Maximum Price to Fill a Bag

Maximum Price to Fill a Bag

1. What is this problem about?

The Maximum Price to Fill a Bag interview question is a variation of the classic Fractional Knapsack problem. You are given a set of items, each with a price and a weight, and a bag with a maximum weight capacity. You want to fill the bag such that the total price of the items is maximized. Unlike the 0/1 Knapsack problem, you can take fractions of an item if needed.

The goal is to determine the highest possible value you can carry given the weight constraint. This problem tests your ability to make locally optimal choices to reach a global optimum.

2. Why is this asked in interviews?

This Maximum Price to Fill a Bag coding problem is a staple at companies like Microsoft because it evaluates a candidate's understanding of greedy strategies. It's a fundamental problem that distinguishes between those who understand optimization and those who might default to more complex but unnecessary algorithms like Dynamic Programming. It also tests your ability to implement custom sorting logic, which is a key skill in day-to-day software engineering.

3. Algorithmic pattern used

The Array, Sorting, Greedy interview pattern is the correct approach for this problem. The greedy choice here is to always pick the item with the highest "price-per-unit-weight" ratio.

1. Calculate the price/weight ratio for each item.
2. Sort the items in descending order of this ratio.
3. Iterate through the sorted items and take as much of each as possible until the bag is full.

4. Example explanation

Bag capacity: 10kg. Items:

• A: Price $60, Weight 20kg (Ratio: 3)
• B: Price $100, Weight 50kg (Ratio: 2)
• C: Price $120, Weight 30kg (Ratio: 4)

Sorted by Ratio: C(4), A(3), B(2).

1. Take item C: Bag can only hold 10kg, so we take 1/3 of C.
2. Price = (1/3) * $120 =$40.
3. Bag is now full (10kg). Total price = $40.

If the capacity was 40kg:

1. Take all of C (30kg, $120). Remaining capacity = 10kg.
2. Take 10kg of A (Ratio 3): Price = 10 * 3 = $30.
3. Total Price = $120 +$30 = $150.

5. Common mistakes candidates make

A frequent mistake in the Maximum Price to Fill a Bag coding problem is trying to use 0/1 Knapsack logic (Dynamic Programming). This is overkill and incorrect for the fractional version. Another error is sorting only by price or only by weight; neither guarantees the maximum value. Candidates also sometimes struggle with precision issues when dealing with floating-point ratios and fractions, so it's important to use appropriate data types.

6. Interview preparation tip

When you see a problem that allows for "fractions" or "proportions" and asks for a maximum/minimum, your first thought should be a Greedy algorithm. Practice writing custom comparators for sorting, as this is the most common way to implement the greedy choice in these types of problems.