Maximum Beauty of an Array After Applying Operation

Medium

Asked by:

18 views

Topics:

ArraysSliding WindowsGreedy Algorithms

You are given a 0-indexed array nums and a non-negative integer k.

In one operation, you can do the following:

Choose an index i that hasn't been chosen before from the range [0, nums.length - 1].
Replace nums[i] with any integer from the range [nums[i] - k, nums[i] + k].

The beauty of the array is the length of the longest subsequence consisting of equal elements.

Return the maximum possible beauty of the array nums after applying the operation any number of times.

Note that you can apply the operation to each index only once.

A subsequence of an array is a new array generated from the original array by deleting some elements (possibly none) without changing the order of the remaining elements.

Example 1:

Input: nums = [4,6,1,2], k = 2
Output: 3
Explanation: In this example, we apply the following operations:
- Choose index 1, replace it with 4 (from range [4,8]), nums = [4,4,1,2].
- Choose index 3, replace it with 4 (from range [0,4]), nums = [4,4,1,4].
After the applied operations, the beauty of the array nums is 3 (subsequence consisting of indices 0, 1, and 3).
It can be proven that 3 is the maximum possible length we can achieve.

Example 2:

Input: nums = [1,1,1,1], k = 10
Output: 4
Explanation: In this example we don't have to apply any operations.
The beauty of the array nums is 4 (whole array).

Constraints:

1 <= nums.length <= 10⁵
0 <= nums[i], k <= 10⁵

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 constraints on the values within the input array? Can they be negative, zero, or very large?
What exactly is the 'operation' we can apply to each number, and what is the allowed range of its parameters?
What constitutes 'beauty' in this context? Can you provide a more precise definition or example?
What is the size limit of the input array? Should I optimize for space or time complexity given a specific range?
If there are multiple ways to achieve the maximum beauty, is any one valid, or is there a specific criterion for choosing among them?

Brute Force Solution

Approach

To find the maximum beauty, the brute force strategy checks every single possible way to apply the given operation to each number in our collection. Then, for each possibility, we calculate the beauty and track the maximum beauty we've seen so far. This guarantees we find the absolute best result, but it might take a very long time.

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

Consider the first number in the collection.
We can either apply the operation to it, or leave it unchanged.
For each of these possibilities, move to the second number.
Again, we can either apply the operation to the second number, or leave it unchanged. This creates two new possibilities for each previous possibility.
Continue this process for every number in the collection, creating many different combinations of applying or not applying the operation.
For each complete combination, calculate the beauty of the resulting collection of numbers according to the defined rules.
Compare the beauty of each combination and keep track of the highest beauty value found.
After checking every single possible combination, the highest beauty value that was tracked is the final answer.

Code Implementation

def maximum_beauty_brute_force(numbers, change):
    max_beauty = 0

    def calculate_beauty(current_numbers):
        beauty = 0
        counts = {}
        for number in current_numbers:
            counts[number] = counts.get(number, 0) + 1
        for count in counts.values():
            beauty += count * count
        return beauty

    def find_maximum_beauty(index, current_numbers):
        nonlocal max_beauty

        # Base case: all numbers have been processed
        if index == len(numbers):
            beauty = calculate_beauty(current_numbers)
            max_beauty = max(max_beauty, beauty)
            return

        # Explore the option of NOT applying the operation
        find_maximum_beauty(index + 1, current_numbers + [numbers[index]])

        # Explore the option of applying the operation
        find_maximum_beauty(index + 1, current_numbers + [numbers[index] + change])

    # Initiate the search with an empty list
    find_maximum_beauty(0, [])

    return max_beauty

Big(O) Analysis

Time Complexity

O(2^n) – The algorithm explores all possible combinations of applying or not applying the operation to each of the n elements in the array. For each element, there are two choices. Therefore, there are 2 * 2 * ... * 2 (n times) = 2^n possible combinations. For each of these 2^n combinations, the beauty is calculated, which takes O(n) time in the worst case if we iterate through the array to find the beauty. Therefore, the overall time complexity is O(n * 2^n). Because the 2^n dominates n as n increases, we can simplify this to O(2^n).

Space Complexity

O(1) – The provided brute force solution, as described, primarily involves iterating through all possible combinations of applying or not applying an operation to each number in the input array. It calculates the beauty for each combination and keeps track of the maximum beauty seen so far. The only extra memory used is for variables to store the current maximum beauty and potentially a few loop counters, which are independent of the input size N (where N is the number of elements in the array). Therefore, the auxiliary space complexity is constant.

Optimal Solution

Approach

The goal is to maximize the number of 'beautiful' elements in the array by applying a simple operation. The clever trick is to sort the numbers and then efficiently count how many numbers can become 'beautiful' based on their difference from the smallest number.

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

First, arrange the numbers in increasing order from smallest to largest. This makes it easier to see relationships between the numbers.
Find the smallest number in the sorted arrangement.
Go through each number in the arrangement. Figure out if we can make that number 'beautiful' by adding to the smallest number so that the number is within a limit.
Count how many numbers can be made 'beautiful'.
The final answer is the total count of the numbers that can become 'beautiful'.

Code Implementation

def maximum_beauty(numbers, allowed_difference):
    numbers.sort()

    smallest_number = numbers[0]

    beautiful_count = 0

    # Iterate and check 'beauty' based on difference from the smallest.
    for current_number in numbers:
        if current_number <= smallest_number + allowed_difference:

            # If the number is within the allowed range, its 'beautiful'.
            beautiful_count += 1

    # The total count is the answer.
    return beautiful_count

Big(O) Analysis

Time Complexity

O(n log n) – The dominant operation is sorting the input array of size n, which typically takes O(n log n) time. The subsequent steps involve iterating through the sorted array once, performing constant-time operations for each element. This linear iteration contributes O(n) time, but it is less significant than the sorting time. Therefore, the overall time complexity is determined by the sorting step, resulting in O(n log n).

Space Complexity

O(1) – The described algorithm primarily involves sorting the input array. The sorting operation is often performed in-place, modifying the original array directly and thus not requiring additional space proportional to the input size. The algorithm then iterates through the sorted array using a constant number of variables to track the smallest element and the count of beautiful elements. Therefore, the auxiliary space used remains constant irrespective of the input array's size (N).

Edge Cases

Case	How to Handle
Null or Empty input array	Return 0 immediately, as there are no elements to form a beauty pair.
Array with only one element	Return 0, as a beauty pair requires at least two elements.
Array with all elements equal and k is 0	The entire array can be considered a beauty array; return the length of the array.
Array with all elements equal and k is not 0	Return 0, because no pair can satisfy the condition of absolute difference being equal to k if k != 0.
Large input array exceeding memory limits	Optimize the solution for space complexity using an in-place algorithm if feasible, or stream the data if possible, otherwise return error.
Integer overflow when calculating the absolute difference	Use a data type capable of holding larger values (e.g., long) or handle the overflow explicitly.
Array contains negative numbers	The absolute difference calculation should correctly handle negative numbers.
Array with duplicate elements and small k value	The algorithm must ensure that same index is not selected for the pair.

Null or Empty input array

How to Handle:

Return 0 immediately, as there are no elements to form a beauty pair.

Array with only one element

How to Handle:

Return 0, as a beauty pair requires at least two elements.

Array with all elements equal and k is 0

How to Handle:

The entire array can be considered a beauty array; return the length of the array.

Array with all elements equal and k is not 0

How to Handle:

Return 0, because no pair can satisfy the condition of absolute difference being equal to k if k != 0.

Large input array exceeding memory limits

How to Handle:

Optimize the solution for space complexity using an in-place algorithm if feasible, or stream the data if possible, otherwise return error.

Integer overflow when calculating the absolute difference

How to Handle:

Use a data type capable of holding larger values (e.g., long) or handle the overflow explicitly.

Array contains negative numbers

How to Handle:

The absolute difference calculation should correctly handle negative numbers.

Array with duplicate elements and small k value

How to Handle:

The algorithm must ensure that same index is not selected for the pair.