#1802

Medium

auto_awesomeBinary search over the valid answer space

LeetCode Problem Workspace

Maximum Value at a Given Index in a Bounded Array

Maximize the value at a given index of an array with constraints using binary search over the valid answer space.

Problem Statement

You are given three positive integers: n, index, and maxSum. Construct an array of length n that satisfies specific conditions. Your goal is to maximize the value of nums[index] while ensuring the sum of all array elements does not exceed maxSum.

The challenge lies in balancing the array construction to both maximize nums[index] and respect the constraints, particularly the total sum. To find the optimal value at nums[index], a binary search strategy is useful to efficiently explore the range of possible values for nums[index].

Examples

Example 1

Input: n = 4, index = 2, maxSum = 6

Output: 2

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

Example details omitted.

Constraints

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

Solution Approach

Binary Search over Possible Values

The main strategy is to perform binary search on the possible values of nums[index], starting from the lowest valid value. At each step, check if the array sum can be within the maxSum constraint with the current value at nums[index].

Calculate the Array Sum Efficiently

For a given nums[index], calculate the sum of the array using a greedy approach where the elements on either side of index are distributed to maximize their values, subject to the sum constraint.

Greedy Array Construction

The array should be constructed greedily around the target value at nums[index]. Distribute the remaining sum to the other elements with a decreasing pattern to minimize the total sum.

Complexity Analysis

Metric	Value
Time	O(\log (\text{maxSum}))
Space	O(1)

The time complexity is O(log(maxSum)), due to the binary search over the valid answer space. Space complexity is O(1) as only a few variables are used to store intermediate values during the computation.

What Interviewers Usually Probe

Candidate should identify the importance of binary search over the answer space to minimize computation time.
Look for clear understanding of how to calculate array sums given the constraints and how to use greedy methods efficiently.
Ensure the candidate is comfortable explaining the trade-off between maximizing nums[index] and respecting the sum constraint.

Common Pitfalls or Variants

Common pitfalls

Failing to handle edge cases, like when nums[index] is at its minimum possible value.
Incorrectly calculating the sum of the array for a given value of nums[index], leading to exceeding the maxSum constraint.
Not considering the effect of the distribution on both sides of the index when constructing the array.

Follow-up variants

What if maxSum is very large compared to n? The binary search will quickly converge to the maximum possible value.
What if nums[index] is set to the minimum value? This could lead to an unbalanced array and suboptimal performance.
What if the index is at the edge of the array (either 0 or n-1)? Special care must be taken in constructing the array around the boundary.

FAQ

How do I approach the "Maximum Value at a Given Index in a Bounded Array" problem?

Use binary search to find the maximum value for nums[index] that satisfies the array sum constraint. Calculate the total array sum efficiently and greedily distribute the sum around the target index.

What is the time complexity of solving the "Maximum Value at a Given Index in a Bounded Array" problem?

The time complexity is O(log(maxSum)) due to the binary search over the possible values of nums[index].

How can I ensure my array construction stays within the maxSum constraint?

By using a greedy approach to distribute the remaining sum around nums[index], and binary search to narrow down the maximum possible value for nums[index].

Can binary search always be applied to problems like this one?

Binary search is effective in problems where you're searching for an optimal value within a range, as long as the problem has a monotonic property. In this case, higher values for nums[index] lead to larger sums, making binary search appropriate.

What should I watch out for when implementing the binary search for this problem?

Make sure to properly calculate the total array sum at each binary search step, ensuring that it does not exceed maxSum. Also, be mindful of how the elements on both sides of nums[index] contribute to the sum.

terminal

Solution

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 $1$.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

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

Continue Practicing

#1739 building boxes #1648 sell diminishing valued colored balls #2358 maximum number of groups entering a competition