3492. Maximum Containers on a Ship

Problem Description

You are given a positive integer n representing an n x n cargo deck on a ship. Each cell on the deck can hold one container with a weight of exactly w.

However, the total weight of all containers, if loaded onto the deck, must not exceed the ship's maximum weight capacity, maxWeight.

Return the maximum number of containers that can be loaded onto the ship.

Intuition

The goal is to determine how many containers can be loaded onto a ship without exceeding its maximum weight capacity. We start by calculating the total possible weight of containers if the deck were filled entirely. This weight is n * n * w, representing the number of containers (n * n since it's a square deck) times the weight of each container (w).

However, this calculated weight might exceed the ship's weight capacity (maxWeight). Therefore, we compare this total potential weight with the maxWeight to decide the actual loading strategy. By taking the minimum of these two values and then dividing by the weight of a single container w, we determine the maximum number of containers that can be placed on the ship. This simple mathematical assessment ensures that neither container nor weight exceeds permissible limits.

Learn more about Math patterns.

Solution Approach

The solution utilizes a straightforward mathematical approach to determine the maximum number of containers that can be positioned on the ship without breaching its maximum weight capacity.

Calculation of Maximum Potential Weight:
- Begin by calculating the theoretical maximum weight that can be carried if every cell on the deck is occupied with a container. This is computed as n * n * w, where n * n gives the total number of possible containers, and w is the weight of each container.
Comparing Against Maximum Weight Capacity:
- Assess whether the calculated potential weight exceeds the ship's maxWeight. We achieve this by evaluating min(n * n * w, maxWeight). This step ensures that we don't consider more weight than permissible by the ship's capacity.
Determining the Maximum Number of Containers:
- Finally, to compute the maximum number of containers that can be loaded without surpassing the weight restriction, divide the result by w. This is expressed as min(n * n * w, maxWeight) // w.

This concise and effective approach involves basic arithmetic operations and cleverly ensures that we respect the constraints while maximizing the number of containers.

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Example Walkthrough

Let's work through a specific example to illustrate the solution approach.

Suppose we have the following parameters:

n = 3: This means we have a 3 x 3 cargo deck, capable of holding up to 9 containers.
w = 2: Each container has a weight of 2 units.
maxWeight = 12: The maximum weight capacity of the ship is 12 units.

Step 1: Calculation of Maximum Potential Weight

Firstly, calculate how much weight the cargo deck would have if every cell were filled with a container. This is given by ( n \times n \times w = 3 \times 3 \times 2 = 18 ) units of weight.

Step 2: Comparing Against Maximum Weight Capacity

Next, compare this potential total weight with the ship’s maximum weight capacity. We take the minimum between these two values. Thus, we evaluate: min(18, 12), which results in 12 units.

Step 3: Determining the Maximum Number of Containers

Finally, to determine the number of containers that can be loaded without exceeding the weight limit, divide the above result by the weight of a single container w. Therefore, we have: ( \text{maxWeightByMass} // w = 12 // 2 = 6 ).

Hence, the maximum number of containers that can be loaded onto the ship while respecting the maximum weight capacity is 6 containers.

Solution Implementation

1class Solution:
2    def maxContainers(self, n: int, w: int, maxWeight: int) -> int:
3        # Calculate the maximum possible total weight of n containers, each with weight w
4        # n * n * w represents the weight of all containers considered if each container could hold an n times weight
5        possible_total_weight = n * n * w
6      
7        # Find the lesser of the calculated total weight and the provided maximum weight capacity
8        limit_weight = min(possible_total_weight, maxWeight)
9      
10        # Determine how many full containers can be loaded without exceeding the weight limit
11        # This is the integer division of the limited weight by the weight of one container
12        return limit_weight // w
13

1class Solution {
2    /**
3     * Calculates the maximum number of containers that can be created.
4     * 
5     * @param n the number representing some quantity of elements
6     * @param w the weight per container
7     * @param maxWeight the maximum allowable weight
8     * @return the maximum number of containers that can be created within the weight limit
9     */
10    public int maxContainers(int n, int w, int maxWeight) {
11        // Calculate the total weight that can be created from n elements with weight w
12        int totalWeight = n * n * w;
13      
14        // Take the minimum between the calculated total weight and the max allowable weight
15        int maxPossibleWeight = Math.min(totalWeight, maxWeight);
16      
17        // Calculate and return the maximum number of containers within the weight limit
18        return maxPossibleWeight / w;
19    }
20}
21

1class Solution {
2public:
3    int maxContainers(int n, int w, int maxWeight) {
4        // Calculate the total potential weight with all containers and weight per container
5        int potentialWeight = n * n * w;
6      
7        // Determine the actual weight that can be handled which is the minimum of potential weight and the maximum allowable weight
8        int actualWeight = std::min(potentialWeight, maxWeight);
9      
10        // Calculate and return the number of containers that can be fully utilized given their individual weight
11        return actualWeight / w;
12    }
13};
14

1/**
2 * Calculates the maximum number of containers that can be filled 
3 * with a given total weight limit.
4 * 
5 * @param n - Number of containers.
6 * @param w - Weight per container.
7 * @param maxWeight - Maximum weight limit that can be accommodated.
8 * @returns Maximum number of completely filled containers.
9 */
10function maxContainers(n: number, w: number, maxWeight: number): number {
11    // Calculate the total possible weight of containers
12    const totalWeight = n * n * w;
13
14    // Determine the effective weight that can be used, which is limited by maxWeight
15    const effectiveWeight = Math.min(totalWeight, maxWeight);
16
17    // Compute the number of completely filled containers
18    // Use bitwise OR with zero to floor the result
19    return (effectiveWeight / w) | 0;
20}
21

Time and Space Complexity

The time complexity of the code is O(1), because the operations performed (multiplication, comparison, and division) take constant time regardless of the input size.

The space complexity is also O(1), as the function only uses a fixed amount of additional space for its operations, independent of the input size.

Learn more about how to find time and space complexity quickly.

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