Minimum Space Wasted From Packaging

Problem Description

You have n packages that you are trying to place in boxes, one package in each box. There are m suppliers that each produce boxes of different sizes (with infinite supply). A package can be placed in a box if the size of the package is less than or equal to the size of the box. The package sizes are given as an integer array packages, and the suppliers are given as a 2D integer array boxes. You want to choose a single supplier and use boxes from them such that the total wasted space is minimized. Return the minimum total wasted space by choosing the box supplier optimally, or -1 if it is impossible to fit all the packages inside boxes. The answer may be large, return it modulo 10^9 + 7.

Key Insights

• Each package can only be placed in a box of equal or larger size.
• The total wasted space is calculated as the difference between the box size and the package size summed across all packages.
• It's crucial to choose the right supplier with box sizes that can accommodate all packages.
• Sorting both packages and box sizes helps in efficiently determining the best fit using binary search.

Space and Time Complexity

Time Complexity: O(n log n + m * k log k) where n is the number of packages, m is the number of suppliers, and k is the average number of box sizes per supplier.
Space Complexity: O(m * k) for storing the boxes.

Solution

To solve this problem, we can follow these steps:

1. Sort the packages in ascending order.
2. For each supplier's box sizes, sort the box sizes.
3. For each package, use binary search to find the smallest box that can accommodate the package.
4. Calculate the total wasted space for each supplier, and track the minimum wasted space across all suppliers.
5. If no supplier can accommodate all packages, return -1.

The data structures used include arrays (for packages and boxes) and a sorted list for binary searching box sizes.

Code Solutions