Allocate Minimum Pages Binary Search on Answer in DSA C++ - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the main goal of the Allocate Minimum Pages problem?

To divide an array of book pages among students such that the maximum pages assigned to a student is minimized.

intermediate

Why do we use binary search in the Allocate Minimum Pages problem?

Because the answer (minimum maximum pages) lies between the maximum single book pages and the sum of all pages, allowing us to search efficiently in this range.

intermediate

What does the 'isPossible' function check in this problem?

It checks if it is possible to allocate books to students such that no student reads more than a given maximum number of pages.

beginner

What is the initial search range for binary search in this problem?

The lower bound is the maximum pages in a single book, and the upper bound is the sum of all pages in all books.

intermediate

How does the binary search update its range during the solution?

If allocation is possible with mid as max pages, it tries to find a smaller max by moving upper bound to mid; otherwise, it moves lower bound to mid + 1.

What is the lower bound for binary search in Allocate Minimum Pages?

A Maximum pages in a single book

B Sum of all pages

C Zero

D Number of students

The minimum maximum pages cannot be less than the largest single book.

What does the 'isPossible' function return if allocation is not feasible?

A True

B Number of students

C False

D Sum of pages

'isPossible' returns false if allocation with given max pages is not possible.

If allocation is possible with mid pages, what should binary search do next?

A Increase upper bound

B Increase lower bound

C Stop searching

D Decrease upper bound

Try to find a smaller maximum by decreasing upper bound.

What is the time complexity of the binary search approach for this problem?

A O(n log(sum of pages))

B O(n^2)

C O(log n)

D O(n)

Binary search over the range of sum of pages with O(n) feasibility check.

What does the final answer represent in Allocate Minimum Pages?

A Number of students

B Maximum pages assigned to any student

C Total pages of all books

D Minimum pages assigned to any student

It is the minimized maximum pages assigned to any student.

Explain how binary search is applied to solve the Allocate Minimum Pages problem.

Think about how to guess the answer and check if it works.

Describe the role of the 'isPossible' function in the Allocate Minimum Pages problem.

It tests if a guess is valid.

Practice

1. What is the main goal of the Allocate Minimum Pages problem using binary search on the answer?

easy

A. To assign pages randomly to students

B. To maximize the total number of pages assigned

C. To find the average number of pages per student

D. To minimize the maximum number of pages assigned to a student

Solution

1. Step 1: Understand the problem objective

   The problem wants to divide book pages among students so that the student with the most pages has as few pages as possible.

2. Step 2: Recognize the use of binary search on answer

   We use binary search on the range of possible maximum pages to find the minimum feasible maximum.

3. Final Answer:

   To minimize the maximum number of pages assigned to a student -> Option D

4. Quick Check:

   Goal = minimize max pages per student [OK]

Hint: Focus on minimizing the largest pages assigned [OK]

Common Mistakes:

• Thinking it maximizes total pages
• Confusing average with maximum
• Ignoring the binary search approach

2. Which of the following is the correct way to initialize the binary search boundaries for the Allocate Minimum Pages problem in C++?

easy

A. int low = max pages in a single book, high = sum of all pages;

B. int low = 0, high = sum of all pages;

C. int low = 0, high = max pages in a single book;

D. int low = sum of all pages, high = max pages in a single book;

Solution

1. Step 1: Determine the minimum and maximum boundaries

   The minimum possible max pages is the largest single book (no student can get less than that), and the maximum is the sum of all pages (one student gets all books).

2. Step 2: Set binary search boundaries accordingly

   So, low = max pages in a single book, high = sum of all pages.

3. Final Answer:

   int low = max pages in a single book, high = sum of all pages; -> Option A

4. Quick Check:

   low = max single book, high = total pages [OK]

Hint: Low = max single book pages, High = sum of all pages [OK]

Common Mistakes:

• Setting low to 0 which is invalid
• Swapping low and high values
• Using average pages as boundary

3. What will be the output of the following C++ code snippet for the Allocate Minimum Pages problem?

int pages[] = {10, 20, 30, 40};
int n = 4, students = 2;

// Function to check feasibility
bool isPossible(int mid) {
    int studentCount = 1, pageSum = 0;
    for (int i = 0; i < n; i++) {
        if (pages[i] > mid) return false;
        if (pageSum + pages[i] > mid) {
            studentCount++;
            pageSum = pages[i];
            if (studentCount > students) return false;
        } else {
            pageSum += pages[i];
        }
    }
    return true;
}

int low = 40, high = 100, result = high;
while (low <= high) {
    int mid = (low + high) / 2;
    if (isPossible(mid)) {
        result = mid;
        high = mid - 1;
    } else {
        low = mid + 1;
    }
}
std::cout << result << std::endl;

medium

A. 50

B. 70

C. 60

D. 40

Solution

1. Step 1: Understand the feasibility check

   The function checks if pages can be allocated so no student gets more than mid pages.

2. Step 2: Binary search to find minimum max pages

   Try mid values between 40 and 100. The minimum feasible max pages is 60 (assign {10,20,30} to student 1 and {40} to student 2).

3. Final Answer:

   60 -> Option C

4. Quick Check:

   Minimum max pages = 60 [OK]

Hint: Check feasibility by simulating allocation [OK]

Common Mistakes:

• Not updating student count correctly
• Choosing mid incorrectly
• Ignoring single book page size

4. Identify the bug in the following C++ code snippet for the Allocate Minimum Pages problem:

bool isPossible(int mid, int pages[], int n, int students) {
    int studentCount = 1, pageSum = 0;
    for (int i = 0; i < n; i++) {
        if (pageSum + pages[i] <= mid) {
            pageSum += pages[i];
        } else {
            studentCount++;
            pageSum = pages[i];
        }
    }
    return studentCount <= students;
}

medium

A. Missing check if a single book has pages more than mid

B. Incorrect increment of studentCount

C. Page sum should reset to zero instead of pages[i]

D. Loop should run from i = 1 instead of 0

Solution

1. Step 1: Analyze the feasibility function

   The function tries to allocate books without exceeding mid pages per student.

2. Step 2: Identify missing validation

   It does not check if any single book has pages more than mid, which makes allocation impossible.

3. Final Answer:

   Missing check if a single book has pages more than mid -> Option A

4. Quick Check:

   Check single book pages <= mid [OK]

Hint: Always check single book pages against mid [OK]

Common Mistakes:

• Ignoring single book size check
• Resetting pageSum incorrectly
• Wrong loop start index

5. Given the books with pages {12, 34, 67, 90} and 2 students, what is the minimum maximum number of pages a student can get when using binary search on answer to allocate books optimally?

hard

A. 90

B. 113

C. 101

D. 67

Solution

1. Step 1: Set binary search boundaries

   Low = max single book = 90, High = sum = 12+34+67+90=203.

2. Step 2: Use binary search to find minimum max pages

   Check mid values. The minimum feasible max pages is 113 (assign {12,34,67}=113 to one student and {90} to the other).

3. Final Answer:

   113 -> Option B

4. Quick Check:

   Minimum max pages = 113 [OK]

Hint: Binary search between max single book and total pages [OK]

Common Mistakes:

• Choosing max single book as answer
• Choosing sum of pages as answer
• Not testing feasibility properly