Maximum Element After Decreasing and Rearranging

Maximum Element After Decreasing and Rearranging

1. What is this problem about?

This problem gives you an array of positive integers and allows you to perform two types of operations: decrease any element to any smaller positive integer, and rearrange the elements in any order. There are two constraints: the first element must be 1, and the absolute difference between any two adjacent elements must be at most 1. You want to find the maximum possible value for the largest element in the final array.

2. Why is this asked in interviews?

The Maximum Element After Decreasing and Rearranging coding problem is popular at Amazon because it tests Greedy logic and Sorting. It's about finding the optimal structure under constraints. It tests whether a candidate can recognize that to maximize the last element, they should make each subsequent element as large as possible relative to the previous one, without exceeding the original value or the distance constraint.

3. Algorithmic pattern used

This uses a Sorting and Greedy interview pattern. The best way to satisfy the adjacency constraint while maximizing the values is to sort the array first. By sorting, you ensure that you are building the 'staircase' of numbers in the most efficient way. You set the first element to 1, and then for each subsequent element, you set it to min(current_value, previous_value + 1).

4. Example explanation

Take the array: [100, 1, 1000].

• Rearrange and Sort: [1, 100, 1000].
• First element must be 1. It is 1.
• Second element (100): We want it to be as big as possible, but max distance from previous (1) is 1. So we decrease 100 to min(100, 1+1) = 2.
• Third element (1000): Max distance from previous (2) is 1. So we decrease 1000 to min(1000, 2+1) = 3.
• Final array: [1, 2, 3]. Max element is 3.

5. Common mistakes candidates make

One mistake is forgetting to set the first element to 1. Another is trying to use complex DP when a simple greedy pass after sorting is sufficient. Some candidates also fail to realize that rearranging is allowed, which is why sorting is a valid first step.

6. Interview preparation tip

When you see 'rearrange' and 'difference between adjacent elements', sorting is almost always the first thing you should consider. It organizes the data in a way that makes greedy decisions much easier to validate.