#1739

Hard

auto_awesomeBinary search over the valid answer space

LeetCode Problem Workspace

Building Boxes

Optimize the number of boxes touching the floor in a cubic room using binary search to minimize floor occupancy.

Problem Statement

You are given a cubic room where each side has a length of n units, and you need to place n unit-sized boxes inside this room. The goal is to minimize the number of boxes that touch the floor, given specific placement rules.

To solve the problem, you need to find the minimum number of boxes that must touch the floor after placing all n boxes in the room. Use binary search on the valid answer space to efficiently compute the result.

Examples

Example 1

Input: n = 3

Output: 3

The figure above is for the placement of the three boxes. These boxes are placed in the corner of the room, where the corner is on the left side.

Example 2

Input: n = 4

Output: 3

The figure above is for the placement of the four boxes. These boxes are placed in the corner of the room, where the corner is on the left side.

Example 3

Input: n = 10

Output: 6

The figure above is for the placement of the ten boxes. These boxes are placed in the corner of the room, where the corner is on the back side.

Constraints

1 <= n <= 109

Solution Approach

Binary Search on Answer Space

Use binary search over the valid range of answers to find the minimum number of boxes touching the floor. The approach explores the space of possible floor-touching box configurations and narrows down the solution efficiently.

Greedy Placement

Once the binary search finds the valid answer range, a greedy strategy helps to maximize the number of boxes placed without touching the floor. This ensures the optimal floor occupancy.

Mathematical Calculation of Layers

Mathematically calculate how many layers can fit without touching the floor based on the dimensions of the cubic room. This gives insight into the optimal distribution of boxes and aids in solving the problem through binary search.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the binary search over the answer space, and the space complexity is influenced by the approach for storing possible configurations. Both complexities are determined by the chosen algorithm and approach to the problem.

What Interviewers Usually Probe

Able to identify the correct application of binary search over the solution space.
Can describe the use of greedy algorithms to optimize floor placement.
Comfortable with applying mathematical reasoning to break down the box placement in a cubic space.

Common Pitfalls or Variants

Common pitfalls

Overlooking the binary search's application to the answer space, leading to inefficiency.
Failing to properly apply greedy placement to maximize the use of space.
Not considering mathematical relationships that govern how boxes can fit in the room.

Follow-up variants

Consider modifying the constraints by increasing n to test scalability.
Explore different ways of calculating box placement, considering different room dimensions.
Alter the number of boxes to be placed in the room to create varied test cases.

FAQ

What is the primary pattern for solving the 'Building Boxes' problem?

The primary pattern involves using binary search over the valid answer space to minimize the number of boxes touching the floor.

How does greedy placement work in this problem?

Greedy placement helps maximize the number of boxes that can be placed without touching the floor after binary search identifies the answer range.

What are the time and space complexities of solving this problem?

The time complexity is mainly influenced by binary search, and the space complexity depends on the storage of configurations.

How can mathematical reasoning be applied to solve this problem?

Mathematical calculations help determine the optimal layers in which boxes can be placed, which is essential for minimizing the number of boxes touching the floor.

What is the best way to test scalability for this problem?

To test scalability, increase the value of n and measure how well the binary search approach performs with larger inputs.

terminal

Solution

Solution 1: Mathematical Rule

According to the problem description, the box with the highest number of layers needs to be placed in the corner of the wall, and the arrangement of the boxes is in a step-like shape, which can minimize the number of boxes touching the ground.

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class Solution:
    def minimumBoxes(self, n: int) -> int:
        s, k = 0, 1
        while s + k * (k + 1) // 2 <= n:
            s += k * (k + 1) // 2
            k += 1
        k -= 1
        ans = k * (k + 1) // 2
        k = 1
        while s < n:
            ans += 1
            s += k
            k += 1
        return ans

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