Maximum Beauty of an Array After Applying Operation

class Solution {
public:
    int maxBeautyValue(vector<int>& elements, int m) {
        if (elements.size() == 1) return 1;
    
        int maxBeauty = 0;
        int highestValue = 0;
    
        for (int el : elements) highestValue = max(highestValue, el);
    
        vector<int> rangeCount(highestValue + 1, 0);
    
        for (int el : elements) {
            rangeCount[max(el - m, 0)]++;
            if (el + m + 1 <= highestValue)
                rangeCount[el + m + 1]--;
        }
    
        int runningSum = 0;
        for (int i : rangeCount) {
            runningSum += i;
            maxBeauty = max(maxBeauty, runningSum);
        }
    
        return maxBeauty;
    }
};

The provided solution in C++ is designed to solve the problem of determining the maximum beauty of an array after applying a specific operation. The operation in consideration allows you to increase the beauty of each element by a certain number of times, up to a maximum limit defined by m. The function maxBeautyValue aims to compute the highest possible beauty of the entire array after these operations are applied optimally.

Here's the logic breakdown:

• Begin by handling an edge case where the length of the vector elements is 1. In such a case, the maximum beauty value is 1.

• Initialize two variables, maxBeauty to track the maximum beauty value calculated and highestValue to store the highest number found in the elements.

• Traverse the elements array to determine the highestValue.

• Create a vector rangeCount with a size of highestValue + 1, initialized to zero. This vector will help in tracking the range of beauties feasible by adjusting every element in the elements with up to m increments or decrements.

• Again traverse through elements. For each element el, increment the position representing el - m by 1 in rangeCount and decrement the position representing el + m + 1 (if it's within bounds) to mark the end of the range affecting el.

• Now, compute the running sum of values in the rangeCount by iterating over it. This sum helps in determining the potentials of various beauty levels as enhanced by the operations. Update maxBeauty each time if the current running sum exceeds it.

• Finally, return the value of maxBeauty which represents the highest beauty achievable using the described operation.

This function efficiently computes the desired output using a range overlap technique without modifying the array in place, ensuring an optimized solution leveraging mathematical calculations and logical reasoning.

class Solution {
    
    public int calculateMaxBeauty(int[] flowers, int range) {
        if (flowers.length == 1) return 1;
    
        int highestBeauty = 0;
        int largestValue = 0;
    
        for (int flower : flowers) largestValue = Math.max(largestValue, flower);
    
        int[] frequency = new int[largestValue + 1];
    
        for (int flower : flowers) {
            frequency[Math.max(flower - range, 0)]++;
            if (flower + range + 1 <= largestValue) {
                frequency[flower + range + 1]--;
            }
        }
    
        int currentFrequency = 0;
        for (int freq : frequency) {
            currentFrequency += freq;
            highestBeauty = Math.max(highestBeauty, currentFrequency);
        }
    
        return highestBeauty;
    }
}

The given Java code solves a problem where you must calculate the maximum beauty of an array after applying a certain operation. The operation adds a beauty value to each element within a specified range of range units. This summary breaks down the steps and the logic used to achieve the result in the provided Java code.

• Initialize Variables: First, the program checks if the array contains only one flower. If so, it returns a beauty of 1 as, with one flower, the beauty can't be less than that. Otherwise, two variables are initialized: highestBeauty to keep track of the maximum beauty observed, and largestValue to store the maximum value in the flowers array.

• Determine the Range of Values: Loop through the flowers array to determine the largest value, which is necessary for setting up the frequency array that later stages of the algorithm depend on.

• Set Up Frequency Changes Within Range: A frequency array is populated based on the values in flowers array adjusted by the allowed range. If a flower’s value, flower, is within range [flower - range, flower + range], the effect is registered in the frequency array:

  • Frequency increment at flower - range (or 0 if the result is negative).
  • Frequency decrement at flower + range + 1 if it remains within the bounds of the largest value observed.

• Calculate Maximum/Current Beauty: A loop goes through the frequency array to continuously add to a currentFrequency counter. This counter keeps track of the accumulated frequency at each point. The highestBeauty is updated whenever currentFrequency exceeds the previously known highestBeauty.

• Return Highest Beauty: The highest value of highestBeauty through the iterations is then returned as the maximum potential beauty of the array after the defined operation.

This approach efficiently calculates the maximum beauty of an array with the application of the range-based operation by leveraging optimized frequency computation within specified ranges.

class Solution:
    def calcMaxBeauty(self, items: list[int], limit: int) -> int:
        # Return 1 for single element scenarios
        if len(items) == 1:
            return 1
    
        highest_value = max(items)  # Determine highest value in the list
        tracker = [0] * (highest_value + 1)  # Store changes in this tracker
    
        # Adjust the tracker values based on the permissible range
        for item in items:
            lower_bound = max(item - limit, 0)  # Lower bound of modification
            tracker[lower_bound] += 1  # Increment at the beginning of the range
            if item + limit + 1 <= highest_value:
                tracker[item + limit + 1] -= 1  # Decrement right after the range
    
        aggregate_beauty = 0
        accumulated = 0  # Cumulative count
    
        # Compute the cumulative sums and find maximum beauty
        for i in tracker:
            accumulated += i
            aggregate_beauty = max(aggregate_beauty, accumulated)
    
        return aggregate_beauty

For the given problem, the Python solution revolves around calculating the maximum beauty of an array after a specified operation that involves adjusting the array's elements within a given limit. The code uses an approach similar to the "prefix sum" technique to determine the largest cumulative increase that can be reached when modifying each element within the specified range constraints.

Here’s how the code functions:

1. Initially, it checks if the array contains only one element. In this scenario, it returns a beauty of 1 as there's no room for modification that could change the array.
2. It identifies the highest value in the array to set up a tracker array. This tracker will note modifications across possible values up to this maximum.
3. For each item in the array, the code calculates possible modifications within the limit. It increases the starting point of modification and decreases the point just beyond the range of modification.
4. The tracker array accumulates these modifications to reflect the total adjustments possible at each point.
5. Finally, it iterates over the accumulated tracker values to find the maximum, representing the maximum beauty.

This approach ensures efficient calculation of possible values by maintaining a record of incremental modifications rather than recomputing possible values for each array item repeatedly.