Allocate minimum number of pages

Given number of pages in n different books and m students. The books are arranged in ascending order of number of pages. Every student is assigned to read some consecutive books. The task is to assign books in such a way that the maximum number of pages assigned to a student is minimum.

Example :

Input : pages[] = {12, 34, 67, 90}
m = 2
Output : 113
Explanation:
There are 2 number of students. Books can be distributed
in following fashion :
1)  and [34, 67, 90]
Max number of pages is allocated to student
2 with 34 + 67 + 90 = 191 pages
2) [12, 34] and [67, 90]
Max number of pages is allocated to student
2 with 67 + 90 = 157 pages
3) [12, 34, 67] and 
Max number of pages is allocated to student
1 with 12 + 34 + 67 = 113 pages

Of the 3 cases, Option 3 has the minimum pages = 113.

Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

The idea is to use Binary Search. We fix a value for the number of pages as mid of current minimum and maximum. We initialize minimum and maximum as 0 and sum-of-all-pages respectively. If a current mid can be a solution, then we search on the lower half, else we search in higher half.

Now the question arises, how to check if a mid value is feasible or not? Basically, we need to check if we can assign pages to all students in a way that the maximum number doesn’t exceed current value. To do this, we sequentially assign pages to every student while the current number of assigned pages doesn’t exceed the value. In this process, if the number of students becomes more than m, then the solution is not feasible. Else feasible.

Below is an implementation of above idea.

C++

 // C++ program for optimal allocation of pages #include using namespace std;    // Utility function to check if current minimum value // is feasible or not. bool isPossible(int arr[], int n, int m, int curr_min) {     int studentsRequired = 1;     int curr_sum = 0;        // iterate over all books     for (int i = 0; i < n; i++)     {         // check if current number of pages are greater         // than curr_min that means we will get the result         // after mid no. of pages         if (arr[i] > curr_min)             return false;            // count how many students are required         // to distribute curr_min pages         if (curr_sum + arr[i] > curr_min)         {             // increment student count             studentsRequired++;                // update curr_sum             curr_sum = arr[i];                // if students required becomes greater             // than given no. of students,return false             if (studentsRequired > m)                 return false;         }            // else update curr_sum         else             curr_sum += arr[i];     }     return true; }    // function to find minimum pages int findPages(int arr[], int n, int m) {     long long sum = 0;        // return -1 if no. of books is less than     // no. of students     if (n < m)         return -1;        // Count total number of pages     for (int i = 0; i < n; i++)         sum += arr[i];        // initialize start as 0 pages and end as     // total pages     int start = 0, end = sum;     int result = INT_MAX;        // traverse until start <= end     while (start <= end)     {         // check if it is possible to distribute         // books by using mid as current minimum         int mid = (start + end) / 2;         if (isPossible(arr, n, m, mid))         {             // if yes then find the minimum distribution             result = min(result, mid);                // as we are finding minimum and books             // are sorted so reduce end = mid -1             // that means             end = mid - 1;         }            else             // if not possible means pages should be             // increased so update start = mid + 1             start = mid + 1;     }        // at-last return minimum no. of  pages     return result; }    // Drivers code int main() {     //Number of pages in books     int arr[] = {12, 34, 67, 90};     int n = sizeof arr / sizeof arr;     int m = 2; //No. of students        cout << "Minimum number of pages = "          << findPages(arr, n, m) << endl;     return 0; }

Java

 // Java program for optimal allocation of pages    public class GFG  {     // Utility method to check if current minimum value     // is feasible or not.     static boolean isPossible(int arr[], int n, int m, int curr_min)     {         int studentsRequired = 1;         int curr_sum = 0;                 // iterate over all books         for (int i = 0; i < n; i++)         {             // check if current number of pages are greater             // than curr_min that means we will get the result             // after mid no. of pages             if (arr[i] > curr_min)                 return false;                     // count how many students are required             // to distribute curr_min pages             if (curr_sum + arr[i] > curr_min)             {                 // increment student count                 studentsRequired++;                         // update curr_sum                 curr_sum = arr[i];                         // if students required becomes greater                 // than given no. of students,return false                 if (studentsRequired > m)                     return false;             }                     // else update curr_sum             else                 curr_sum += arr[i];         }         return true;     }             // method to find minimum pages     static int findPages(int arr[], int n, int m)     {         long sum = 0;                 // return -1 if no. of books is less than         // no. of students         if (n < m)             return -1;                 // Count total number of pages         for (int i = 0; i < n; i++)             sum += arr[i];                 // initialize start as 0 pages and end as         // total pages         int start = 0, end = (int) sum;         int result = Integer.MAX_VALUE;                 // traverse until start <= end         while (start <= end)         {             // check if it is possible to distribute             // books by using mid is current minimum             int mid = (start + end) / 2;             if (isPossible(arr, n, m, mid))             {                 // if yes then find the minimum distribution                 result = Math.min(result, mid);                         // as we are finding minimum and books                 // are sorted so reduce end = mid -1                 // that means                 end = mid - 1;             }                     else                 // if not possible means pages should be                 // increased so update start = mid + 1                 start = mid + 1;         }                 // at-last return minimum no. of  pages         return result;     }            // Driver Method     public static void main(String[] args)     {         //Number of pages in books         int arr[] = {12, 34, 67, 90};                   int m = 2; //No. of students                 System.out.println("Minimum number of pages = " +                           findPages(arr, arr.length, m));     } }

Python3

 # Python3 program for optimal allocation of pages    # Utility function to check if  # current minimum value is feasible or not. def isPossible(arr, n, m, curr_min):     studentsRequired = 1     curr_sum = 0        # iterate over all books     for i in range(n):            # check if current number of pages are          # greater than curr_min that means          # we will get the result after         # mid no. of pages         if (arr[i] > curr_min):             return False            # count how many students are required         # to distribute curr_min pages         if (curr_sum + arr[i] > curr_min):                # increment student count             studentsRequired += 1                # update curr_sum             curr_sum = arr[i]                # if students required becomes greater             # than given no. of students, return False             if (studentsRequired > m):                 return False            # else update curr_sum         else:             curr_sum += arr[i]        return True    # function to find minimum pages def findPages(arr, n, m):        sum = 0        # return -1 if no. of books is      # less than no. of students     if (n < m):         return -1        # Count total number of pages     for i in range(n):         sum += arr[i]        # initialize start as 0 pages and      # end as total pages     start, end = 0, sum     result = 10**9        # traverse until start <= end     while (start <= end):            # check if it is possible to distribute         # books by using mid as current minimum         mid = (start + end) // 2         if (isPossible(arr, n, m, mid)):                # if yes then find the minimum distribution             result = min(result, mid)                # as we are finding minimum and books             # are sorted so reduce end = mid -1             # that means             end = mid - 1            else:             # if not possible means pages should be             # increased so update start = mid + 1             start = mid + 1        # at-last return minimum no. of pages     return result    # Driver Code    # Number of pages in books arr = [12, 34, 67, 90] n = len(arr) m = 2   # No. of students    print("Minimum number of pages = ",                findPages(arr, n, m))    # This code is contributed by Mohit Kumar

C#

 // C# program for optimal // allocation of pages using System;    class GFG {    // Utility function to check  // if current minimum value  // is feasible or not. static bool isPossible(int []arr, int n,                         int m, int curr_min) {     int studentsRequired = 1;     int curr_sum = 0;        // iterate over all books     for (int i = 0; i < n; i++)     {         // check if current number of          // pages are greater than curr_min         // that means we will get the          // result after mid no. of pages         if (arr[i] > curr_min)             return false;            // count how many students          // are required to distribute         // curr_min pages         if (curr_sum + arr[i] > curr_min)         {             // increment student count             studentsRequired++;                // update curr_sum             curr_sum = arr[i];                // if students required becomes              // greater than given no. of              // students, return false             if (studentsRequired > m)                 return false;         }            // else update curr_sum         else             curr_sum += arr[i];     }     return true; }    // function to find minimum pages static int findPages(int []arr,                       int n, int m) {     long sum = 0;        // return -1 if no. of books      // is less than no. of students     if (n < m)         return -1;        // Count total number of pages     for (int i = 0; i < n; i++)         sum += arr[i];        // initialize start as 0 pages      // and end as total pages     int start = 0, end = (int)sum;     int result = int.MaxValue;        // traverse until start <= end     while (start <= end)     {         // check if it is possible to          // distribute books by using          // mid as current minimum         int mid = (start + end) / 2;         if (isPossible(arr, n, m, mid))         {             // if yes then find the             // minimum distribution             result = Math.Min(result, mid);                // as we are finding minimum and              // books are sorted so reduce             // end = mid -1 that means             end = mid - 1;         }            else             // if not possible means pages              // should be increased so update             // start = mid + 1             start = mid + 1;     }        // at-last return      // minimum no. of pages     return result; }    // Drivers code static public void Main () {        //Number of pages in books int []arr = {12, 34, 67, 90}; int n = arr.Length; int m = 2; // No. of students    Console.WriteLine("Minimum number of pages = " +                           findPages(arr, n, m)); } }    // This code is contributed by anuj_67.

PHP

 \$curr_min)             return false;            // count how many students are         // required to distribute         // curr_min pages         if (\$curr_sum + \$arr[\$i] > \$curr_min)         {             // increment student count             \$studentsRequired++;                // update curr_sum             \$curr_sum = \$arr[\$i];                // if students required becomes              // greater than given no. of              // students, return false             if (\$studentsRequired > \$m)                 return false;         }            // else update curr_sum         else             \$curr_sum += \$arr[\$i];     }     return true; }    // function to find minimum pages function findPages(\$arr, \$n, \$m) {     \$sum = 0;        // return -1 if no. of books is      // less than no. of students     if (\$n < \$m)         return -1;        // Count total number of pages     for (\$i = 0; \$i < \$n; \$i++)         \$sum += \$arr[\$i];        // initialize start as 0 pages      // and end as total pages     \$start = 0;     \$end = \$sum;     \$result = PHP_INT_MAX;        // traverse until start <= end     while (\$start <= \$end)     {         // check if it is possible to          // distribute books by using          // mid as current minimum         \$mid = (int)(\$start + \$end) / 2;         if (isPossible(\$arr, \$n, \$m, \$mid))         {             // if yes then find the minimum              // distribution             \$result = min(\$result, \$mid);                // as we are finding minimum and              // books are sorted so reduce              // end = mid -1 that means             \$end = \$mid - 1;         }            else             // if not possible means pages              // should be increased so update              // start = mid + 1             \$start = \$mid + 1;     }        // at-last return minimum     // no. of pages     return \$result; }    // Driver code    // Number of pages in books \$arr = array(12, 34, 67, 90); \$n = count(\$arr);  \$m = 2; // No. of students    echo "Minimum number of pages = ",     findPages(\$arr, \$n, \$m), "\n";    // This code is contributed by Sach_Code ?>

Output :

Minimum number of pages = 113

This article is contributed by Sahil Chhabra (akku). If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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Improved By : vt_m, Sach_Code, mohit kumar 29