Maximum Element After Decreasing and Rearranging

Problem Description

You are given an array of positive integers arr. Perform some operations (possibly none) on arr so that it satisfies these conditions:

• The value of the first element in arr must be 1.
• The absolute difference between any 2 adjacent elements must be less than or equal to 1.

There are 2 types of operations that you can perform any number of times:

• Decrease the value of any element of arr to a smaller positive integer.
• Rearrange the elements of arr to be in any order.

Return the maximum possible value of an element in arr after performing the operations to satisfy the conditions.

Key Insights

• The first element must be 1, which sets a starting point for the arrangement.
• The absolute difference constraint implies that the elements must form a contiguous range of integers.
• The maximum possible value of the last element is constrained by the number of elements in the array and the requirement that adjacent elements differ by at most 1.
• Sorting the array helps in determining the maximum possible values that can be achieved after satisfying the conditions.

Space and Time Complexity

Time Complexity: O(n log n) - due to sorting the array
Space Complexity: O(1) - no additional space needed beyond input storage

Solution

To solve this problem, we can follow these steps:

1. Sort the array to allow for easy arrangement.
2. Start from the first element (which we set to 1) and incrementally build up the rest of the array while ensuring that the difference between adjacent elements does not exceed 1.
3. The last element of the modified array will give us the maximum value possible.

We will use a sorted array and iterate through it, adjusting the values according to the defined rules.

Code Solutions