#1383

Hard

auto_awesomeGreedy choice plus invariant validation

LeetCode Problem Workspace

Maximum Performance of a Team

Maximize the performance of a team by selecting up to k engineers with the highest performance based on speed and efficiency.

array greedy sorting heap priority queue

Problem Statement

You are given two integer arrays, speed and efficiency, each of length n, representing the speeds and efficiencies of n engineers. You are also given an integer k, which represents the maximum number of engineers that can be selected to form a team. Your task is to select a subset of at most k engineers to maximize the team's performance.

The performance of a team is calculated as the sum of the speeds of the selected engineers, multiplied by the minimum efficiency of those engineers. The goal is to select the k engineers that maximize this value. To achieve this, a greedy approach is used by sorting the engineers by efficiency in descending order.

Examples

Example 1

Input: n = 6, speed = [2,10,3,1,5,8], efficiency = [5,4,3,9,7,2], k = 2

Output: 60

We have the maximum performance of the team by selecting engineer 2 (with speed=10 and efficiency=4) and engineer 5 (with speed=5 and efficiency=7). That is, performance = (10 + 5) * min(4, 7) = 60.

Example 2

Input: n = 6, speed = [2,10,3,1,5,8], efficiency = [5,4,3,9,7,2], k = 3

Output: 68

This is the same example as the first but k = 3. We can select engineer 1, engineer 2 and engineer 5 to get the maximum performance of the team. That is, performance = (2 + 10 + 5) * min(5, 4, 7) = 68.

Example 3

Input: n = 6, speed = [2,10,3,1,5,8], efficiency = [5,4,3,9,7,2], k = 4

Output: 72

Example details omitted.

Constraints

1 <= k <= n <= 105
speed.length == n
efficiency.length == n
1 <= speed[i] <= 105
1 <= efficiency[i] <= 108

Solution Approach

Sort engineers by efficiency

Sort the engineers in decreasing order based on their efficiency. This ensures that when engineers are selected, the team is always formed with the highest efficiency first, maximizing the potential performance.

Use a min-heap for speed sum

Maintain a min-heap to keep track of the engineers' speeds. As engineers are added, the heap ensures that we always have the k engineers with the highest total speed, as long as we don't exceed the limit of k engineers.

Calculate performance dynamically

For each engineer added, calculate the performance dynamically by multiplying the sum of the speeds by the current minimum efficiency. Keep track of the maximum performance encountered.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity is dominated by the sorting step, which is O(n log n), and the heap operations, which are O(log k). Thus, the overall time complexity is O(n log n). The space complexity is O(k) due to the min-heap storing up to k engineers' speeds at a time.

What Interviewers Usually Probe

Tests understanding of greedy algorithms and invariant maintenance.
Assesses ability to optimize performance while handling large inputs.
Evaluates proficiency with heap data structures for efficient selection and tracking.

Common Pitfalls or Variants

Common pitfalls

Not maintaining the correct number of engineers in the heap, which can lead to incorrect performance calculations.
Failing to sort the engineers by efficiency in decreasing order, which could lead to suboptimal selections.
Overcomplicating the problem by not recognizing the importance of a greedy choice in maximizing performance.

Follow-up variants

The problem could involve different performance formulas, such as weighted averages or more complex functions of speed and efficiency.
Instead of k engineers, you might need to select the best team given a fixed budget or constraints on individual engineer costs.
Consider an extended version where the engineers have different skill levels or other attributes affecting the performance calculation.

FAQ

How do I approach the Maximum Performance of a Team problem?

Use a greedy approach by sorting engineers by efficiency and tracking the highest speeds using a min-heap.

What is the time complexity of the Maximum Performance of a Team problem?

The time complexity is O(n log n), dominated by the sorting step and heap operations.

How do I ensure the team performance is maximized in this problem?

Maximize the team performance by always selecting engineers with the highest efficiencies and speeds, ensuring the sum of speeds is optimal for the minimum efficiency.

What is the primary pattern used to solve this problem?

This problem uses a greedy choice combined with invariant validation, focusing on maximizing team performance based on sorted efficiencies and a heap for speed tracking.

What is the role of the min-heap in the solution?

The min-heap helps track the sum of the selected engineers' speeds, ensuring that the k engineers with the highest total speed are selected at each step.

terminal

Solution

Solution 1

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class Solution:
    def maxPerformance(
        self, n: int, speed: List[int], efficiency: List[int], k: int
    ) -> int:
        t = sorted(zip(speed, efficiency), key=lambda x: -x[1])
        ans = tot = 0
        mod = 10**9 + 7
        h = []
        for s, e in t:
            tot += s
            ans = max(ans, tot * e)
            heappush(h, s)
            if len(h) == k:
                tot -= heappop(h)
        return ans % mod

Continue Practicing

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