Allocate Minimum Pages Binary Search on Answer in DSA Typescript - 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 on the answer 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 efficient searching in this range.

intermediate

What does the 'isPossible' function check in the Allocate Minimum Pages problem?

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

beginner

In the context of Allocate Minimum Pages, what is the significance of the lower bound in binary search?

The lower bound is the maximum number of pages in a single book, since no student can get less than that.

advanced

What is the time complexity of the Allocate Minimum Pages solution using binary search on answer?

O(N * log(S)), where N is the number of books and S is the sum of all pages.

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

A Sum of all pages

B Maximum pages in a single book

C Zero

D Number of students

The minimum possible maximum pages is at least the largest single book's pages.

What does the 'isPossible' function return if the current max pages limit is too small?

A False

B True

C Throws error

D Returns number of students

If the max pages limit is too small, allocation is not possible, so it returns false.

Which technique is used to find the minimum maximum pages efficiently?

A Linear search

B Depth-first search

C Breadth-first search

D Binary search on answer

Binary search on the answer range is used to efficiently find the minimum maximum pages.

If the number of students is more than the number of books, what is the result?

A Allocation is impossible

B Each student gets at least one book

C Some students get no books

D Maximum pages is zero

Some students will get no books if students exceed books.

What is the upper bound for binary search in this problem?

A Sum of all pages

B Zero

C Number of students

D Maximum pages in a single book

The maximum possible maximum pages is the sum of all pages if one student reads all books.

Explain how binary search on the answer works in the Allocate Minimum Pages problem.

Think about narrowing down the possible maximum pages by checking if allocation is possible.

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

It tests if a given max pages limit can work for allocation.

Practice

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

easy

A. To assign pages randomly among 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 asks to allocate books to students such that the maximum pages assigned to any student is minimized.

2. Step 2: Identify the binary search approach

   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 [OK]

Hint: Minimize max pages per student using binary search [OK]

Common Mistakes:

• Trying to maximize pages instead of minimizing
• Ignoring the maximum pages constraint
• Confusing total pages with max pages per student

2. Which of the following is the correct TypeScript syntax to declare the binary search boundaries for the Allocate Minimum Pages problem given an array pages?

easy

A. let low = pages.length; let high = Math.max(...pages);

B. let low = Math.min(...pages); let high = pages.length;

C. let low = 0; let high = pages.reduce((a, b) => a + b, 0);

D. let low = 0; let high = Math.min(...pages);

Solution

1. Step 1: Determine low and high boundaries

   Low is minimum possible max pages (0 or max single book pages), high is sum of all pages (max total pages).

2. Step 2: Check correct syntax for sum

   Using reduce sums all pages correctly for high boundary.

3. Final Answer:

   let low = 0; let high = pages.reduce((a, b) => a + b, 0); -> Option C

4. Quick Check:

   low = 0, high = sum pages [OK]

Hint: Sum array for high, zero or max page for low [OK]

Common Mistakes:

• Setting low as pages.length instead of page count
• Using min pages as high boundary
• Incorrect sum calculation syntax

3. Consider the following TypeScript code snippet for checking feasibility in Allocate Minimum Pages problem:

function isPossible(pages: number[], students: number, maxPages: number): boolean {
  let requiredStudents = 1;
  let currentSum = 0;
  for (let page of pages) {
    if (page > maxPages) return false;
    if (currentSum + page > maxPages) {
      requiredStudents++;
      currentSum = page;
    } else {
      currentSum += page;
    }
  }
  return requiredStudents <= students;
}

What is the output of isPossible([10, 20, 30, 40], 2, 50)?

medium

A. false

B. SyntaxError

C. true

D. RuntimeError

Solution

1. Step 1: Trace allocation with maxPages = 50

   Assign pages: 10+20=30 (still <=50), add 30 would exceed 50, so new student starts with 30, then 40 exceeds 50 if added, so new student starts with 40.

2. Step 2: Count required students

   We used 3 students but only 2 allowed, so requiredStudents=3 > 2 means return false. But check carefully: 10+20=30, next 30 alone fits in second student, 40 alone needs third student.

3. Final Answer:

   false -> Option A

4. Quick Check:

   Required students = 3 > 2, so false [OK]

Hint: Count students needed, compare with allowed [OK]

Common Mistakes:

• Ignoring that a single book can exceed maxPages
• Miscounting students when sum exceeds maxPages
• Assuming sum of all pages fits in one student

4. The following TypeScript function attempts to find the minimum maximum pages but has a bug. Identify the bug.

function findMinPages(pages: number[], students: number): number {
  let low = Math.min(...pages);
  let high = pages.reduce((a, b) => a + b, 0);
  let result = high;
  while (low <= high) {
    let mid = Math.floor((low + high) / 2);
    if (isPossible(pages, students, mid)) {
      result = mid;
      high = mid - 1;
    } else {
      low = mid + 1;
    }
  }
  return result;
}

medium

A. High should be min page, not sum of pages

B. Low should start from max page, not min page

C. Mid calculation causes infinite loop

D. Result variable is not updated

Solution

1. Step 1: Analyze low boundary initialization

   Low should be the maximum single book pages, because no student can get less than the largest book.

2. Step 2: Identify impact of wrong low

   Starting low at minimum page can cause incorrect feasibility checks and longer search.

3. Final Answer:

   Low should start from max page, not min page -> Option B

4. Quick Check:

   Low = max page [OK]

Hint: Low boundary = max single book pages [OK]

Common Mistakes:

• Setting low to min page instead of max page
• Confusing high boundary with low
• Not updating result inside loop

5. Given books with pages [12, 34, 67, 90] and 3 students, what is the minimum maximum pages allocated to any student using binary search on answer?

hard

A. 90

B. 113

C. 67

D. 34

Solution

1. Step 1: Set search boundaries

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

2. Step 2: Binary search feasibility checks

   Mid = (90+203)//2=146. Check if possible with maxPages=146: allocate [12+34+67=113] to student1, 90 to student2, only 2 students needed <=3, possible. So high=145.

3. Step 3: Narrow down to minimum max pages

   Continuing: mid=103 possible [12+34=46,67,90] 3 students. mid=96 possible. mid=92 possible. mid=90: [12+34=46], [67], [90] all <=90, 3 students <=3 feasible, high=89. Loop ends with result=90.

4. Step 4: Re-evaluate minimum max pages

   Check if 113 is feasible with 3 students: [12+34+67=113], [90] 2 students needed <=3, so 113 is feasible but not minimum.

5. Step 5: Check if 90 is feasible

   Allocations: [12+34=46], [67], [90] 3 students needed <=3, feasible.

6. Final Answer:

   90 -> Option A

7. Quick Check:

   Minimum max pages = 90 [OK]

Hint: Binary search between max page and sum pages [OK]

Common Mistakes:

• Misallocating at max=90 thinking 4 students needed
• Miscounting students during allocation
• Confusing with 2 students (113)