Minimum Number of Boats in Python (Time Complexity Analysis Included)

You are given an array weights where weights[i] is the weight of the i^th student, and an infinite number of boats where each boat can carry a maximum weight of maxWeight. Each boat carries at most two people at the same time, provided the sum of the weight of those people is at most maxWeight and the absolute difference of their weights shouldn't exceed maxDiff.

Return the minimum number of boats to carry every given student.

Example 1:

Input: weights = [2, 4, 2, 1], maxWeight = 6, maxDiff = 1
Output: 3
Explanation: 3 boats: (1, 2), (2) and (4)

Example 2:

Input: weights = [81, 37, 32, 88, 55, 93, 45, 72],
    maxWeight = 100, maxDiff = 10
Output: 6
Explanation: 6 boats (81), (37, 32), (88), (55, 45), (93), (72)

Understanding the Problem

The core challenge of this problem is to minimize the number of boats required to transport all students given the constraints on weight and weight difference. This problem is significant in scenarios like evacuation planning, resource allocation, and load balancing.

Potential pitfalls include not considering the weight difference constraint or not optimizing the pairing of students to minimize the number of boats.

Approach

To solve this problem, we can use a greedy algorithm. The idea is to sort the weights and try to pair the lightest and heaviest students that can share a boat. This approach ensures that we minimize the number of boats used.

Initial naive solution might involve trying all possible pairs, but this would be computationally expensive. Instead, sorting the array and using a two-pointer technique is more efficient.

Optimized Solution

1. **Sort the weights array.**

2. **Use two pointers:** one starting at the beginning (lightest) and one at the end (heaviest).

3. **Try to pair the lightest and heaviest students:** If they can share a boat, move both pointers inward. If not, move only the pointer for the heaviest student.

4. **Count the number of boats used.**

Algorithm

Here is a step-by-step breakdown of the algorithm:

Sort the weights array.
Initialize two pointers: `left` at the start and `right` at the end of the array.
Initialize a counter for the number of boats.
While `left` is less than or equal to `right`:
- Check if the students at `left` and `right` can share a boat.
- If they can, increment `left` and decrement `right`.
- If they cannot, decrement `right` only.
- Increment the boat counter.
Return the boat counter.

Code Implementation

def min_boats(weights, maxWeight, maxDiff):
    # Sort the weights array
    weights.sort()
    
    # Initialize two pointers
    left, right = 0, len(weights) - 1
    boats = 0
    
    # Use two-pointer technique to find the minimum number of boats
    while left <= right:
        # If the lightest and heaviest can share a boat
        if weights[left] + weights[right] <= maxWeight and abs(weights[left] - weights[right]) <= maxDiff:
            left += 1  # Move the left pointer inward
        # Move the right pointer inward in both cases
        right -= 1
        # Increment the boat counter
        boats += 1
    
    return boats

# Example usage
print(min_boats([2, 4, 2, 1], 6, 1))  # Output: 3
print(min_boats([81, 37, 32, 88, 55, 93, 45, 72], 100, 10))  # Output: 6

Complexity Analysis

The time complexity of this approach is O(n log n) due to the sorting step. The space complexity is O(1) as we are using only a few extra variables.

Compared to a naive O(n^2) approach, this is significantly more efficient.

Edge Cases

Potential edge cases include:

All students having the same weight.
Weights such that no two students can share a boat.
Empty weights array.

These cases are handled by the algorithm as it inherently checks the constraints and processes the array accordingly.

Testing

To test the solution comprehensively, consider:

Simple cases with a few students.
Cases with maximum constraints.
Edge cases as mentioned above.

Using a testing framework like `unittest` in Python can help automate and validate these tests.

Thinking and Problem-Solving Tips

When approaching such problems:

Break down the problem into smaller parts.
Consider edge cases and constraints early.
Think about different approaches and their trade-offs.
Practice similar problems to improve problem-solving skills.

Conclusion

Understanding and solving this problem helps in developing skills in greedy algorithms and two-pointer techniques. These are widely applicable in optimization problems.

Practice and exploration of similar problems can further enhance your problem-solving abilities.

Additional Resources

For further reading and practice:

LeetCode - Practice problems on greedy algorithms.
GeeksforGeeks - Tutorials and problems on two-pointer techniques.
Python unittest Documentation - For writing and running tests.