Minimize Maximum of Array

Problem Description

You are given a 0-indexed array nums comprising of n non-negative integers. In one operation, you must choose an integer i such that 1 <= i < n and nums[i] > 0, decrease nums[i] by 1, and increase nums[i - 1] by 1. Return the minimum possible value of the maximum integer of nums after performing any number of operations.

Key Insights

• The goal is to minimize the maximum value in the array after a series of allowed operations.
• Operations allow shifting values to the left, which means you can effectively distribute larger numbers to the leftmost part of the array.
• The maximum value cannot be less than the average value of the array (rounded up) due to the constraints of the operations.
• A binary search can be employed to efficiently find the minimum maximum value.

Space and Time Complexity

Time Complexity: O(n log(max(nums)))
Space Complexity: O(1)

Solution

To solve the problem, we can use a binary search approach. The idea is to determine the minimum possible maximum value we can achieve by performing the allowed operations.

1. Set the left boundary of the search to the maximum element in the array and the right boundary to the sum of the array divided by the length of the array (rounded up).
2. Use binary search to check if a mid value can be the maximum value after performing the operations.
3. For each mid value, simulate the operations to see if we can make all elements less than or equal to mid by shifting values to the left.
4. Adjust the boundaries based on whether the mid value is feasible or not.

The data structures used include a simple array for nums and integer variables for the binary search process.

Code Solutions