Minimize Deviation in Array

Let’s understand the problem of minimizing the deviation in an array and break down the solution into simpler steps.

The deviation of an array is the difference between the maximum and minimum elements. We can perform two operations on the elements: divide by 2 if the element is even, or multiply by 2 if the element is odd. The goal is to find the minimum deviation possible.

Multiply all odd numbers by 2 so that all numbers are even. This makes it easier to decrease the numbers consistently.
Push all numbers into a max-heap.
Repeatedly divide the maximum number by 2 and update the heap.
Keep track of the maximum and minimum numbers, and update the result whenever the current deviation is smaller.

Algorithm

Initialize: Multiply all odd numbers by 2, then put all numbers into a max-heap.
Initialize Variables: Set maxNum as the maximum number in the heap, and minNum as the minimum number in the initial array (remember all numbers are even now).
Process Heap: While the maximum number in the heap is even: a. Pop the maximum number from the heap. b. Divide it by 2 and push it back into the heap. c. Update minNum as the minimum of minNum and the new number. d. Update the result if the current deviation (maxNum - minNum) is smaller.
Return Result: The result is the minimum deviation found.

Implementation

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import heapq

class Solution:
    def minimumDeviation(self, nums: List[int]) -> int:
        max_heap = []
        minNum = float('inf')

        # Make all numbers even and push to max heap
        for num in nums:
            if num % 2 == 1:
                num *= 2
            max_heap.append(-num)
            minNum = min(minNum, num)

        heapq.heapify(max_heap)
        result = float('inf')

        while len(max_heap) == len(nums):
            maxNum = -heapq.heappop(max_heap)
            result = min(result, maxNum - minNum)

            if maxNum % 2 == 0:
                minNum = min(minNum, maxNum // 2)
                heapq.heappush(max_heap, -maxNum // 2)

        return result

Insights and Key Takeaways

The use of max-heap allows efficient retrieval and update of the maximum number.
By making all numbers even initially, we can uniformly decrease them by dividing by 2.
The time complexity is O(N * log(maxNum) * log(N)) where N is the number of elements, and log(maxNum) is the number of division by 2 operations needed for the maximum number.
The space complexity is O(N) for the heap.
This solution leverages the constraints of the operations to find a global minimum deviation.

10 Prerequisite LeetCode Problems

“Minimize Deviation in Array” requires understanding heaps, properties of numbers (even and odd numbers), and the ability to handle complex logical conditions. Here are 10 simpler problems to build these skills:

“Kth Largest Element in an Array” (LeetCode Problem #215): This problem provides an introduction to the concept of heaps.
“Top K Frequent Elements” (LeetCode Problem #347): This problem also involves using a heap, but adds the complexity of frequency counts.
“Find K Pairs with Smallest Sums” (LeetCode Problem #373): This problem uses heaps to solve a more complex problem, and introduces you to combining heap usage with other data structures.
“Sliding Window Maximum” (LeetCode Problem #239): This problem introduces the concept of a sliding window, which is a crucial step to solve the “Minimize Deviation in Array” problem.
“Odd Even Linked List” (LeetCode Problem #328): This problem deals with the properties of odd and even numbers, which is important for the “Minimize Deviation in Array” problem.
“Number of Recent Calls” (LeetCode Problem #933): This problem provides practice in managing and updating data in a queue, which is similar to heap operations.
“Sum of Subarray Minimums” (LeetCode Problem #907): This problem provides practice with arrays and dealing with minimums, which can be beneficial in “Minimize Deviation in Array”.
“Shortest Subarray with Sum at Least K” (LeetCode Problem #862): This problem presents a more complex array manipulation challenge and will help to improve problem-solving skills for similar problems.
“Design a Stack With Increment Operation” (LeetCode Problem #1381): This problem provides practice with designing data structures with specific properties, which will be helpful in “Minimize Deviation in Array”.
“Last Stone Weight” (LeetCode Problem #1046): This problem practices the use of heap data structures and managing elements within it.

Problem Classification

Problem Statement: You are given an array nums of n positive integers.

You can perform two types of operations on any element of the array any number of times:

If the element is even, divide it by 2. For example, if the array is [1,2,3,4], then you can do this operation on the last element, and the array will be [1,2,3,2]. If the element is odd, multiply it by 2. For example, if the array is [1,2,3,4], then you can do this operation on the first element, and the array will be [2,2,3,4]. The deviation of the array is the maximum difference between any two elements in the array.

Return the minimum deviation the array can have after performing some number of operations.

Example 1:

Input: nums = [1,2,3,4] Output: 1 Explanation: You can transform the array to [1,2,3,2], then to [2,2,3,2], then the deviation will be 3 - 2 = 1. Example 2:

Input: nums = [4,1,5,20,3] Output: 3 Explanation: You can transform the array after two operations to [4,2,5,5,3], then the deviation will be 5 - 2 = 3. Example 3:

Input: nums = [2,10,8] Output: 3

Constraints:

n == nums.length 2 <= n <= 5 * 104 1 <= nums[i] <= 109

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?

Alternatively, if you’re working on a specific problem, you might ask something like:

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.