Minimum Cost to Hire K Workers

The problem requires hiring exactly ( k ) workers with the minimum total wage, subject to certain rules regarding the payment. We can solve this problem using a greedy algorithm by sorting the workers based on the wage-to-quality ratio. Here’s the code to achieve this:

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import heapq

class Solution:
    def mincostToHireWorkers(self, quality: List[int], wage: List[int], k: int) -> float:
        workers = sorted([float(w) / q, q] for w, q in zip(wage, quality))
        res = float('inf')
        qsum = 0
        heap = []

        for ratio, q in workers:
            heapq.heappush(heap, -q)
            qsum += q

            if len(heap) > k:
                qsum += heapq.heappop(heap)

            if len(heap) == k:
                res = min(res, ratio * qsum)

        return res

Explanation:

We start by creating a list of tuples representing each worker, with the ratio of wage to quality and the quality itself.
We then sort this list by the ratio, ensuring that we consider workers in increasing order of their wage-to-quality ratio.
We use a min-heap to maintain the ( k ) largest quality values as we iterate through the sorted list.
We keep track of the sum of quality values (qsum) in the current group and update it accordingly as we add or remove workers from the heap.
When the heap size reaches ( k ), we calculate the total wage for the current group using the current ratio and the sum of quality values. We update the result if this wage is lower than the current minimum.
The final result is the minimum wage found during the iteration.

This code returns the least amount of money needed to form a paid group of ( k ) workers. Its time complexity is ( O(n \log n) ), where ( n ) is the number of workers, and the space complexity is ( O(n) ).

10 Prerequisite LeetCode Problems

“857. Minimum Cost to Hire K Workers” involves heaps (priority queues), sorting, and greedy algorithms. Here are some problems to understand these concepts:

LeetCode 215. Kth Largest Element in an Array
- This problem helps you understand how to use a heap to solve problems related to finding the Kth largest or smallest element.
LeetCode 347. Top K Frequent Elements
- Another problem that requires understanding of heaps. This problem asks you to find the K most frequent elements in an array.
LeetCode 703. Kth Largest Element in a Stream
- A more complex problem involving heaps, this problem requires you to find the Kth largest element in a stream of integers.
LeetCode 75. Sort Colors
- This is a problem that helps you understand in-place sorting and will strengthen your understanding of sorting algorithms.
LeetCode 451. Sort Characters By Frequency
- This problem requires you to sort characters in a string based on their frequency. It’ll help you understand how to apply sorting based on conditions.
LeetCode 56. Merge Intervals
- This problem asks you to merge all overlapping intervals in a list. This will help you understand sorting with custom comparison function.
LeetCode 435. Non-overlapping Intervals
- This problem involves the concept of intervals and greedy algorithms. It will help you understand how to use greedy algorithms to solve problems involving intervals.
LeetCode 452. Minimum Number of Arrows to Burst Balloons
- This problem requires understanding of greedy algorithms. It asks you to find the minimum number of arrows required to burst all balloons.
LeetCode 406. Queue Reconstruction by Height
- This problem involves sorting and greedy algorithm to arrange people based on their heights and the number of people in front who have greater or equal height.
LeetCode 253. Meeting Rooms II
- This problem involves the use of a priority queue (min-heap) and will help you understand how to schedule tasks/events using a heap.

Problem Classification

Problem Statement:There are n workers. You are given two integer arrays quality and wage where quality[i] is the quality of the ith worker and wage[i] is the minimum wage expectation for the ith worker.

We want to hire exactly k workers to form a paid group. To hire a group of k workers, we must pay them according to the following rules:

Every worker in the paid group should be paid in the ratio of their quality compared to other workers in the paid group. Every worker in the paid group must be paid at least their minimum wage expectation. Given the integer k, return the least amount of money needed to form a paid group satisfying the above conditions. Answers within 10-5 of the actual answer will be accepted.

Example 1:

Input: quality = [10,20,5], wage = [70,50,30], k = 2 Output: 105.00000 Explanation: We pay 70 to 0th worker and 35 to 2nd worker. Example 2:

Input: quality = [3,1,10,10,1], wage = [4,8,2,2,7], k = 3 Output: 30.66667 Explanation: We pay 4 to 0th worker, 13.33333 to 2nd and 3rd workers separately.

Constraints:

n == quality.length == wage.length 1 <= k <= n <= 104 1 <= quality[i], wage[i] <= 104

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

How did you infer from the problem statement that this problem can be solved using ?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Thought Process

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Coding Constructs

Consider the following piece of complex software code.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.