Maximum Units on a Truck

To solve this problem, you would follow a simple “greedy” strategy: you should always load boxes with more units per box first, because that way you can maximize the total number of units that can be put on the truck.

Python solution:

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class Solution:
    def maximumUnits(self, boxTypes: List[List[int]], truckSize: int) -> int:
        # Sort the box types in descending order of the number of units per box
        boxTypes.sort(key=lambda x: -x[1])

        units = 0
        for numberOfBoxes, numberOfUnitsPerBox in boxTypes:
            if truckSize <= 0:
                break

            # Determine the number of boxes of this type to load
            boxesToLoad = min(numberOfBoxes, truckSize)

            # Load the boxes onto the truck and update the total number of units
            units += boxesToLoad * numberOfUnitsPerBox

            # Update the remaining size of the truck
            truckSize -= boxesToLoad

        return units

In this solution, we first sort the boxTypes in descending order of numberOfUnitsPerBox using the built-in Python sort method and a custom sorting key that negates numberOfUnitsPerBox (this results in a descending order sort because Python’s sort method sorts in ascending order by default). We then iterate over the sorted boxTypes and for each type, we load as many boxes of that type onto the truck as we can (without exceeding truckSize). We update the total number of units and the remaining size of the truck accordingly. Finally, we return the total number of units.

This solution uses a greedy approach.

In a greedy approach, we make the choice that seems to be the best at each step. Here, at each step, we’re picking the box type with the maximum units per box that we can put on the truck. This is a “greedy” choice because it seems like the best choice at each step without considering the future.

In many cases, the greedy approach might not lead to an optimal solution. However, in this problem, it guarantees the optimal solution because picking boxes with more units first will always lead to a maximum total number of units. It’s crucial to understand when a problem can be solved using a greedy approach and when it can’t, and this understanding often comes with practice.

The reason we need to sort is that we want to prioritize the box types that have the most units per box. By sorting the list in descending order based on the number of units per box, we can ensure that we’re always choosing the box types that contribute the most units to our total.

This aligns with our greedy strategy: at each step, we want to make the decision that appears to be the best. In this case, the “best” decision is to always choose the box type with the most units per box, hence the sorting.

By sorting and then iterating from the start of the sorted list, we’re effectively prioritizing boxes with more units. We continue picking from this sorted list until our truck is full (i.e., we’ve reached our truckSize limit), ensuring that we’ve maximized the number of units on the truck.

Abstract Representation of the Problem

We have a set of groups, each having a quantity and a value. We also have a limit on the total quantity we can select. Our task is to select groups such that the total quantity does not exceed the limit and the total value is maximized.

Each group can be represented as a tuple (quantity, value). The groups are sorted in descending order by value. We iterate through the sorted groups, adding the quantity from each group to a running total until we reach the limit. As we add the quantity from each group, we also add the corresponding value to a total value counter. The goal is to maximize the total value.

This abstract problem is a common pattern in optimization problems, particularly in resource allocation where we want to maximize or minimize some value given certain constraints. This specific problem is a variant of the knapsack problem.

Identifying Problem Isomorphism

“Maximum Units on a Truck” can be approximately mapped to “Fractional Knapsack Problem”.

In both problems, we have a set of items with individual quantities and values, and a limit on the total quantity that can be taken. The goal is to maximize the total value within the given constraints.

“Fractional Knapsack Problem” is a simpler problem compared to “Maximum Units on a Truck”. In the “Fractional Knapsack Problem”, you can take fractions of items to reach the optimal solution, which is not allowed in “Maximum Units on a Truck”. In “Maximum Units on a Truck”, we have to take whole items (or boxes), which can make the problem more complex, especially if the total quantity limit is not a multiple of the item quantities.