#2439

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Minimize Maximum of Array

Minimize Maximum of Array involves finding the smallest possible maximum value after applying a series of operations on an array.

array binary search dynamic programming greedy prefix sum

Problem Statement

You are given a 0-indexed array nums consisting of n non-negative integers. In one operation, you choose an index i, and increase all elements from index 0 to i by 1. The goal is to return the minimum possible value of the maximum integer of nums after performing any number of operations.

For example, in the array nums = [3, 7, 1, 6], applying a series of operations can minimize the largest number. The task is to find the minimum possible maximum value efficiently.

Examples

Example 1

Input: nums = [3,7,1,6]

Output: 5

One set of optimal operations is as follows:

Choose i = 1, and nums becomes [4,6,1,6].
Choose i = 3, and nums becomes [4,6,2,5].
Choose i = 1, and nums becomes [5,5,2,5]. The maximum integer of nums is 5. It can be shown that the maximum number cannot be less than 5. Therefore, we return 5.

Example 2

Input: nums = [10,1]

Output: 10

It is optimal to leave nums as is, and since 10 is the maximum value, we return 10.

Constraints

n == nums.length
2 <= n <= 105
0 <= nums[i] <= 109

Solution Approach

Binary Search for Optimization

We perform a binary search on the possible maximum values of the array. The search range is between the current maximum value and the minimum value achievable. By checking if we can achieve a candidate value through a series of operations, we can narrow down the search.

Greedy Validation of Feasibility

For each candidate maximum value in the binary search, we validate it by greedily performing operations. This ensures that we do not exceed the candidate value, which helps refine the binary search process to converge to the optimal result.

Prefix Sum for Efficient Calculation

A prefix sum approach can help in quickly computing the result of applying operations on any subarray. This allows us to efficiently determine if a certain candidate maximum can be achieved for each binary search iteration.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity is O(n log M), where n is the length of the array and M is the difference between the maximum and minimum values in nums. Space complexity is O(n) due to the prefix sum array used for validation during each binary search step.

What Interviewers Usually Probe

The candidate is familiar with binary search applications in optimization problems.
The candidate demonstrates understanding of greedy algorithms in combination with binary search.
The candidate efficiently manages the prefix sum to validate the feasibility of each operation.

Common Pitfalls or Variants

Common pitfalls

Misunderstanding the operation limits may lead to incorrect binary search bounds.
Failing to efficiently validate each candidate maximum value within the binary search process.
Not using prefix sums or an alternative efficient method for checking feasibility in large arrays.

Follow-up variants

In some variants, the number of allowed operations may be restricted, requiring additional constraints handling.
The problem may be extended to multi-dimensional arrays, increasing the complexity of both binary search and validation.
Instead of greedy operations, some variants may involve dynamic programming or different optimization methods for determining the result.

FAQ

What is the optimal approach for solving the 'Minimize Maximum of Array' problem?

The optimal approach involves using binary search to narrow down the possible maximum values combined with greedy operations to validate each candidate maximum.

How can binary search be applied in the 'Minimize Maximum of Array' problem?

Binary search is applied by testing different candidate maximum values and checking if it's feasible to achieve them using a series of operations.

What is the time complexity of the solution?

The time complexity is O(n log M), where n is the length of the array and M is the range between the minimum and maximum values in nums.

Why is a prefix sum important in this problem?

Prefix sums are used to efficiently compute the cumulative effect of operations on the array, allowing fast feasibility checks during binary search iterations.

What are some potential pitfalls when solving this problem?

Common pitfalls include mismanaging binary search bounds, failing to validate candidate maximum values properly, and not using an efficient method like prefix sums for large arrays.

terminal

Solution

Solution 1: Binary Search

To minimize the maximum value of the array, it is intuitive to use binary search. We binary search for the maximum value $mx$ of the array, and find the smallest $mx$ that satisfies the problem requirements.

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class Solution:
    def minimizeArrayValue(self, nums: List[int]) -> int:
        def check(mx):
            d = 0
            for x in nums[:0:-1]:
                d = max(0, d + x - mx)
            return nums[0] + d <= mx

        left, right = 0, max(nums)
        while left < right:
            mid = (left + right) >> 1
            if check(mid):
                right = mid
            else:
                left = mid + 1
        return left

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