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Book Allocation Problem

Hard

Minimise the largest allocation

Problem

Given an array nums of n integers, where nums[i] represents the number of pages in the i-th book, and an integer m representing the number of students, allocate all the books to the students so that each student gets at least one book, each book is allocated to only one student, and the allocation is contiguous. Allocate the books to m students in such a way that the maximum number of pages assigned to a student is minimized. If the allocation of books is not possible, return -1.

In simple words

Binary-search the page limit: check how many students it needs, shrinking until it fits.

The idea

Binary search the maximum number of pages any student may receive, and greedily hand out books in order counting how many students that limit needs. Fewer students needed than allowed means the limit can be tightened.

The trick

  • Search range: max(pages) to sum(pages).
  • Impossible when there are more students than books.
  • This 'minimise the maximum' shape covers painter's partition and split-array too.
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candidates114

Step 1 of 26. Brute force: try every page-limit until the books fit 2 students. Values: 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203. candidates 114.

1/26
Brute force
timeO(sum·n)spaceO(1)

Try every limit.

1let best = -1;2for (let cap = maxPage; cap <= sum && best < 0; cap++)3  if (studentsNeeded(cap) <= students) best = cap;4return best;

Input

array
[90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, …]

Memory

try
—
candidates
114
limit
—

Output

limit
—
answer
—

Check yourself

3 quick questions about this walkthrough. A wrong answer costs nothing.

Examples

Example 1

Input:
pages = [12, 34, 67, 90], students = 2
Output:
113
Explanation:
Split so the busiest student reads 113 pages.

Example 2

Input:
pages = [25, 46, 28, 49, 24], students = 4
Output:
71
Explanation:
Fairest max load is 71.

Example 3

Input:
pages = [5, 17, 100, 11], students = 4
Output:
100
Explanation:
One book of 100 forces max 100.

Constraints

  • 1 <= n, m <= 10^4
  • 1 <= nums[i] <= 10^5

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