Find the minimum cost to cross the River

Given an integer N which is the number of villagers who need to cross a river but there is only one boat on which a maximum of 2 person can travel. Each person i has to pay some specific price Pi to travel alone in the boat. If two person i, j travel in the boat then they have to pay max(Pi, Pj). The task is to find the minimum amount all the villagers have to pay to cross the river.

Examples:

Input: Price[] = {30, 40, 60, 70}
Output: 220
P1 and P2 go together (which costs 40)
and P1 comes back (total cost 70 now).
Now P3 and P4 go (total cost 140) and
P2 comes back (total cost 180) and
finally P1 and P2 go together (total cost 220).



Input: Price[] = {892, 124}
Output: 892

Approach: There are two ways for the two most costly person to cross the river:

  • Both of them cross the river with the cheapest person turn by turn. So, the total cost will be the cost of the two costly persons + 2 * (cost of the cheapest person) (due to coming back).
  • The two cheapest person cross the river and the cheapest person comes back. Now, the 2 most costly person cross the river and 2nd cheapest person comes back. So, the total cost will be the cost of the cheapest and costliest person plus 2 * cost of the second cheapest person.
  • Total cost will be the minimum of the above two ways.

Let’s consider the example we used above to understand the approach:
P1 = 30, P2 = 40, P3 = 60, P4 = 70

According to first method, P4 goes with P1 and P1 comes back (cost is P4+P1).
Now, P3 goes with P1 and P1 comes back (cost for this ride will be P4+P1).
So, total cost for sending two most costly person according to method 1 is P4+2*P1+P3 = 190

According to second method, P2 goes with P1 and P1 comes back (cost is P2+P1).
Now, P3 goes with P4 and P2 comes back (cost for this ride will be P4+P2).
So, total cost for sending two most costly person according to method 2 is P4+2*P2+P1 = 180

Hence, cost for sending P3 and P4 will be minimum of 2 methods, i.e., 180.

Now, we are left with P1 and P2 whom we have to send together and cost will
be P2 = 40.

So, total cost for travelling is 180 + 40 = 220.

Below is the implementation of the above approach:

CPP

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
#define ll long long int
  
// Function to return the minimum cost
int minimumCost(ll price[], int n)
{
  
    // Sort the price array
    sort(price, price + n);
    ll totalCost = 0;
  
    // Calcualte minimum price
    // of n-2 most costly person
    for (int i = n - 1; i > 1; i -= 2) {
        if (i == 2) {
            totalCost += price[2] + price[0];
        }
        else {
  
            // Both the ways as discussed above
            ll price_first = price[i] + price[0] + 2 * price[1];
            ll price_second = price[i] + price[i - 1] + 2 * price[0];
            totalCost += min(price_first, price_second);
        }
    }
  
    // Calculate the minimum price
    // of the two cheapest person
    if (n == 1) {
        totalCost += price[0];
    }
    else {
        totalCost += price[1];
    }
  
    return totalCost;
}
  
// Driver code
int main()
{
    ll price[] = { 30, 40, 60, 70 };
    int n = sizeof(price) / sizeof(price[0]);
  
    cout << minimumCost(price, n);
  
    return 0;
}

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Java

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// Java implementation of the approach
import java.util.*;
  
class GFG
{
      
    // Function to return the minimum cost
    static long minimumCost(long price[], int n)
    {
      
        // Sort the price array
        Arrays.sort(price);
          
        long totalCost = 0;
      
        // Calcualte minimum price
        // of n-2 most costly person
        for (int i = n - 1; i > 1; i -= 2)
        {
            if (i == 2)
            {
                totalCost += price[2] + price[0];
            }
            else
            {
      
                // Both the ways as discussed above
                long price_first = price[i] + price[0] + 2 * price[1];
                long price_second = price[i] + price[i - 1] + 2 * price[0];
                totalCost += Math.min(price_first, price_second);
            }
        }
      
        // Calculate the minimum price
        // of the two cheapest person
        if (n == 1)
        {
            totalCost += price[0];
        }
        else
        {
            totalCost += price[1];
        }
      
        return totalCost;
    }
      
    // Driver code
    public static void main (String[] args)
    {
        long price[] = { 30, 40, 60, 70 };
        int n = price.length;
      
        System.out.println(minimumCost(price, n));
    }
}
  
// This code is contributed by AnkitRai01

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Python

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# Python3 implementation of the approach
  
# Function to return the minimum cost
def minimumCost(price, n):
  
    # Sort the price array
    price = sorted(price)
    totalCost = 0
  
    # Calcualte minimum price
    # of n-2 most costly person
    for i in range(n - 1, 1, -2):
        if (i == 2):
            totalCost += price[2] + price[0]
  
        else:
  
            # Both the ways as discussed above
            price_first = price[i] + price[0] + 2 * price[1]
            price_second = price[i] + price[i - 1] + 2 * price[0]
            totalCost += min(price_first, price_second)
  
    # Calculate the minimum price
    # of the two cheapest person
    if (n == 1):
        totalCost += price[0]
  
    else:
        totalCost += price[1]
  
    return totalCost
  
# Driver code
  
price = [30, 40, 60, 70]
n = len(price)
  
print(minimumCost(price, n))
  
# This code is contributed by mohit kumar 29

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C#

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// C# implementation of the approach
using System;
  
class GFG
{
      
    // Function to return the minimum cost
    static long minimumCost(long []price, int n)
    {
      
        // Sort the price array
        Array.Sort(price);
          
        long totalCost = 0;
      
        // Calcualte minimum price
        // of n-2 most costly person
        for (int i = n - 1; i > 1; i -= 2)
        {
            if (i == 2)
            {
                totalCost += price[2] + price[0];
            }
            else
            {
      
                // Both the ways as discussed above
                long price_first = price[i] + price[0] + 2 * price[1];
                long price_second = price[i] + price[i - 1] + 2 * price[0];
                totalCost += Math.Min(price_first, price_second);
            }
        }
      
        // Calculate the minimum price
        // of the two cheapest person
        if (n == 1)
        {
            totalCost += price[0];
        }
        else
        {
            totalCost += price[1];
        }
      
        return totalCost;
    }
      
    // Driver code
    public static void Main ()
    {
        long []price = { 30, 40, 60, 70 };
        int n = price.Length;
      
        Console.WriteLine(minimumCost(price, n));
    }
}
  
// This code is contributed by AnkitRai01

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Output:

220



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Improved By : mohit kumar 29, AnkitRai01