Minimum time required to transport all the boxes from source to the destination under the given constraints

Given two arrays box[] and truck[] where box[i] represents the weight of the ith box and truck[i] represents the maximum load that the ith truck can carry. Now each truck takes 1 hour to transport a box from source to destination and another one hour to come back. Now given that all the boxes are kept at the source, the task is to find the minimum time required to transport all the boxes from source to the destination. Note that there will always be some time in which the boxes can be transported and only a single box can be carried by a truck at any instance of time.

Examples:

Input: box[] = {7, 6, 5, 4, 3}, truck[] = {10, 3}
Output: 7
1st hour: truck[0] carries box[0] and truck[1] carries box[4]
2nd hour: Both truck are back on the source location.
Now, truck[1] cannot carry anymore boxes as all the remaining boxes
have weights more than the capacity of truck[1].
So, truck[0] will carry box[1] and box[2]
in total of four hours. (source-destination and then destination-source)
And finally box[3] will take another hour to reach the destination.
So, total time taken = 2 + 4 + 1 = 7

Input: box[] = {10, 2, 16, 19}, truck[] = {29, 25}
Output: 3



Approach: The idea is to use binary search and sort the two arrays, here the lower bound will be 0 and upper bound will be 2 * size of box[] because in the worst case the amount of time required to transport all the boxes will be 2 * size of box array, now compute the mid value, and for each mid value check if all the boxes can be transported by the loaders in time = mid. If yes, then update upper bound as mid – 1 and if no then update lower bound as mid + 1.

Below is the implementation of the above approach:

C++

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// Function that returns true if it is
// possible to transport all the boxes
// in the given amount of time
bool isPossible(int box[], int truck[],
                int n, int m, int min_time)
{
    int temp = 0;
    int count = 0;
  
    while (count < m) {
        for (int j = 0; j < min_time
                        && temp < n
                        && truck[count] >= box[temp];
             j += 2)
            temp++;
  
        count++;
    }
  
    // If all the boxes can be
    // transported in the given time
    if (temp == n)
        return true;
  
    // If all the boxes can't be
    // transported in the given time
    return false;
}
  
// Function to return the minimum time required
int minTime(int box[], int truck[], int n, int m)
{
  
    // Sort the two arrays
    sort(box, box + n);
    sort(truck, truck + m);
  
    int l = 0;
    int h = 2 * n;
  
    // Stores minimum time in which
    // all the boxes can be transported
    int min_time = 0;
  
    // Check for the minimum time in which
    // all the boxes can be transported
    while (l <= h) {
        int mid = (l + h) / 2;
  
        // If it is possible to transport all
        // the boxes in mid amount of time
        if (isPossible(box, truck, n, m, mid)) {
            min_time = mid;
            h = mid - 1;
        }
        else
            l = mid + 1;
    }
  
    return min_time;
}
  
// Driver code
int main()
{
    int box[] = { 10, 2, 16, 19 };
    int truck[] = { 29, 25 };
  
    int n = sizeof(box) / sizeof(int);
    int m = sizeof(truck) / sizeof(int);
  
    printf("%d", minTime(box, truck, n, m));
  
    return 0;
}

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Java














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// Java implementation of the approach 


import java.util.Arrays; 


  


class GFG 


{ 


  


// Function that returns true if it is 


// possible to transport all the boxes 


// in the given amount of time 


static boolean isPossible(int box[], int truck[], 


                int n, int m, int min_time) 


{ 


    int temp = 0; 


    int count = 0; 


  


    while (count < m) 


    { 


        for (int j = 0; j < min_time 


                        && temp < n 


                        && truck[count] >= box[temp]; 


            j += 2) 


            temp++; 


  


        count++; 


    } 


  


    // If all the boxes can be 


    // transported in the given time 


    if (temp == n) 


        return true; 


  


    // If all the boxes can't be 


    // transported in the given time 


    return false; 


} 


  


// Function to return the minimum time required 


static int minTime(int box[], int truck[], int n, int m) 


{ 


  


    // Sort the two arrays 


    Arrays.sort(box); 


    Arrays.sort(truck); 


  


    int l = 0; 


    int h = 2 * n; 


  


    // Stores minimum time in which 


    // all the boxes can be transported 


    int min_time = 0; 


  


    // Check for the minimum time in which 


    // all the boxes can be transported 


    while (l <= h) { 


        int mid = (l + h) / 2; 


  


        // If it is possible to transport all 


        // the boxes in mid amount of time 


        if (isPossible(box, truck, n, m, mid)) 


        { 


            min_time = mid; 


            h = mid - 1; 


        } 


        else


            l = mid + 1; 


    } 


  


    return min_time; 


} 


  


// Driver code 


public static void main(String[] args) 


{ 


    int box[] = { 10, 2, 16, 19 }; 


    int truck[] = { 29, 25 }; 


  


    int n = box.length; 


    int m = truck.length; 


  


    System.out.printf("%d", minTime(box, truck, n, m)); 


} 


} 


  


/* This code contributed by PrinciRaj1992 */



















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# Python3 implementation of the approach  


  


# Function that returns true if it is  


# possible to transport all the boxes  


# in the given amount of time  


def isPossible(box, truck, n, m, min_time) : 


      


    temp = 0


    count = 0


  


    while (count < m) : 


        j = 0


        while (j < min_time and temp < n and 


                    truck[count] >= box[temp] ): 


            temp +=1


            j += 2


  


        count += 1


  


    # If all the boxes can be  


    # transported in the given time  


    if (temp == n) : 


        return True


  


    # If all the boxes can't be  


    # transported in the given time  


    return False


  


# Function to return the minimum time required  


def minTime(box, truck, n, m) :  


  


    # Sort the two arrays  


    box.sort();  


    truck.sort();  


  


    l = 0


    h = 2 * n  


  


    # Stores minimum time in which  


    # all the boxes can be transported  


    min_time = 0


  


    # Check for the minimum time in which  


    # all the boxes can be transported  


    while (l <= h) : 


        mid = (l + h) // 2


  


        # If it is possible to transport all  


        # the boxes in mid amount of time  


        if (isPossible(box, truck, n, m, mid)) : 


            min_time = mid  


            h = mid - 1


      


        else : 


              


            l = mid + 1


  


    return min_time 


  


# Driver code  


if __name__ == "__main__" :  


  


    box = [ 10, 2, 16, 19 ]  


    truck = [ 29, 25 ]  


  


    n = len(box)  


    m = len(truck) 


  


    print(minTime(box, truck, n, m)) 


      


# This code is contributed by Ryuga 



















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C#














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// C# implementation of the approach 


using System; 


      


class GFG 


{ 


  


// Function that returns true if it is 


// possible to transport all the boxes 


// in the given amount of time 


static bool isPossible(int []box, int []truck, 


                int n, int m, int min_time) 


{ 


    int temp = 0; 


    int count = 0; 


  


    while (count < m) 


    { 


        for (int j = 0; j < min_time 


                        && temp < n 


                        && truck[count] >= box[temp]; 


            j += 2) 


            temp++; 


  


        count++; 


    } 


  


    // If all the boxes can be 


    // transported in the given time 


    if (temp == n) 


        return true; 


  


    // If all the boxes can't be 


    // transported in the given time 


    return false; 


} 


  


// Function to return the minimum time required 


static int minTime(int []box, int []truck, int n, int m) 


{ 


  


    // Sort the two arrays 


    Array.Sort(box); 


    Array.Sort(truck); 


  


    int l = 0; 


    int h = 2 * n; 


  


    // Stores minimum time in which 


    // all the boxes can be transported 


    int min_time = 0; 


  


    // Check for the minimum time in which 


    // all the boxes can be transported 


    while (l <= h) 


    { 


        int mid = (l + h) / 2; 


  


        // If it is possible to transport all 


        // the boxes in mid amount of time 


        if (isPossible(box, truck, n, m, mid)) 


        { 


            min_time = mid; 


            h = mid - 1; 


        } 


        else


            l = mid + 1; 


    } 


  


    return min_time; 


} 


  


// Driver code 


public static void Main(String[] args) 


{ 


    int[] box = { 10, 2, 16, 19 }; 


    int []truck = { 29, 25 }; 


  


    int n = box.Length; 


    int m = truck.Length; 


  


    Console.WriteLine("{0}", minTime(box, truck, n, m)); 


} 


} 


  


/* This code contributed by PrinciRaj1992 */



















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<?php 


  


// PHP implementation of the approach 


  


// Function that returns true if it is 


// possible to transport all the boxes 


// in the given amount of time 


function isPossible($box, $truck, 


                $n, $m, $min_time) 


{ 


    $temp = 0; 


    $count = 0; 


  


    while ($count < $m) 


    { 


        for ( $j = 0; $j < $min_time


                        && $temp < $n


                        && $truck[$count] >= $box[$temp]; 


            $j += 2) 


            $temp++; 


  


        $count++; 


    } 


  


    // If all the boxes can be 


    // transported in the given time 


    if ($temp == $n) 


        return true; 


  


    // If all the boxes can't be 


    // transported in the given time 


    return false; 


} 


  


// Function to return the minimum time required 


function minTime( $box, $truck, $n, $m) 


{ 


  


    // Sort the two arrays 


    sort($box); 


    sort($truck); 


  


    $l = 0; 


    $h = 2 * $n; 


  


    // Stores minimum time in which 


    // all the boxes can be transported 


    $min_time = 0; 


  


    // Check for the minimum time in which 


    // all the boxes can be transported 


    while ($l <= $h) { 


        $mid = intdiv(($l + $h) , 2); 


  


        // If it is possible to transport all 


        // the boxes in mid amount of time 


        if (isPossible($box, $truck, $n, $m, $mid))  


        { 


            $min_time = $mid; 


            $h = $mid - 1; 


        } 


        else


            $l = $mid + 1; 


    } 


  


    return $min_time; 


} 


  


// Driver code 


$box = array( 10, 2, 16, 19 ); 


$truck = array( 29, 25 ); 


  


$n = sizeof($box); 


$m = sizeof($truck); 


  


echo minTime($box, $truck, $n, $m); 


  


  


// This code is contributed by ihritik 


  


?> 



















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