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Minimize Maximum of Array

Medium

Asked by:

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Topics:

ArraysBinary SearchGreedy Algorithms

You are given a 0-indexed array nums comprising of n non-negative integers.

In one operation, you must:

Choose an integer i such that 1 <= i < n and nums[i] > 0.
Decrease nums[i] by 1.
Increase nums[i - 1] by 1.

Return the minimum possible value of the maximum integer of nums after performing any number of operations.

Example 1:

Input: nums = [3,7,1,6]
Output: 5
Explanation:
One set of optimal operations is as follows:
1. Choose i = 1, and nums becomes [4,6,1,6].
2. Choose i = 3, and nums becomes [4,6,2,5].
3. Choose i = 1, and nums becomes [5,5,2,5].
The maximum integer of nums is 5. It can be shown that the maximum number cannot be less than 5.
Therefore, we return 5.

Example 2:

Input: nums = [10,1]
Output: 10
Explanation:
It is optimal to leave nums as is, and since 10 is the maximum value, we return 10.

Constraints:

n == nums.length
2 <= n <= 10⁵
0 <= nums[i] <= 10⁹

Try coding it in LeetCode

Minimize Maximum of Array

Medium

Asked by:

33 views

Topics:

ArraysBinary SearchGreedy Algorithms

You are given a 0-indexed array nums comprising of n non-negative integers.

In one operation, you must:

Choose an integer i such that 1 <= i < n and nums[i] > 0.
Decrease nums[i] by 1.
Increase nums[i - 1] by 1.

Return the minimum possible value of the maximum integer of nums after performing any number of operations.

Example 1:

Input: nums = [3,7,1,6]
Output: 5
Explanation:
One set of optimal operations is as follows:
1. Choose i = 1, and nums becomes [4,6,1,6].
2. Choose i = 3, and nums becomes [4,6,2,5].
3. Choose i = 1, and nums becomes [5,5,2,5].
The maximum integer of nums is 5. It can be shown that the maximum number cannot be less than 5.
Therefore, we return 5.

Example 2:

Input: nums = [10,1]
Output: 10
Explanation:
It is optimal to leave nums as is, and since 10 is the maximum value, we return 10.

Constraints:

n == nums.length
2 <= n <= 10⁵
0 <= nums[i] <= 10⁹

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What are the possible ranges for the integer values within the array?
Can the array be empty, or can it contain negative numbers?
If the maximum possible value cannot be minimized, what should I return?
Is the length of the array guaranteed to be within a specific range?
Are we allowed to modify the original input array, or do we need to operate on a copy?

Brute Force Solution

Approach

The brute force strategy aims to find the smallest possible maximum value in the array after redistributing values between adjacent elements. It achieves this by systematically exploring all conceivable redistributions of values and identifying the one that minimizes this maximum.

Here's how the algorithm would work step-by-step:

Consider all possible ways to transfer values from one position to another, always moving values to the right.
For each such arrangement, calculate the maximum value present in the modified array.
Remember the smallest maximum value you encounter across all these arrangements.
After trying every possibility, the smallest remembered maximum is your answer.

Code Implementation

def minimize_maximum_of_array_brute_force(numbers):
    minimum_maximum_value = float('inf')

    def redistribute(index, current_array):
        nonlocal minimum_maximum_value

        # Base case: Reached the end of the array
        if index == len(numbers) - 1:
            maximum_value = max(current_array)
            minimum_maximum_value = min(minimum_maximum_value, maximum_value)
            return

        # Explore all possible redistributions.
        for amount_to_transfer in range(current_array[index] + 1):
            new_array = current_array[:]
            new_array[index] -= amount_to_transfer
            new_array[index + 1] += amount_to_transfer

            # Recursive call to the next index
            redistribute(index + 1, new_array)

    # Start the redistribution process from the first element
    redistribute(0, numbers[:])

    return minimum_maximum_value

Big(O) Analysis

Time Complexity

O(n!) – The proposed brute force approach considers all possible ways to transfer values between elements. Each element could potentially donate to any element to its right. This leads to exploring a vast number of redistributions. In the worst case, the number of such arrangements grows factorially with the size of the input array 'n', meaning we explore permutations. Therefore, the number of computations grows proportionally to n!, dominating the time complexity, so the algorithm has O(n!) time complexity.

Space Complexity

O(1) – The brute force strategy, as described, explores all possible redistributions by modifying the input array in place. While it considers numerous arrangements and calculates maximum values, it does not explicitly create any auxiliary data structures like temporary arrays, hash maps, or recursion stacks to store intermediate redistributions. The algorithm only maintains a variable to track the smallest maximum encountered, which requires constant space irrespective of the array's size N. Therefore, the auxiliary space complexity is O(1).

Optimal Solution

Approach

The goal is to lower the highest value in the group of numbers. We can do this by sharing values from larger numbers to smaller ones to even things out, always from left to right.

Here's how the algorithm would work step-by-step:

Think of it like balancing buckets of water; if one bucket is overflowing, pour some water into the next bucket to its right.
Go through the numbers one by one, adding the value of the previous bucket (number) to the current bucket.
After adding the previous value to the current bucket, take the average of all the buckets so far (from the start until the current bucket).
Round this average up to the nearest whole number; this becomes the new value of the current bucket, and it represents the maximum you've seen so far while balancing the values.
Keep track of the largest maximum you've encountered during this process. This is your final answer, the smallest possible value of the largest number.

Code Implementation

import math

def minimize_maximum_of_array(numbers):
    current_sum = 0
    maximum_value = 0

    for index, number in enumerate(numbers):
        current_sum += number

        # Calculate the average of the array up to the current index.
        average = current_sum / (index + 1)

        # Round the average up to the nearest integer
        current_maximum = math.ceil(average)

        # Keep track of the overall maximum
        maximum_value = max(maximum_value, current_maximum)

    return maximum_value

Big(O) Analysis

Time Complexity

O(n) – The algorithm iterates through the input array nums of size n exactly once. Inside the loop, a running sum is maintained, and the average up to the current element is calculated. The average calculation and rounding operations take constant time. Therefore, the dominant operation is the single pass through the array, making the time complexity directly proportional to the input size n.

Space Complexity

O(1) – The algorithm iterates through the input array in place, modifying the array values directly. It maintains a running sum and a maximum value seen so far, both of which are stored in single variables. These variables consume constant space regardless of the size of the input array (N). Therefore, the auxiliary space complexity is O(1).

Edge Cases

Case	How to Handle
Empty array	Return 0 immediately as there are no elements to minimize the maximum of.
Single element array	Return the single element itself as it is both the minimum and the maximum.
Array with large numbers causing potential integer overflow during intermediate calculations (e.g., sum)	Use long data type to prevent potential integer overflow.
Array with all zeros	Return 0 as the minimized maximum will also be 0.
Array with very large size (performance bottleneck)	Consider an algorithm with O(n) or O(n log n) time complexity for efficient scaling.
Array with negative numbers	The binary search lower bound may need to be adjusted if negative values are permitted.
Array where no operation reduces the maximum element	The algorithm should still converge and return the original maximum value as the best possible result.
Extreme positive and negative numbers together	The average calculation within the algorithm should handle this appropriately without overflow due to the addition of very large and very small values.

Empty array

How to Handle:

Return 0 immediately as there are no elements to minimize the maximum of.

Single element array

How to Handle:

Return the single element itself as it is both the minimum and the maximum.

Array with large numbers causing potential integer overflow during intermediate calculations (e.g., sum)

How to Handle:

Use long data type to prevent potential integer overflow.

Array with all zeros

How to Handle:

Return 0 as the minimized maximum will also be 0.

Array with very large size (performance bottleneck)

How to Handle:

Consider an algorithm with O(n) or O(n log n) time complexity for efficient scaling.

Array with negative numbers

How to Handle:

The binary search lower bound may need to be adjusted if negative values are permitted.

Array where no operation reduces the maximum element

How to Handle:

The algorithm should still converge and return the original maximum value as the best possible result.

Extreme positive and negative numbers together

How to Handle:

The average calculation within the algorithm should handle this appropriately without overflow due to the addition of very large and very small values.

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What are the possible ranges for the integer values within the array?
Can the array be empty, or can it contain negative numbers?
If the maximum possible value cannot be minimized, what should I return?
Is the length of the array guaranteed to be within a specific range?
Are we allowed to modify the original input array, or do we need to operate on a copy?

Brute Force Solution

Approach

Here's how the algorithm would work step-by-step:

Consider all possible ways to transfer values from one position to another, always moving values to the right.
For each such arrangement, calculate the maximum value present in the modified array.
Remember the smallest maximum value you encounter across all these arrangements.
After trying every possibility, the smallest remembered maximum is your answer.

Code Implementation

def minimize_maximum_of_array_brute_force(numbers):
    minimum_maximum_value = float('inf')

    def redistribute(index, current_array):
        nonlocal minimum_maximum_value

        # Base case: Reached the end of the array
        if index == len(numbers) - 1:
            maximum_value = max(current_array)
            minimum_maximum_value = min(minimum_maximum_value, maximum_value)
            return

        # Explore all possible redistributions.
        for amount_to_transfer in range(current_array[index] + 1):
            new_array = current_array[:]
            new_array[index] -= amount_to_transfer
            new_array[index + 1] += amount_to_transfer

            # Recursive call to the next index
            redistribute(index + 1, new_array)

    # Start the redistribution process from the first element
    redistribute(0, numbers[:])

    return minimum_maximum_value

Big(O) Analysis

Time Complexity

Space Complexity

Optimal Solution

Approach

The goal is to lower the highest value in the group of numbers. We can do this by sharing values from larger numbers to smaller ones to even things out, always from left to right.

Here's how the algorithm would work step-by-step:

Think of it like balancing buckets of water; if one bucket is overflowing, pour some water into the next bucket to its right.
Go through the numbers one by one, adding the value of the previous bucket (number) to the current bucket.
After adding the previous value to the current bucket, take the average of all the buckets so far (from the start until the current bucket).
Round this average up to the nearest whole number; this becomes the new value of the current bucket, and it represents the maximum you've seen so far while balancing the values.
Keep track of the largest maximum you've encountered during this process. This is your final answer, the smallest possible value of the largest number.

Code Implementation

import math

def minimize_maximum_of_array(numbers):
    current_sum = 0
    maximum_value = 0

    for index, number in enumerate(numbers):
        current_sum += number

        # Calculate the average of the array up to the current index.
        average = current_sum / (index + 1)

        # Round the average up to the nearest integer
        current_maximum = math.ceil(average)

        # Keep track of the overall maximum
        maximum_value = max(maximum_value, current_maximum)

    return maximum_value

Big(O) Analysis

Time Complexity

Space Complexity

Edge Cases

Case	How to Handle
Empty array	Return 0 immediately as there are no elements to minimize the maximum of.
Single element array	Return the single element itself as it is both the minimum and the maximum.
Array with large numbers causing potential integer overflow during intermediate calculations (e.g., sum)	Use long data type to prevent potential integer overflow.
Array with all zeros	Return 0 as the minimized maximum will also be 0.
Array with very large size (performance bottleneck)	Consider an algorithm with O(n) or O(n log n) time complexity for efficient scaling.
Array with negative numbers	The binary search lower bound may need to be adjusted if negative values are permitted.
Array where no operation reduces the maximum element	The algorithm should still converge and return the original maximum value as the best possible result.
Extreme positive and negative numbers together	The average calculation within the algorithm should handle this appropriately without overflow due to the addition of very large and very small values.

Empty array

How to Handle:

Return 0 immediately as there are no elements to minimize the maximum of.

Single element array

How to Handle:

Return the single element itself as it is both the minimum and the maximum.

Array with large numbers causing potential integer overflow during intermediate calculations (e.g., sum)

How to Handle:

Use long data type to prevent potential integer overflow.

Array with all zeros

How to Handle:

Return 0 as the minimized maximum will also be 0.

Array with very large size (performance bottleneck)

How to Handle:

Consider an algorithm with O(n) or O(n log n) time complexity for efficient scaling.

Array with negative numbers

How to Handle:

The binary search lower bound may need to be adjusted if negative values are permitted.

Array where no operation reduces the maximum element

How to Handle:

The algorithm should still converge and return the original maximum value as the best possible result.

Extreme positive and negative numbers together

How to Handle:

The average calculation within the algorithm should handle this appropriately without overflow due to the addition of very large and very small values.