Path with maximum average value

Given a square matrix of size N*N, where each cell is associated with a specific cost. A path is defined as a specific sequence of cells that starts from the top-left cell move only right or down and ends on bottom right cell. We want to find a path with the maximum average over all existing paths. Average is computed as total cost divided by the number of cells visited in the path. 

Examples: 

Input : Matrix = [1, 2, 3
                  4, 5, 6
                  7, 8, 9]
Output : 5.8
Path with maximum average is, 1 -> 4 -> 7 -> 8 -> 9
Sum of the path is 29 and average is 29/5 = 5.8

One interesting observation is, the only allowed moves are down and right, we need N-1 down moves and N-1 right moves to reach the destination (bottom rightmost). So any path from top left corner to bottom right corner requires 2N – 1 cells. In average value, the denominator is fixed and we need to just maximize numerator. Therefore we basically need to find the maximum sum path. Calculating maximum sum of path is a classic dynamic programming problem, if dp[i][j] represents maximum sum till cell (i, j) from (0, 0) then at each cell (i, j), we update dp[i][j] as below,

for all i, 1 <= i <= N
   dp[i][0] = dp[i-1][0] + cost[i][0];    
for all j, 1 <= j <= N
   dp[0][j] = dp[0][j-1] + cost[0][j];            
otherwise    
   dp[i][j] = max(dp[i-1][j], dp[i][j-1]) + cost[i][j]; 

Once we get maximum sum of all paths we will divide this sum by (2N – 1) and we will get our maximum average. 

Implementation:

//C/C++ program to find maximum average cost path
#include <bits/stdc++.h>
using namespace std;

// Maximum number of rows and/or columns
const int M = 100;

// method returns maximum average of all path of
// cost matrix
double maxAverageOfPath(int cost[M][M], int N)
{
    int dp[N+1][N+1];
    dp[0][0] = cost[0][0];

    /* Initialize first column of total cost(dp) array */
    for (int i = 1; i < N; i++)
        dp[i][0] = dp[i-1][0] + cost[i][0];

    /* Initialize first row of dp array */
    for (int j = 1; j < N; j++)
        dp[0][j] = dp[0][j-1] + cost[0][j];

    /* Construct rest of the dp array */
    for (int i = 1; i < N; i++)
        for (int j = 1; j <= N; j++)
            dp[i][j] = max(dp[i-1][j],
                          dp[i][j-1]) + cost[i][j];

    // divide maximum sum by constant path
    // length : (2N - 1) for getting average
    return (double)dp[N-1][N-1] / (2*N-1);
}

/* Driver program to test above functions */
int main()
{
    int cost[M][M] = { {1, 2, 3},
        {6, 5, 4},
        {7, 3, 9}
    };
    printf("%f", maxAverageOfPath(cost, 3));
    return 0;
}

// JAVA Code for Path with maximum average
// value
import java.io.*;

class GFG {
    
    // method returns maximum average of all
    // path of cost matrix
    public static double maxAverageOfPath(int cost[][],
                                               int N)
    {
        int dp[][] = new int[N+1][N+1];
        dp[0][0] = cost[0][0];
     
        /* Initialize first column of total cost(dp)
           array */
        for (int i = 1; i < N; i++)
            dp[i][0] = dp[i-1][0] + cost[i][0];
     
        /* Initialize first row of dp array */
        for (int j = 1; j < N; j++)
            dp[0][j] = dp[0][j-1] + cost[0][j];
     
        /* Construct rest of the dp array */
        for (int i = 1; i < N; i++)
            for (int j = 1; j < N; j++)
                dp[i][j] = Math.max(dp[i-1][j],
                           dp[i][j-1]) + cost[i][j];
     
        // divide maximum sum by constant path
        // length : (2N - 1) for getting average
        return (double)dp[N-1][N-1] / (2 * N - 1);
    }
    
    /* Driver program to test above function */
    public static void main(String[] args) 
    {
        int cost[][] = {{1, 2, 3},
                        {6, 5, 4},
                        {7, 3, 9}};
                
        System.out.println(maxAverageOfPath(cost, 3));
    }
}
// This code is contributed by Arnav Kr. Mandal.

// C# Code for Path with maximum average
// value
using System;
class GFG {
    
    // method returns maximum average of all
    // path of cost matrix
    public static double maxAverageOfPath(int [,]cost,
                                               int N)
    {
        int [,]dp = new int[N+1,N+1];
        dp[0,0] = cost[0,0];
    
        /* Initialize first column of total cost(dp)
           array */
        for (int i = 1; i < N; i++)
            dp[i, 0] = dp[i - 1,0] + cost[i, 0];
    
        /* Initialize first row of dp array */
        for (int j = 1; j < N; j++)
            dp[0, j] = dp[0,j - 1] + cost[0, j];
    
        /* Construct rest of the dp array */
        for (int i = 1; i < N; i++)
            for (int j = 1; j < N; j++)
                dp[i, j] = Math.Max(dp[i - 1, j],
                        dp[i,j - 1]) + cost[i, j];
    
        // divide maximum sum by constant path
        // length : (2N - 1) for getting average
        return (double)dp[N - 1, N - 1] / (2 * N - 1);
    }
    
    // Driver Code
    public static void Main() 
    {
        int [,]cost = {{1, 2, 3},
                       {6, 5, 4},
                       {7, 3, 9}};
                
        Console.Write(maxAverageOfPath(cost, 3));
    }
}

// This code is contributed by nitin mittal.

<script>

    // JavaScript Code for Path with maximum average value
    
    // method returns maximum average of all
    // path of cost matrix
    function maxAverageOfPath(cost, N)
    {
        let dp = new Array(N+1);
        for (let i = 0; i < N + 1; i++)
        {
            dp[i] = new Array(N + 1);
            for (let j = 0; j < N + 1; j++)
            {
                dp[i][j] = 0;
            }
        }
        dp[0][0] = cost[0][0];
       
        /* Initialize first column of total cost(dp)
           array */
        for (let i = 1; i < N; i++)
            dp[i][0] = dp[i-1][0] + cost[i][0];
       
        /* Initialize first row of dp array */
        for (let j = 1; j < N; j++)
            dp[0][j] = dp[0][j-1] + cost[0][j];
       
        /* Construct rest of the dp array */
        for (let i = 1; i < N; i++)
            for (let j = 1; j < N; j++)
                dp[i][j] = Math.max(dp[i-1][j],
                           dp[i][j-1]) + cost[i][j];
       
        // divide maximum sum by constant path
        // length : (2N - 1) for getting average
        return dp[N-1][N-1] / (2 * N - 1);
    }
    
    let cost = [[1, 2, 3],
                [6, 5, 4],
                [7, 3, 9]];
                  
      document.write(maxAverageOfPath(cost, 3));

</script>

<?php
// Php program to find maximum average cost path 

// method returns maximum average of all path of 
// cost matrix 
function maxAverageOfPath($cost, $N) 
{ 
    $dp = array(array()) ;
    $dp[0][0] = $cost[0][0]; 

    /* Initialize first column of total cost(dp) array */
    for ($i = 1; $i < $N; $i++) 
        $dp[$i][0] = $dp[$i-1][0] + $cost[$i][0]; 

    /* Initialize first row of dp array */
    for ($j = 1; $j < $N; $j++) 
        $dp[0][$j] = $dp[0][$j-1] + $cost[0][$j]; 

    /* Construct rest of the dp array */
    for ($i = 1; $i < $N; $i++) 
    {
        for ($j = 1; $j <= $N; $j++) 
            $dp[$i][$j] = max($dp[$i-1][$j],$dp[$i][$j-1]) + $cost[$i][$j]; 
    }
    // divide maximum sum by constant path 
    // length : (2N - 1) for getting average 
    return $dp[$N-1][$N-1] / (2*$N-1); 
} 

    // Driver code
    $cost = array(array(1, 2, 3), 
            array( 6, 5, 4), 
            array(7, 3, 9) ) ;
            
    echo maxAverageOfPath($cost, 3) ; 

    // This code is contributed by Ryuga
?>

# Python program to find 
# maximum average cost path

# Maximum number of rows 
# and/or columns
M = 100

# method returns maximum average of 
# all path of cost matrix
def maxAverageOfPath(cost, N):
    
    dp = [[0 for i in range(N + 1)] for j in range(N + 1)]
    dp[0][0] = cost[0][0]

    # Initialize first column of total cost(dp) array
    for i in range(1, N):
        dp[i][0] = dp[i - 1][0] + cost[i][0]

    # Initialize first row of dp array
    for j in range(1, N):
        dp[0][j] = dp[0][j - 1] + cost[0][j]

    # Construct rest of the dp array
    for i in range(1, N):
        for j in range(1, N):
            dp[i][j] = max(dp[i - 1][j],
                        dp[i][j - 1]) + cost[i][j]

    # divide maximum sum by constant path
    # length : (2N - 1) for getting average
    return dp[N - 1][N - 1] / (2 * N - 1)

# Driver program to test above function
cost = [[1, 2, 3],
        [6, 5, 4],
        [7, 3, 9]]

print(maxAverageOfPath(cost, 3))

# This code is contributed by Soumen Ghosh.

5.200000

Time complexity: O(N²) for  given input N
Auxiliary Space: O(N²) for given input N.

Method – 2: Without using Extra N*N space 

We can use input cost array as a dp to store the ans. so this way we don’t need an extra dp array or no that extra space.\

One observation is that the only allowed moves are down and right, we need N-1 down moves and N-1 right moves to reach the destination (bottom rightmost). So any path from top left corner to bottom right corner requires 2N – 1 cell. In average value, the denominator is fixed and we need to just maximize numerator. Therefore we basically need to find the maximum sum path. Calculating maximum sum of path is a classic dynamic programming problem, also we don’t need any prev cost[i][j] value after calculating dp[i][j] so we can modifies the cost[i][j] value such that we don’t need extra space for dp[i][j].

for all i, 1 <= i < N
  cost[i][0] = cost[i-1][0] + cost[i][0];    
for all j, 1 <= j < N
  cost[0][j] = cost[0][j-1] + cost[0][j];            
otherwise    
  cost[i][j] = max(cost[i-1][j], cost[i][j-1]) + cost[i][j]; 

Below is the implementation of the above approach:

// C++ program to find maximum average cost path
#include <bits/stdc++.h>
using namespace std;

// Method returns maximum average of all path of cost matrix
double maxAverageOfPath(vector<vector<int>>cost)
{
    int N = cost.size();

    // Initialize first column of total cost array
    for (int i = 1; i < N; i++)
        cost[i][0] = cost[i][0] + cost[i - 1][0];

    // Initialize first row of array
    for (int j = 1; j < N; j++)
        cost[0][j] = cost[0][j - 1] + cost[0][j];

    // Construct rest of the array
    for (int i = 1; i < N; i++)
        for (int j = 1; j <= N; j++)
            cost[i][j] = max(cost[i - 1][j], cost[i][j - 1]) + cost[i][j];

    // divide maximum sum by constant path
    // length : (2N - 1) for getting average
    return (double)cost[N - 1][N - 1] / (2 * N - 1);
}

// Driver program
int main()
{
    vector<vector<int>> cost = {{1, 2, 3},
        {6, 5, 4},
        {7, 3, 9}
    };
    cout << maxAverageOfPath(cost);
    return 0;
}

// Java program to find maximum average cost path
import java.io.*;

class GFG {

  // Method returns maximum average of all path of cost
  // matrix
  static double maxAverageOfPath(int[][] cost)
  {
    int N = cost.length;

    // Initialize first column of total cost array
    for (int i = 1; i < N; i++)
      cost[i][0] = cost[i][0] + cost[i - 1][0];

    // Initialize first row of array
    for (int j = 1; j < N; j++)
      cost[0][j] = cost[0][j - 1] + cost[0][j];

    // Construct rest of the array
    for (int i = 1; i < N; i++)
      for (int j = 1; j < N; j++)
        cost[i][j] = Math.max(cost[i - 1][j],
                              cost[i][j - 1])
        + cost[i][j];

    // divide maximum sum by constant path
    // length : (2N - 1) for getting average
    return (double)cost[N - 1][N - 1] / (2 * N - 1);
  }

  // Driver program
  public static void main(String[] args)
  {
    int[][] cost
      = { { 1, 2, 3 }, { 6, 5, 4 }, { 7, 3, 9 } };
    System.out.println(maxAverageOfPath(cost));
  }
}

// This code is contributed by karandeep1234

// C# program to find maximum average cost path
using System;
class GFG {

  // Method returns maximum average of all path of cost
  // matrix
  static double maxAverageOfPath(int[, ] cost)
  {
    int N = cost.GetLength(0);

    // Initialize first column of total cost array
    for (int i = 1; i < N; i++)
      cost[i, 0] = cost[i, 0] + cost[i - 1, 0];

    // Initialize first row of array
    for (int j = 1; j < N; j++)
      cost[0, j] = cost[0, j - 1] + cost[0, j];

    // Construct rest of the array
    for (int i = 1; i < N; i++)
      for (int j = 1; j < N; j++)
        cost[i, j] = Math.Max(cost[i - 1, j],
                              cost[i, j - 1])
        + cost[i, j];

    // divide maximum sum by constant path
    // length : (2N - 1) for getting average
    return (double)cost[N - 1, N - 1] / (2 * N - 1);
  }

  // Driver program
  static void Main(string[] args)
  {
    int[, ] cost
      = { { 1, 2, 3 }, { 6, 5, 4 }, { 7, 3, 9 } };
    Console.WriteLine(maxAverageOfPath(cost));
  }
}

// This code is contributed by karandeep1234

// Method returns maximum average of all path of cost matrix
function maxAverageOfPath(cost)
{
    let N = cost.length;

    // Initialize first column of total cost array
    for (let i = 1; i < N; i++)
        cost[i][0] = cost[i][0] + cost[i - 1][0];

    // Initialize first row of array
    for (let j = 1; j < N; j++)
        cost[0][j] = cost[0][j - 1] + cost[0][j];

    // Construct rest of the array
    for (let i = 1; i < N; i++)
        for (let j = 1; j <= N; j++)
            cost[i][j] = Math.max(cost[i - 1][j], cost[i][j - 1]) + cost[i][j];

    // divide maximum sum by constant path
    // length : (2N - 1) for getting average
    return (cost[N - 1][N - 1]) / (2.0 * N - 1);
}

// Driver program
let cost = [[1, 2, 3],
        [6, 5, 4],
        [7, 3, 9]];
console.log(maxAverageOfPath(cost))

// This code is contributed by karandeep1234.

# Python program to find maximum average cost path
from typing import List

def maxAverageOfPath(cost: List[List[int]]) -> float:
    N = len(cost)

    # Initialize first column of total cost array
    for i in range(1, N):
        cost[i][0] = cost[i][0] + cost[i - 1][0]

    # Initialize first row of array
    for j in range(1, N):
        cost[0][j] = cost[0][j - 1] + cost[0][j]

    # Construct rest of the array
    for i in range(1, N):
        for j in range(1, N):
            cost[i][j] = max(cost[i - 1][j], cost[i][j - 1]) + cost[i][j]

    # divide maximum sum by constant path
    # length : (2N - 1) for getting average
    return cost[N - 1][N - 1] / (2 * N - 1)

# Driver program
def main():
    cost = [[1, 2, 3],
            [6, 5, 4],
            [7, 3, 9]]
    print(maxAverageOfPath(cost))

if __name__ == '__main__':
    main()

   

5.2

Time Complexity: O(N*N)
Auxiliary Space: O(1)