Maximum Coins Heroes Can Collect

Maximum Coins Heroes Can Collect

What is this problem about?

The "Maximum Coins Heroes Can Collect" problem is an engaging scenario that involves matching different entities based on their strengths and rewards. You are typically given two main groups: a set of heroes, each with a specific power level, and a set of monsters, each defined by a power level and a reward in the form of coins. A hero can defeat a monster if their power is greater than or equal to the monster's power. The goal is to determine the maximum number of coins each hero can collect by defeating all the monsters they are capable of overcoming.

Why is this asked in interviews?

This problem is a favorite at companies like Deutsche Bank because it tests a candidate's ability to efficiently process and relate two different datasets. The Maximum Coins Heroes Can Collect interview question evaluates your understanding of sorting, searching, and pre-calculation techniques. Instead of a brute-force approach, which would be too slow for large inputs, you must demonstrate how to optimize the matching process using binary search and prefix sums. It’s a great way to measure a candidate's focus on algorithmic efficiency.

Algorithmic pattern used?

The core algorithmic pattern used in this problem is a combination of Sorting, Binary Search, and Prefix Sums. First, you sort the monsters by their power level. Then, you compute a prefix sum array of the monster's coins, so that the total coins for the first $n$ monsters can be retrieved in $O(1)$ time. Finally, for each hero, you use Binary Search to find the most powerful monster they can defeat in the sorted list. This allows you to quickly fetch the total coins they collect from the prefix sum array.

Example explanation?

Imagine you have heroes with powers [10, 5] and monsters with (power, coins) pairs: [(3, 100), (12, 500), (8, 200)].

1. Sort monsters by power: [(3, 100), (8, 200), (12, 500)].
2. Calculate prefix sums of coins: [100, 300, 800].
3. Hero 1 (Power 10): Binary search finds the monster with power 8. Total coins = 300 (sum of 100 + 200).
4. Hero 2 (Power 5): Binary search finds the monster with power 3. Total coins = 100. By using this approach, we avoid checking every hero against every monster, which is the heart of the Maximum Coins Heroes Can Collect coding problem.

Common mistakes candidates make?

A common error is trying to solve the problem using a nested loop, which leads to $O(H \times M)$ time complexity, where $H$ is the number of heroes and $M$ is the number of monsters. This will likely time out. Another mistake is forgetting to handle cases where a hero is weaker than all monsters, resulting in zero coins. Candidates also sometimes forget that they need to sort the coins along with the monster powers, or they apply the prefix sum on the unsorted list of coins, which leads to incorrect totals.

Interview preparation tip

When you encounter problems involving "matching" or "thresholds," think about the binary search interview pattern. Always consider if pre-processing one of the arrays (like sorting and prefix sums) can make the queries faster. This "Pre-process once, query many times" strategy is a fundamental concept in high-performance software engineering.