#2789

Medium

auto_awesomeGreedy choice plus invariant validation

LeetCode Problem Workspace

Largest Element in an Array after Merge Operations

This problem focuses on applying greedy choices and merging elements to find the largest element in an array.

Problem Statement

You are given a 0-indexed array nums consisting of positive integers. The task is to apply the following operation: Pick two adjacent elements, merge them by adding them together, and replace them with their sum. Repeat this operation until only one element remains. The goal is to return the largest element that can be obtained in the final array.

The solution requires making greedy choices in merging the elements. Starting from the end of the array and merging adjacent elements ensures that the resulting sum will be as large as possible. You will need to determine the optimal sequence of merges to maximize the final value in the array.

Examples

Example 1

Input: nums = [2,3,7,9,3]

Output: 21

We can apply the following operations on the array:

Choose i = 0. The resulting array will be nums = [5,7,9,3].
Choose i = 1. The resulting array will be nums = [5,16,3].
Choose i = 0. The resulting array will be nums = [21,3]. The largest element in the final array is 21. It can be shown that we cannot obtain a larger element.

Example 2

Input: nums = [5,3,3]

Output: 11

We can do the following operations on the array:

Choose i = 1. The resulting array will be nums = [5,6].
Choose i = 0. The resulting array will be nums = [11]. There is only one element in the final array, which is 11.

Constraints

1 <= nums.length <= 105
1 <= nums[i] <= 106

Solution Approach

Greedy Merging Strategy

The optimal approach is to merge adjacent elements starting from the end of the array, working backwards. This ensures that you maximize the sum of merged elements as you reduce the array. It leverages greedy decisions by always choosing the best local operation (merging the largest possible adjacent elements) to ensure the largest global result.

Invariant Validation

During the merging process, it’s crucial to keep track of the valid array state. Each step must be validated to ensure that the merge operation is consistently yielding the largest sum possible at each stage. This helps to prevent errors that could arise from premature or non-optimal merges.

Iterative Update of Array

The array must be updated iteratively as merges happen. After each merge, the array reduces in size, and each element's position will change. The process continues until one element remains, which is the final result. Efficient tracking of the largest value at each iteration is key to finding the solution.

Complexity Analysis

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

The time complexity of this problem depends on how you approach the merging process. A direct implementation of merging from the end may require O(n) operations. The space complexity is determined by the size of the array, and a well-optimized solution may achieve O(1) space usage by merging in place.

What Interviewers Usually Probe

Look for the candidate's ability to identify and implement greedy choices effectively.
Candidates who start from the end of the array and merge adjacent elements will demonstrate a good understanding of the problem.
Watch for the candidate's explanation of validating each merge operation to avoid suboptimal choices.

Common Pitfalls or Variants

Common pitfalls

Failing to correctly merge adjacent elements starting from the end of the array leads to suboptimal results.
Not keeping track of the largest possible value during each merge operation can result in missed opportunities to maximize the final element.
Overcomplicating the merge strategy or trying to apply complex algorithms that deviate from the greedy approach can slow down the solution.

Follow-up variants

Varying the number of merges allowed (limiting merges or setting maximum operations)
Using different array sizes and testing edge cases like minimal and maximal input sizes
Exploring the problem with non-integer elements or arrays with a mix of large and small values

FAQ

What is the pattern used in the Largest Element in an Array after Merge Operations problem?

The pattern used is greedy choice with invariant validation, ensuring that the best local choices result in the optimal global outcome.

How do you ensure the largest element in the array?

By merging adjacent elements starting from the end of the array, you ensure that the merged sums are as large as possible, leading to the largest final value.

What are the time and space complexities for this problem?

Time complexity depends on the merge operations but can be O(n), and space complexity depends on the array size, with an optimized solution reaching O(1) space.

How does starting from the end of the array improve the solution?

Starting from the end ensures that larger sums are formed early, and adjacent merges combine the largest values possible, which optimizes the final result.

What is a common mistake when solving this problem?

A common mistake is not properly merging adjacent elements from the end of the array or failing to track the largest sum at each step, leading to suboptimal results.

terminal

Solution

Solution 1: Merge in Reverse Order

According to the problem description, in order to maximize the maximum element in the merged array, we should merge the elements on the right first, making the elements on the right as large as possible, so as to perform as many merge operations as possible and finally get the maximum element.

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class Solution:
    def maxArrayValue(self, nums: List[int]) -> int:
        for i in range(len(nums) - 2, -1, -1):
            if nums[i] <= nums[i + 1]:
                nums[i] += nums[i + 1]
        return max(nums)

Continue Practicing

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