time Complexity: O(row * col)

Auxiliary Space: O(1)

Alternate Solution
We can also use the Dijkstra’s shortest path algorithm. Below is the implementation of the approach:

C++

 /* Minimum Cost Path using Dijkstra’s shortest path    algorithm with Min Heap by dinglizeng */ #include #include #include using namespace std;   /* define the number of rows and the number of columns */ #define R 4 #define C 5   /* 8 possible moves */ int dx[] = {1,-1, 0, 0, 1, 1,-1,-1}; int dy[] = {0, 0, 1,-1, 1,-1, 1,-1};   /* The data structure to store the coordinates of \\   the unit square and the cost of path from the top left. */ struct Cell{     int x;     int y;     int cost; };   /* The compare class to be used by a Min Heap.  * The greater than condition is used as this    is for a Min Heap based on priority_queue.  */ class mycomparison { public:   bool operator() (const Cell &lhs, const Cell &rhs) const   {     return (lhs.cost > rhs.cost);   } };   /* To verify whether a move is within the boundary. */ bool isSafe(int x, int y){     return x >= 0 && x < R && y >= 0 && y < C; }   /* This solution is based on Dijkstra’s shortest path algorithm  * For each unit square being visited, we examine all     possible next moves in 8 directions,  *    calculate the accumulated cost of path for each      next move, adjust the cost of path of the adjacent      units to the minimum as needed.  *    then add the valid next moves into a Min Heap.  * The Min Heap pops out the next move with the minimum    accumulated cost of path.  * Once the iteration reaches the last unit at the lower    right corner, the minimum cost path will be returned.  */ int minCost(int cost[R][C], int m, int n) {       /* the array to store the accumulated cost        of path from top left corner */     int dp[R][C];       /* the array to record whether a unit        square has been visited */     bool visited[R][C];          /* Initialize these two arrays, set path cost       to maximum integer value, each unit as not visited */     for(int i = 0; i < R; i++) {         for(int j = 0; j < C; j++) {             dp[i][j] = INT_MAX;             visited[i][j] = false;         }     }           /* Define a reverse priority queue.      * Priority queue is a heap based implementation.      * The default behavior of a priority queue is         to have the maximum element at the top.      * The compare class is used in the definition of the Min Heap.      */     priority_queue, mycomparison> pq;           /* initialize the starting top left unit with the       cost and add it to the queue as the first move. */     dp = cost;     pq.push({0, 0, cost});           while(!pq.empty()) {           /* pop a move from the queue, ignore the units            already visited */         Cell cell = pq.top();         pq.pop();         int x = cell.x;         int y = cell.y;         if(visited[x][y]) continue;           /* mark the current unit as visited */         visited[x][y] = true;           /* examine all non-visited adjacent units in 8 directions           * calculate the accumulated cost of path for            each next move from this unit,          * adjust the cost of path for each next adjacent            units to the minimum if possible.          */         for(int i = 0; i < 8; i++) {             int next_x = x + dx[i];             int next_y = y + dy[i];             if(isSafe(next_x, next_y) && !visited[next_x][next_y]) {                 dp[next_x][next_y] = min(dp[next_x][next_y],                    dp[x][y] + cost[next_x][next_y]);                 pq.push({next_x, next_y, dp[next_x][next_y]});             }         }     }       /* return the minimum cost path at the lower        right corner */     return dp[m][n];        }   /* Driver program to test above functions */ int main() {    int cost[R][C] = { {1, 8, 8, 1, 5},                       {4, 1, 1, 8, 1},                       {4, 2, 8, 8, 1},                       {1, 5, 8, 8, 1} };    printf(" %d ", minCost(cost, 3, 4));    return 0; }

Output

7

Using a reverse priority queue in this solution can reduce the time complexity compared with a full scan looking for the node with minimum path cost. The overall Time Complexity of the DP implementation is O(mn) without consideration of priority queue in use, which is much better than Naive Recursive implementation.

?list=PLqM7alHXFySEQDk2MDfbwEdjd2svVJH9p