1802. Maximum Value at a Given Index in a Bounded Array 

Description

You are given three positive integers: n, index, and maxSum. You want to construct an array nums (0-indexed) that satisfies the following conditions:

nums.length == n
nums[i] is a positive integer where 0 <= i < n.
abs(nums[i] - nums[i+1]) <= 1 where 0 <= i < n-1.
The sum of all the elements of nums does not exceed maxSum.
nums[index] is maximized.

Return nums[index] of the constructed array.

Note that abs(x) equals x if x >= 0, and -x otherwise.

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

Solutions

Solution 1: Binary Search

According to the problem description, if we determine the value of $nums[index]$ as $x$, we can find a minimum array sum. That is, the elements on the left side of $index$ in the array decrease from $x-1$ to $1$, and if there are remaining elements, the remaining elements are all $1$; similarly, the elements at $index$ and on the right side of the array decrease from $x$ to $1$, and if there are remaining elements, the remaining elements are all $1$.

In this way, we can calculate the sum of the array. If the sum is less than or equal to $maxSum$, then the current $x$ is valid. As $x$ increases, the sum of the array will also increase, so we can use the binary search method to find the maximum $x$ that meets the conditions.

To facilitate the calculation of the sum of the elements on the left and right sides of the array, we define a function $sum(x, cnt)$, which represents the sum of an array with $cnt$ elements and a maximum value of $x$. The function $sum(x, cnt)$ can be divided into two cases:

If $x \geq cnt$, then the sum of the array is $\frac{(x + x - cnt + 1) \times cnt}{2}$
If $x \lt cnt$, then the sum of the array is $\frac{(x + 1) \times x}{2} + cnt - x$

Next, define the left boundary of the binary search as $left = 1$, the right boundary as $right = maxSum$, and then binary search for the value $mid$ of $nums[index]$. If $sum(mid - 1, index) + sum(mid, n - index) \leq maxSum$, then the current $mid$ is valid, we can update $left$ to $mid$, otherwise we update $right$ to $mid - 1$.

Finally, return $left$ as the answer.

The time complexity is $O(\log M)$, where $M=maxSum$. The space complexity is $O(1)$.

Python3

class Solution:
    def maxValue(self, n: int, index: int, maxSum: int) -> int:
        def sum(x, cnt):
            return (
                (x + x - cnt + 1) * cnt // 2 if x >= cnt else (x + 1) * x // 2 + cnt - x
            )

        left, right = 1, maxSum
        while left < right:
            mid = (left + right + 1) >> 1
            if sum(mid - 1, index) + sum(mid, n - index) <= maxSum:
                left = mid
            else:
                right = mid - 1
        return left

Java

class Solution {
    public int maxValue(int n, int index, int maxSum) {
        int left = 1, right = maxSum;
        while (left < right) {
            int mid = (left + right + 1) >>> 1;
            if (sum(mid - 1, index) + sum(mid, n - index) <= maxSum) {
                left = mid;
            } else {
                right = mid - 1;
            }
        }
        return left;
    }

    private long sum(long x, int cnt) {
        return x >= cnt ? (x + x - cnt + 1) * cnt / 2 : (x + 1) * x / 2 + cnt - x;
    }
}

C++

class Solution {
public:
    int maxValue(int n, int index, int maxSum) {
        auto sum = [](long x, int cnt) {
            return x >= cnt ? (x + x - cnt + 1) * cnt / 2 : (x + 1) * x / 2 + cnt - x;
        };
        int left = 1, right = maxSum;
        while (left < right) {
            int mid = (left + right + 1) >> 1;
            if (sum(mid - 1, index) + sum(mid, n - index) <= maxSum) {
                left = mid;
            } else {
                right = mid - 1;
            }
        }
        return left;
    }
};

Go

func maxValue(n int, index int, maxSum int) int {
	sum := func(x, cnt int) int {
		if x >= cnt {
			return (x + x - cnt + 1) * cnt / 2
		}
		return (x+1)*x/2 + cnt - x
	}
	return sort.Search(maxSum, func(x int) bool {
		x++
		return sum(x-1, index)+sum(x, n-index) > maxSum
	})
}