Maximum Value at a Given Index in a Bounded Array

Problem Statement

The task is to design an array, nums, with a specific set of criteria geared towards maximizing the value at a given index while adhering to overall sum limitations. In detail, for given integers n, index, and maxSum, construct an array nums of length n, where each element is a positive integer. The requirements are:

• Each element, nums[i], must be a positive integer within the array of length n.
• The absolute difference between consecutive elements, nums[i] and nums[i+1], must be at most 1.
• The sum of all array elements must not exceed the value maxSum.
• Among all possible arrays that satisfy these conditions, the element at the specified index should be as large as possible.

The goal is to determine the maximum possible value of nums[index] given these constraints.

Examples

Example 1

Input:

n = 4, index = 2, maxSum = 6

Output:

2

Explanation:

nums = [1,2,2,1] is one array that satisfies all the conditions.
There are no arrays that satisfy all the conditions and have nums[2] == 3, so 2 is the maximum nums[2].

Example 2

Input:

n = 6, index = 1, maxSum = 10

Output:

3

Constraints

• 1 <= n <= maxSum <= 109
• 0 <= index < n

Approach and Intuition

The problem can be approached by trying to balance the array such that the value at index is maximized without violating the sum constraint. Here's the conceptual approach:

1. Initialization:

   • Start by assigning the lowest value, which is 1, to each element of the array. This ensures all elements are positive and the array initially meets the sum constraint minimally.

2. Maximum Potential Value:

   • Calculate the theoretical maximum value at nums[index] assuming there are no other constraints aside from the total sum. This can be an initial high guess like maxSum - (n - 1) (subtracting the minimum necessary to keep others at least 1).

3. Distribution Adjustment:

   • Once a high guess is found, adjust the values of other elements starting from nums[index] and spreading outwards. Increase neighbors (nums[index-1], nums[index+1]) by 1 iteratively if possible within the remaining sum after placing the maximum estimate at index.

4. Validation and Adjustment:

   • Ensure that after maximizing nums[index], the array still meets the sum limitation. If not, adjust downwards and redistribute. This might involve iterative trials, decreasing from the initial high estimate until the array validates.

5. Final Computation and Output:

   • Through this redistribute-validate loop, the algorithm zeroes in on the maximum feasible value for nums[index] with the given constraints.

By understanding and executing through this structured approach, you can derive the algorithm to find the maximal element at a specified index which is bounded by the value differences and sum limitation. The examples provided highlight how the redistribution and maximization function towards attaining these requirements.

Solutions

class Solution {
    private long calculateSum(int idx, int val, int size) {
        long total = 0;
    
        if (val > idx) {
            total += (long)(val + val - idx) * (idx + 1) / 2;
        } else {
            total += (long)(val + 1) * val / 2 + idx - val + 1;
        };
    
        if (val >= size - idx) {
            total += (long)(val + val - size + 1 + idx) * (size - idx) / 2;
        } else {
            total += (long)(val + 1) * val / 2 + size - idx - val;
        }   
            
        return total - val;
            
    }
    public int findMaxValue(int size, int idx, int maxSum) {
        int start = 1, end = maxSum;
        while (start < end) {
            int mid = (start + end + 1) / 2;
            if (calculateSum(idx, mid, size) <= maxSum) {
                start = mid;
            } else {
                end = mid - 1;
            }
        }
            
        return start;
    }
}