Maximum Performance of a Team

Maximum Performance of a Team

1. What is this problem about?

The Maximum Performance of a Team interview question is a classic optimization problem that involves selecting a subset of individuals to maximize a specific metric. You are given two arrays: speed and efficiency, where the i-th person has a specific speed and a specific efficiency. You need to pick at most k engineers to form a team. The "performance" of a team is defined as the sum of their speeds multiplied by the minimum efficiency among the team members.

The challenge lies in the fact that the team's overall efficiency is bottlenecked by the least efficient member. This means adding a very fast but low-efficiency person might actually decrease the total performance, even though the total speed increases.

2. Why is this asked in interviews?

This Maximum Performance of a Team coding problem is a favorite at companies like Google and Meta because it tests your ability to handle multi-variable optimization. It requires a solid grasp of greedy algorithms and efficient data structures. Interviewers are looking for your ability to recognize that by fixing one variable (efficiency), you can transform the problem into a simpler one (selecting the best speeds). It also evaluates your proficiency with Priority Queues (Heaps) to maintain a dynamic set of top elements.

3. Algorithmic pattern used

The Array, Sorting, Heap (Priority Queue), Greedy interview pattern is the standard approach for this problem. First, you sort the engineers by their efficiency in descending order. This allows you to iterate through each engineer, treating their efficiency as the "minimum efficiency" for a potential team. As you iterate, you maintain a Min-Heap of the speeds of the engineers you've considered so far. If the heap size exceeds k, you remove the smallest speed, ensuring you always keep the fastest k engineers for the current minimum efficiency.

4. Example explanation

Suppose you need to pick a team of at most 2 members (k=2). Engineers:

• A: Speed 10, Efficiency 4
• B: Speed 5, Efficiency 7
• C: Speed 8, Efficiency 5

1. Sort by Efficiency: B(7), C(5), A(4).
2. Consider B: Min Efficiency = 7, Speeds = [5], Total Speed = 5. Performance = 5 * 7 = 35.
3. Consider C: Min Efficiency = 5, Speeds = [5, 8], Total Speed = 13. Performance = 13 * 5 = 65.
4. Consider A: Min Efficiency = 4. We can only have 2 members, so we keep the top 2 speeds from {5, 8, 10}, which are 8 and 10. Total Speed = 18. Performance = 18 * 4 = 72.

Max Performance is 72.

5. Common mistakes candidates make

A frequent error in the Maximum Performance of a Team coding problem is trying to use dynamic programming. While it might look like a DP problem, the state space is too large. Another mistake is sorting by speed instead of efficiency, which makes it much harder to track the minimum efficiency. Candidates also often forget to use a 64-bit integer (like long in Java or C++) for the performance calculation, leading to integer overflow errors since the product can be very large.

6. Interview preparation tip

When you see a problem where the result is determined by a "minimum" or "maximum" of one property and a "sum" of another, think about sorting by the "minimum/maximum" property. This "locks" that property for your current iteration, allowing you to focus on optimizing the "sum" using a heap or another greedy strategy. Mastering the use of a Min-Heap to keep track of the "top K" elements is a high-value skill for many greedy optimization problems.